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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... %*********************************************************************** %***** ***** %***** Package F I D E - Test Examples Ver. 1.1.2 May 29,1995 ***** %***** ***** %*********************************************************************** %*********************************************************************** %***** ***** %***** T e s t Examples --- Module E X P R E S ***** %***** ***** %*********************************************************************** let cos th**2=1 - sin th**2, cos fi**2=1 - sin fi**2; factor df; on rat; for all x,y let diff(x,y)=df(x,y); depend u,r,th,fi; depend v,r,th,fi; depend f,r,th,fi; depend w,r,th,fi; % Spherical coordinate system scalefactors 3,r*sin th*cos fi,r*sin th*sin fi,r*cos th,r,th,fi; tensor a1,a2,a3,a4,a5; vectors u,v; dyads w; a1:=grad f; a1:= ( df(f,r) , df(f,th) ---------- , r df(f,fi) ----------- ) sin(th)*r a2:=div u; df(u(3),fi) df(u(2),th) cos(th)*u(2) + 2*sin(th)*u(1) a2:=------------- + ------------- + df(u(1),r) + ------------------------------- sin(th)*r r sin(th)*r a3:=curl v; df(v(3),th) - df(v(2),fi) cos(th)*v(3) a3:= ( ------------- + ---------------- + -------------- , r sin(th)*r sin(th)*r df(v(1),fi) - v(3) - df(v(3),r) + ------------- + --------- , sin(th)*r r - df(v(1),th) v(2) df(v(2),r) + ---------------- + ------ ) r r a4:=lapl v; - 2*df(v(3),fi) - 2*df(v(2),th) df(v(1),fi,2) a4:= ( ------------------ + ------------------ + --------------- + df(v(1),r,2) 2 2 2 2 sin(th)*r r sin(th) *r 2*df(v(1),r) df(v(1),th,2) df(v(1),th)*cos(th) + -------------- + --------------- + --------------------- r 2 2 r sin(th)*r 2*(cos(th)*v(2) + sin(th)*v(1)) - --------------------------------- , 2 sin(th)*r - 2*df(v(3),fi)*cos(th) df(v(2),fi,2) -------------------------- + --------------- + df(v(2),r,2) 2 2 2 2 sin(th) *r sin(th) *r 2*df(v(2),r) df(v(2),th,2) df(v(2),th)*cos(th) + -------------- + --------------- + --------------------- r 2 2 r sin(th)*r 2*df(v(1),th) - v(2) + --------------- + ------------- , 2 2 2 r sin(th) *r df(v(3),fi,2) 2*df(v(3),r) df(v(3),th,2) --------------- + df(v(3),r,2) + -------------- + --------------- 2 2 r 2 sin(th) *r r df(v(3),th)*cos(th) 2*df(v(2),fi)*cos(th) 2*df(v(1),fi) + --------------------- + ----------------------- + --------------- 2 2 2 2 sin(th)*r sin(th) *r sin(th)*r - v(3) + ------------- ) 2 2 sin(th) *r a3:=2*a3+a4; - 2*df(v(3),fi) 2*df(v(3),th) - 2*df(v(2),fi) a3:= ( ------------------ + --------------- + ------------------ 2 r sin(th)*r sin(th)*r - 2*df(v(2),th) df(v(1),fi,2) 2*df(v(1),r) + ------------------ + --------------- + df(v(1),r,2) + -------------- 2 2 2 r r sin(th) *r df(v(1),th,2) df(v(1),th)*cos(th) + --------------- + --------------------- 2 2 r sin(th)*r 2*(cos(th)*v(3)*r - cos(th)*v(2) - sin(th)*v(1)) + -------------------------------------------------- , 2 sin(th)*r - 2*df(v(3),fi)*cos(th) df(v(2),fi,2) -------------------------- - 2*df(v(3),r) + --------------- 2 2 2 2 sin(th) *r sin(th) *r 2*df(v(2),r) df(v(2),th,2) + df(v(2),r,2) + -------------- + --------------- r 2 r df(v(2),th)*cos(th) 2*df(v(1),fi) 2*df(v(1),th) + --------------------- + --------------- + --------------- 2 sin(th)*r 2 sin(th)*r r 2 - 2*sin(th) *v(3)*r - v(2) + ----------------------------- , 2 2 sin(th) *r df(v(3),fi,2) 2*df(v(3),r) df(v(3),th,2) --------------- + df(v(3),r,2) + -------------- + --------------- 2 2 r 2 sin(th) *r r df(v(3),th)*cos(th) 2*df(v(2),fi)*cos(th) + --------------------- + ----------------------- + 2*df(v(2),r) 2 2 2 sin(th)*r sin(th) *r 2 2*df(v(1),fi) - 2*df(v(1),th) 2*sin(th) *v(2)*r - v(3) + --------------- + ------------------ + -------------------------- ) 2 r 2 2 sin(th)*r sin(th) *r a5:=lapl f; df(f,fi,2) 2*df(f,r) df(f,th,2) df(f,th)*cos(th) a5:=------------- + df(f,r,2) + ----------- + ------------ + ------------------ 2 2 r 2 2 sin(th) *r r sin(th)*r a1:=a1+div w; df(w(3,1),fi) df(w(2,1),th) a1:= ( --------------- + --------------- + df(w(1,1),r) + df(f,r) sin(th)*r r cos(th)*w(2,1) - sin(th)*w(3,3) - sin(th)*w(2,2) + 2*sin(th)*w(1,1) + --------------------------------------------------------------------- sin(th)*r , df(w(3,2),fi) df(w(2,2),th) df(f,th) --------------- + --------------- + df(w(1,2),r) + ---------- + sin(th)*r r r - cos(th)*w(3,3) + cos(th)*w(2,2) + sin(th)*w(2,1) + 2*sin(th)*w(1,2) ------------------------------------------------------------------------ sin(th)*r , df(w(3,3),fi) df(w(2,3),th) df(f,fi) --------------- + --------------- + df(w(1,3),r) + ----------- sin(th)*r r sin(th)*r cos(th)*w(3,2) + cos(th)*w(2,3) + sin(th)*w(3,1) + 2*sin(th)*w(1,3) + --------------------------------------------------------------------- sin(th)*r ) a1:=u.dyad((a,0,1),(1,b,3),(0,c,d)); a1:= ( u(2) + u(1)*a , u(3)*c + u(2)*b , u(3)*d + 3*u(2) + u(1) ) a2:=vect(a,b,c); a2:= ( a , b , c ) a1.a2; 2 2 u(3)*b*c + u(3)*c*d + u(2)*a + u(2)*b + 3*u(2)*c + u(1)*a + u(1)*c % Scalar product u.v; u(3)*v(3) + u(2)*v(2) + u(1)*v(1) % Vector product u?v; ( - u(3)*v(2) + u(2)*v(3) , u(3)*v(1) - u(1)*v(3) , - u(2)*v(1) + u(1)*v(2) ) % Dyadic u&v; ( ( u(1)*v(1) , u(1)*v(2) , u(1)*v(3) ) , ( u(2)*v(1) , u(2)*v(2) , u(2)*v(3) ) , ( u(3)*v(1) , u(3)*v(2) , u(3)*v(3) ) ) % Directional derivative dirdf(u,v); df(v(1),fi)*u(3) df(v(1),th)*u(2) ( ------------------ + df(v(1),r)*u(1) + ------------------ sin(th)*r r u(3)*v(3) + u(2)*v(2) - ----------------------- , r df(v(2),fi)*u(3) df(v(2),th)*u(2) ------------------ + df(v(2),r)*u(1) + ------------------ sin(th)*r r - cos(th)*u(3)*v(3) + sin(th)*u(2)*v(1) + ------------------------------------------ , sin(th)*r df(v(3),fi)*u(3) df(v(3),th)*u(2) ------------------ + df(v(3),r)*u(1) + ------------------ sin(th)*r r u(3)*(cos(th)*v(2) + sin(th)*v(1)) + ------------------------------------ ) sin(th)*r clear a1,a2,a3,a4,a5,u,v,w; for all x,y clear diff(x,y); clear cos th**2, cos fi**2; remfac df; off rat; scalefactors 3,x,y,z,x,y,z; %*********************************************************************** %***** ***** %***** T e s t Examples --- Module I I M E T ***** %***** ***** %*********************************************************************** % Example I.1 - 1-D Lagrangian Hydrodynamics off exp; factor diff; on rat,eqfu; % Declare which indexes will be given to coordinates coordinates x,t into j,m; % Declares uniform grid in x coordinate grid uniform,x; % Declares dependencies of functions on coordinates dependence eta(t,x),v(t,x),eps(t,x),p(t,x); % Declares p as known function given p; same eta,v,p; iim a, eta,diff(eta,t)-eta*diff(v,x)=0, v,diff(v,t)+eta/ro*diff(p,x)=0, eps,diff(eps,t)+eta*p/ro*diff(v,x)=0; ***************************** ***** Program ***** IIMET Ver 1.1.2 ***************************** Partial Differential Equations ============================== diff(eta,t) - diff(v,x)*eta = 0 diff(p,x)*eta --------------- + diff(v,t) = 0 ro diff(v,x)*eta*p diff(eps,t) + ----------------- = 0 ro Backtracking needed in grid optimalization 0 interpolations are needed in x coordinate Equation for eta variable is integrated in half grid point Equation for v variable is integrated in half grid point Equation for eps variable is integrated in half grid point 0 interpolations are needed in t coordinate Equation for eta variable is integrated in half grid point Equation for v variable is integrated in half grid point Equation for eps variable is integrated in half grid point Equations after Discretization Using IIM : ========================================== (4*(eta(j,m + 1) - eta(j,m) - eta(j + 1,m) + eta(j + 1,m + 1))*hx - ( (eta(j + 1,m + 1) + eta(j,m + 1))*(v(j + 1,m + 1) - v(j,m + 1)) + (eta(j + 1,m) + eta(j,m))*(v(j + 1,m) - v(j,m)))*(ht(m + 1) + ht(m)))/(4 *(ht(m + 1) + ht(m))*hx) = 0 (4*(v(j,m + 1) - v(j,m) - v(j + 1,m) + v(j + 1,m + 1))*hx*ro + ( (eta(j + 1,m + 1) + eta(j,m + 1))*(p(j + 1,m + 1) - p(j,m + 1)) + (eta(j + 1,m) + eta(j,m))*(p(j + 1,m) - p(j,m)))*(ht(m + 1) + ht(m)))/(4 *(ht(m + 1) + ht(m))*hx*ro) = 0 (4*(eps(j,m + 1) - eps(j,m) - eps(j + 1,m) + eps(j + 1,m + 1))*hx*ro + ( (eta(j + 1,m + 1)*p(j + 1,m + 1) + eta(j,m + 1)*p(j,m + 1)) *(v(j + 1,m + 1) - v(j,m + 1)) + (eta(j + 1,m)*p(j + 1,m) + eta(j,m)*p(j,m))*(v(j + 1,m) - v(j,m))) *(ht(m + 1) + ht(m)))/(4*(ht(m + 1) + ht(m))*hx*ro) = 0 clear a; clearsame; cleargiven; %*********************************************************************** % Example I.2 - How other functions (here sin, cos) can be used in % discretized terms diffunc sin,cos; difmatch all,diff(u*sin x,x),u=one,2,(u(i+1)*sin x(i+1)-u(i-1) *sin x(i-1))/(dim1+dip1), u=half,0,(u(i+1/2)*sin x(i+1/2)-u(i-1/2)*sin x(i-1/2)) /di; difmatch all,cos x*diff(u,x,2),u=one,0,cos x i*(u(i+1)-2*u(i)+u(i-1)) /di^2, u=half,3,(u(i+3/2)-u(i+1/2))/dip2/2 - (u(i-1/2)-u(i-3/2))/dim2/2; off exp; coordinates x,t into j,m; grid uniform,x,t; dependence u(x,t),v(x,t); iim a,u,diff(u,t)+diff(u,x)+cos x*diff(v,x,2)=0, v,diff(v,t)+diff(sin x*u,x)=0; ***************************** ***** Program ***** IIMET Ver 1.1.2 ***************************** Partial Differential Equations ============================== diff(u,t) + diff(u,x) + diff(v,x,2)*cos(x) = 0 diff(sin(x)*u,x) + diff(v,t) = 0 0 interpolations are needed in x coordinate Equation for u variable is integrated in half grid point Equation for v variable is integrated in half grid point 0 interpolations are needed in t coordinate Equation for u variable is integrated in half grid point Equation for v variable is integrated in half grid point Equations after Discretization Using IIM : ========================================== 2*j + 1 2*j + 1 2*j + 3 - ((2*(v(---------,m + 1) + v(---------,m)) - v(---------,m) 2 2 2 2*j + 3 2*j - 1 2*j - 1 - v(---------,m + 1) - v(---------,m) - v(---------,m + 1)) 2 2 2 2*j + 1 *cos(x(---------))*ht + ( 2 (u(j,m + 1) + u(j,m) - u(j + 1,m) - u(j + 1,m + 1))*ht 2 - (u(j,m + 1) - u(j,m) - u(j + 1,m) + u(j + 1,m + 1))*hx)*hx)/(2*ht*hx ) = 0 ( - ((u(j,m + 1) + u(j,m))*sin(x(j))*ht 2*j + 1 2*j + 1 - 2*(v(---------,m + 1) - v(---------,m))*hx) 2 2 + (u(j + 1,m + 1) + u(j + 1,m))*sin(x(j + 1))*ht)/(2*ht*hx) = 0 clear a; %*********************************************************************** % Example I.3 - Schrodinger equation factor diff; coordinates t,x into m,j; grid uniform,x,t; dependence ur(x,t),ui(x,t); same ui,ur; iim a,ur,-diff(ui,t)+1/2*diff(ur,x,2)+(ur**2+ui**2)*ur=0, ui,diff(ur,t)+1/2*diff(ui,x,2)+(ur**2+ui**2)*ui=0; ***************************** ***** Program ***** IIMET Ver 1.1.2 ***************************** Partial Differential Equations ============================== diff(ur,x,2) 2 2 - diff(ui,t) + -------------- = - ur*(ui + ur ) 2 diff(ui,x,2) 2 2 -------------- + diff(ur,t) = - ui*(ui + ur ) 2 0 interpolations are needed in t coordinate Equation for ur variable is integrated in half grid point Equation for ui variable is integrated in half grid point 0 interpolations are needed in x coordinate Equation for ur variable is integrated in one grid point Equation for ui variable is integrated in one grid point Equations after Discretization Using IIM : ========================================== ((ur(m,j + 1) - 2*ur(m,j) + ur(m,j - 1) - 2*ur(m + 1,j) + ur(m + 1,j + 1) 2 2 + ur(m + 1,j - 1))*ht - 4*(ui(m + 1,j) - ui(m,j))*hx )/(4*ht*hx ) = 3 3 2 2 ur(m + 1,j) + ur(m,j) + ui(m,j) *ur(m,j) + ui(m + 1,j) *ur(m + 1,j) - ----------------------------------------------------------------------- 2 ((ui(m,j + 1) - 2*ui(m,j) + ui(m,j - 1) - 2*ui(m + 1,j) + ui(m + 1,j + 1) 2 2 + ui(m + 1,j - 1))*ht + 4*(ur(m + 1,j) - ur(m,j))*hx )/(4*ht*hx ) = 2 2 3 2 (ui(m,j) + ur(m,j) )*ui(m,j) + ui(m + 1,j) + ui(m + 1,j)*ur(m + 1,j) - ------------------------------------------------------------------------- 2 clear a; clearsame; %*********************************************************************** % Example I.4 - Vector calculus in p.d.e. input % cooperation with expres module % 2-D hydrodynamics scalefactors 2,x,y,x,y; vectors u; off exp,twogrid; on eqfu; factor diff,ht,hx,hy; coordinates x,y,t into j,i,m; grid uniform,x,y,t; dependence n(t,x,y),u(t,x,y),p(t,x,y); iim a,n,diff(n,t)+u.grad n+n*div u=0, u,m*n*(diff(u,t)+u.grad u)+grad p=vect(0,0), p,3/2*(diff(p,t)+u.grad p)+5/2*p*div u=0; ***************************** ***** Program ***** IIMET Ver 1.1.2 ***************************** Partial Differential Equations ============================== diff(n,t) + diff(n,x)*u1 + diff(n,y)*u2 + diff(u1,x)*n + diff(u2,y)*n = 0 diff(p,x) + diff(u1,t)*m*n + diff(u1,x)*m*n*u1 + diff(u1,y)*m*n*u2 = 0 diff(p,y) + diff(u2,t)*m*n + diff(u2,x)*m*n*u1 + diff(u2,y)*m*n*u2 = 0 3*diff(p,t) 3*diff(p,x)*u1 3*diff(p,y)*u2 5*diff(u1,x)*p ------------- + ---------------- + ---------------- + ---------------- 2 2 2 2 5*diff(u2,y)*p + ---------------- = 0 2 0 interpolations are needed in x coordinate Equation for n variable is integrated in half grid point Equation for u1 variable is integrated in half grid point Equation for u2 variable is integrated in half grid point Equation for p variable is integrated in half grid point 0 interpolations are needed in y coordinate Equation for n variable is integrated in half grid point Equation for u1 variable is integrated in half grid point Equation for u2 variable is integrated in half grid point Equation for p variable is integrated in half grid point 0 interpolations are needed in t coordinate Equation for n variable is integrated in half grid point Equation for u1 variable is integrated in half grid point Equation for u2 variable is integrated in half grid point Equation for p variable is integrated in half grid point Equations after Discretization Using IIM : ========================================== -1 -1 (hy *hx *(n(j + 1,i + 1,m + 1)*u1(j + 1,i + 1,m + 1)*hy + n(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1)*hx + n(j + 1,i + 1,m)*u1(j + 1,i + 1,m)*hy + n(j + 1,i + 1,m)*u2(j + 1,i + 1,m)*hx + n(j + 1,i,m + 1)*u1(j + 1,i,m + 1)*hy - n(j + 1,i,m + 1)*u2(j + 1,i,m + 1)*hx + n(j + 1,i,m)*u1(j + 1,i,m)*hy - n(j + 1,i,m)*u2(j + 1,i,m)*hx - n(j,i + 1,m + 1)*u1(j,i + 1,m + 1)*hy + n(j,i + 1,m + 1)*u2(j,i + 1,m + 1)*hx - n(j,i + 1,m)*u1(j,i + 1,m)*hy + n(j,i + 1,m)*u2(j,i + 1,m)*hx - n(j,i,m + 1)*u1(j,i,m + 1)*hy - n(j,i,m + 1)*u2(j,i,m + 1)*hx -1 - n(j,i,m)*u1(j,i,m)*hy - n(j,i,m)*u2(j,i,m)*hx))/4 + (ht *( n(j,i,m + 1) - n(j,i,m) - n(j,i + 1,m) + n(j,i + 1,m + 1) - n(j + 1,i,m) + n(j + 1,i,m + 1) - n(j + 1,i + 1,m) + n(j + 1,i + 1,m + 1)))/4 = 0 -1 (hx *((n(j + 1,i,m + 1)*u1(j + 1,i,m + 1) + n(j,i,m + 1)*u1(j,i,m + 1)) *(u1(j + 1,i,m + 1) - u1(j,i,m + 1)) + (n(j + 1,i,m)*u1(j + 1,i,m) + n(j,i,m)*u1(j,i,m)) *(u1(j + 1,i,m) - u1(j,i,m)) + (n(j + 1,i + 1,m)*u1(j + 1,i + 1,m) + n(j,i + 1,m)*u1(j,i + 1,m)) *(u1(j + 1,i + 1,m) - u1(j,i + 1,m)) + ( n(j + 1,i + 1,m + 1)*u1(j + 1,i + 1,m + 1) + n(j,i + 1,m + 1)*u1(j,i + 1,m + 1)) -1 -1 -1 *(u1(j + 1,i + 1,m + 1) - u1(j,i + 1,m + 1)))*m)/8 + (hy *hx *ht *((( (n(j,i + 1,m + 1) + n(j,i + 1,m)) *(u1(j,i + 1,m + 1) - u1(j,i + 1,m)) + (n(j,i,m + 1) + n(j,i,m))*(u1(j,i,m + 1) - u1(j,i,m)) + (n(j + 1,i,m + 1) + n(j + 1,i,m)) *(u1(j + 1,i,m + 1) - u1(j + 1,i,m)) + (n(j + 1,i + 1,m + 1) + n(j + 1,i + 1,m)) *(u1(j + 1,i + 1,m + 1) - u1(j + 1,i + 1,m)))*hx*m + 2*( p(j + 1,i,m + 1) + p(j + 1,i,m) + p(j + 1,i + 1,m) + p(j + 1,i + 1,m + 1) - (p(j,i,m + 1) + p(j,i,m) + p(j,i + 1,m) + p(j,i + 1,m + 1)))*ht) *hy + ((n(j,i + 1,m + 1)*u2(j,i + 1,m + 1) + n(j,i,m + 1)*u2(j,i,m + 1)) *(u1(j,i + 1,m + 1) - u1(j,i,m + 1)) + (n(j,i + 1,m)*u2(j,i + 1,m) + n(j,i,m)*u2(j,i,m)) *(u1(j,i + 1,m) - u1(j,i,m)) + (n(j + 1,i + 1,m)*u2(j + 1,i + 1,m) + n(j + 1,i,m)*u2(j + 1,i,m)) *(u1(j + 1,i + 1,m) - u1(j + 1,i,m)) + ( n(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1) + n(j + 1,i,m + 1)*u2(j + 1,i,m + 1)) *(u1(j + 1,i + 1,m + 1) - u1(j + 1,i,m + 1)))*ht*hx*m))/8 = 0 -1 -1 (hy *hx *(((n(j + 1,i,m + 1)*u1(j + 1,i,m + 1) + n(j,i,m + 1)*u1(j,i,m + 1)) *(u2(j + 1,i,m + 1) - u2(j,i,m + 1)) + (n(j + 1,i,m)*u1(j + 1,i,m) + n(j,i,m)*u1(j,i,m)) *(u2(j + 1,i,m) - u2(j,i,m)) + (n(j + 1,i + 1,m)*u1(j + 1,i + 1,m) + n(j,i + 1,m)*u1(j,i + 1,m)) *(u2(j + 1,i + 1,m) - u2(j,i + 1,m)) + ( n(j + 1,i + 1,m + 1)*u1(j + 1,i + 1,m + 1) + n(j,i + 1,m + 1)*u1(j,i + 1,m + 1)) *(u2(j + 1,i + 1,m + 1) - u2(j,i + 1,m + 1)))*hy + ( (n(j,i + 1,m + 1)*u2(j,i + 1,m + 1) + n(j,i,m + 1)*u2(j,i,m + 1)) *(u2(j,i + 1,m + 1) - u2(j,i,m + 1)) + (n(j,i + 1,m)*u2(j,i + 1,m) + n(j,i,m)*u2(j,i,m)) *(u2(j,i + 1,m) - u2(j,i,m)) + (n(j + 1,i + 1,m)*u2(j + 1,i + 1,m) + n(j + 1,i,m)*u2(j + 1,i,m)) *(u2(j + 1,i + 1,m) - u2(j + 1,i,m)) + ( n(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1) + n(j + 1,i,m + 1)*u2(j + 1,i,m + 1)) -1 *(u2(j + 1,i + 1,m + 1) - u2(j + 1,i,m + 1)))*hx)*m)/8 + ( - hy -1 *ht *(2*(p(j,i,m + 1) + p(j,i,m) - p(j,i + 1,m) - p(j,i + 1,m + 1) + p(j + 1,i,m) + p(j + 1,i,m + 1) - p(j + 1,i + 1,m) - p(j + 1,i + 1,m + 1))*ht - ((n(j,i + 1,m + 1) + n(j,i + 1,m)) *(u2(j,i + 1,m + 1) - u2(j,i + 1,m)) + (n(j,i,m + 1) + n(j,i,m))*(u2(j,i,m + 1) - u2(j,i,m)) + (n(j + 1,i,m + 1) + n(j + 1,i,m)) *(u2(j + 1,i,m + 1) - u2(j + 1,i,m)) + (n(j + 1,i + 1,m + 1) + n(j + 1,i + 1,m)) *(u2(j + 1,i + 1,m + 1) - u2(j + 1,i + 1,m)))*hy*m))/8 = 0 -1 -1 (hy *hx *(3*((p(j + 1,i,m + 1) - p(j,i,m + 1)) *(u1(j + 1,i,m + 1) + u1(j,i,m + 1)) + (p(j + 1,i,m) - p(j,i,m))*(u1(j + 1,i,m) + u1(j,i,m)) + (p(j + 1,i + 1,m) - p(j,i + 1,m)) *(u1(j + 1,i + 1,m) + u1(j,i + 1,m)) + (p(j + 1,i + 1,m + 1) - p(j,i + 1,m + 1)) *(u1(j + 1,i + 1,m + 1) + u1(j,i + 1,m + 1)))*hy + 2*( 4*p(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1) - p(j + 1,i + 1,m + 1)*u2(j + 1,i,m + 1) + 4*p(j + 1,i + 1,m)*u2(j + 1,i + 1,m) - p(j + 1,i + 1,m)*u2(j + 1,i,m) + p(j + 1,i,m + 1)*u2(j + 1,i + 1,m + 1) - 4*p(j + 1,i,m + 1)*u2(j + 1,i,m + 1) + p(j + 1,i,m)*u2(j + 1,i + 1,m) - 4*p(j + 1,i,m)*u2(j + 1,i,m) + 4*p(j,i + 1,m + 1)*u2(j,i + 1,m + 1) - p(j,i + 1,m + 1)*u2(j,i,m + 1) + 4*p(j,i + 1,m)*u2(j,i + 1,m) - p(j,i + 1,m)*u2(j,i,m) + p(j,i,m + 1)*u2(j,i + 1,m + 1) - 4*p(j,i,m + 1)*u2(j,i,m + 1) + p(j,i,m)*u2(j,i + 1,m) - 4*p(j,i,m)*u2(j,i,m))*hx + 5*( (p(j + 1,i,m + 1) + p(j,i,m + 1)) *(u1(j + 1,i,m + 1) - u1(j,i,m + 1)) + (p(j + 1,i,m) + p(j,i,m))*(u1(j + 1,i,m) - u1(j,i,m)) + (p(j + 1,i + 1,m) + p(j,i + 1,m)) *(u1(j + 1,i + 1,m) - u1(j,i + 1,m)) + (p(j + 1,i + 1,m + 1) + p(j,i + 1,m + 1)) -1 *(u1(j + 1,i + 1,m + 1) - u1(j,i + 1,m + 1)))*hy))/16 + (3*ht *( p(j,i,m + 1) - p(j,i,m) - p(j,i + 1,m) + p(j,i + 1,m + 1) - p(j + 1,i,m) + p(j + 1,i,m + 1) - p(j + 1,i + 1,m) + p(j + 1,i + 1,m + 1)))/8 = 0 clear a,u; %*********************************************************************** % Example I.5 - 1-D hydrodynamics up to 3-rd moments (heat flow) coordinates x,t into j,m; grid uniform,x,t; dependence n(x,t),u(x,t),tt(x,t),p(x,t),q(x,t); iim a, n,diff(n,t)+u*diff(n,x)+diff(u,x)=0, u,n*m*(diff(u,t)+u*diff(u,x))+k*diff(n*tt,x)+diff(p,x)=0, tt,3/2*k*n*(diff(tt,t)+u*diff(tt,x))+n*k*tt*diff(u,x)+1/2*p *diff(u,x)+diff(q,x)=0, p,diff(p,t)+u*diff(p,x)+p*diff(u,x)+n*k*tt*diff(u,x)+2/5*diff(q,x) =0, q,diff(q,t)+u*diff(q,x)+q*diff(u,x)+5/2*n*k**2*tt/m*diff(tt,x)+n*k *tt*diff(p,x)-p*diff(p,x)=0; ***************************** ***** Program ***** IIMET Ver 1.1.2 ***************************** Partial Differential Equations ============================== diff(n,t) + diff(n,x)*u + diff(u,x) = 0 diff(n*tt,x)*k + diff(p,x) + diff(u,t)*m*n + diff(u,x)*m*n*u = 0 3*diff(tt,t)*k*n 3*diff(tt,x)*k*n*u diff(q,x) + ------------------ + -------------------- 2 2 diff(u,x)*(2*k*n*tt + p) + -------------------------- = 0 2 2*diff(q,x) diff(p,t) + diff(p,x)*u + ------------- + diff(u,x)*(k*n*tt + p) = 0 5 2 5*diff(tt,x)*k *n*tt diff(p,x)*(k*n*tt - p) + diff(q,t) + diff(q,x)*u + ---------------------- 2*m + diff(u,x)*q = 0 0 interpolations are needed in x coordinate Equation for n variable is integrated in half grid point Equation for u variable is integrated in half grid point Equation for tt variable is integrated in half grid point Equation for p variable is integrated in half grid point Equation for q variable is integrated in half grid point 0 interpolations are needed in t coordinate Equation for n variable is integrated in half grid point Equation for u variable is integrated in half grid point Equation for tt variable is integrated in half grid point Equation for p variable is integrated in half grid point Equation for q variable is integrated in half grid point Equations after Discretization Using IIM : ========================================== -1 (hx *((n(j + 1,m + 1) - n(j,m + 1))*(u(j + 1,m + 1) + u(j,m + 1)) + (n(j + 1,m) - n(j,m))*(u(j + 1,m) + u(j,m)) + 2*(u(j + 1,m + 1) + u(j + 1,m) - (u(j,m + 1) + u(j,m)))))/4 -1 ht *(n(j,m + 1) - n(j,m) - n(j + 1,m) + n(j + 1,m + 1)) + ---------------------------------------------------------- = 0 2 -1 ( - hx *(n(j,m + 1)*tt(j,m + 1) + n(j,m)*tt(j,m) - n(j + 1,m)*tt(j + 1,m) -1 -1 - n(j + 1,m + 1)*tt(j + 1,m + 1))*k)/2 + (hx *ht *(( (n(j + 1,m + 1) + n(j + 1,m))*(u(j + 1,m + 1) - u(j + 1,m)) + (n(j,m + 1) + n(j,m))*(u(j,m + 1) - u(j,m)))*hx*m + 2*(p(j + 1,m + 1) + p(j + 1,m) - (p(j,m + 1) + p(j,m)))*ht + ( (n(j + 1,m + 1)*u(j + 1,m + 1) + n(j,m + 1)*u(j,m + 1)) *(u(j + 1,m + 1) - u(j,m + 1)) + (n(j + 1,m)*u(j + 1,m) + n(j,m)*u(j,m))*(u(j + 1,m) - u(j,m)))*ht*m) )/4 = 0 -1 (hx *((n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1)) *(u(j + 1,m + 1) - u(j,m + 1)) + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(u(j + 1,m) - u(j,m)))*k)/4 -1 -1 + (hx *ht *(((p(j + 1,m + 1) + p(j,m + 1))*(u(j + 1,m + 1) - u(j,m + 1)) + (p(j + 1,m) + p(j,m))*(u(j + 1,m) - u(j,m)) + 4*(q(j + 1,m + 1) + q(j + 1,m) - (q(j,m + 1) + q(j,m))))*ht + 3*((n(j + 1,m + 1) + n(j + 1,m))*(tt(j + 1,m + 1) - tt(j + 1,m)) + (n(j,m + 1) + n(j,m))*(tt(j,m + 1) - tt(j,m)))*hx*k + 3*( (n(j + 1,m + 1)*u(j + 1,m + 1) + n(j,m + 1)*u(j,m + 1)) *(tt(j + 1,m + 1) - tt(j,m + 1)) + (n(j + 1,m)*u(j + 1,m) + n(j,m)*u(j,m))*(tt(j + 1,m) - tt(j,m)) )*ht*k))/8 = 0 -1 (hx *(5*((n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1)) *(u(j + 1,m + 1) - u(j,m + 1)) + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(u(j + 1,m) - u(j,m)))*k + 2*(5*p(j + 1,m + 1)*u(j + 1,m + 1) + 5*p(j + 1,m)*u(j + 1,m) - 5*p(j,m + 1)*u(j,m + 1) - 5*p(j,m)*u(j,m) + 2*q(j + 1,m + 1) + 2*q(j + 1,m) - 2*q(j,m + 1) - 2*q(j,m))))/20 -1 ht *(p(j,m + 1) - p(j,m) - p(j + 1,m) + p(j + 1,m + 1)) + ---------------------------------------------------------- = 0 2 -1 ( - hx *(2*((p(j + 1,m + 1) + p(j,m + 1))*(p(j + 1,m + 1) - p(j,m + 1)) + (p(j + 1,m) + p(j,m))*(p(j + 1,m) - p(j,m)) - 2*( q(j + 1,m + 1)*u(j + 1,m + 1) + q(j + 1,m)*u(j + 1,m) - q(j,m + 1)*u(j,m + 1) - q(j,m)*u(j,m)))*m - 5*( (n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1)) *(tt(j + 1,m + 1) - tt(j,m + 1)) + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(tt(j + 1,m) - tt(j,m))) 2 *k - 2*((n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1)) *(p(j + 1,m + 1) - p(j,m + 1)) + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(p(j + 1,m) - p(j,m))) *k*m))/(8*m) -1 ht *(q(j,m + 1) - q(j,m) - q(j + 1,m) + q(j + 1,m + 1)) + ---------------------------------------------------------- = 0 2 clear a; remfac diff,ht,hx,hy; on exp; off rat; %*********************************************************************** %***** ***** %***** T e s t Examples --- Module A P P R O X ***** %***** ***** %*********************************************************************** % Example A.1 coordinates x,t into j,n; maxorder t=2,x=3; functions u,v; approx( (u(n+1/2)-u(n-1/2))/ht=(v(n+1/2,j+1/2)-v(n+1/2,j-1/2) +v(n-1/2,j+1/2)-v(n-1/2,j-1/2))/(2*hx) ); Difference scheme approximates differential equation df(u,t)=df(v,x) with orders of approximation: 2 hx 2 ht % Example A.2 maxorder t=3,x=3; approx( (u(n+1)-u(n))/ht=(u(n+1,j+1/2)-u(n+1,j-1/2) +u(n,j+1/2)-u(n,j-1/2))/(2*hx) ); Difference scheme approximates differential equation df(u,t)=df(u,x) with orders of approximation: 2 hx ht % Example A.3 maxorder t=2,x=3; center t=1/2; approx( (u(n+1)-u(n))/ht=(v(n+1,j+1/2)-v(n+1,j-1/2) +v(n,j+1/2)-v(n,j-1/2))/(2*hx) ); Difference scheme approximates differential equation df(u,t)=df(v,x) with orders of approximation: 2 hx 2 ht % Example A.4 approx( u(n+1)/ht=(v(n+1,j+1/2)-v(n+1,j-1/2) +v(n,j+1/2)-v(n,j-1/2))/(2*hx) ); Reformulate difference scheme, grid steps remain in denominators Difference scheme approximates differential equation 0=df(v,x) with orders of approximation: 2 hx -1 ht % Example A.5 maxorder t=3,x=3; approx( (u(n+1)-u(n))/ht=(u(n+1,j+1/2)-u(n+1,j-1/2))/hx); Difference scheme approximates differential equation df(u,t)=df(u,x) with orders of approximation: 2 hx ht % Example A.6 approx( (u(n+1)-u(n))/ht=(u(n+1/2,j+1/2)-u(n+1/2,j-1/2))/hx); Difference scheme approximates differential equation df(u,t)=df(u,x) with orders of approximation: 2 hx 2 ht % Example A.7; maxorder x=4; approx((u(n+1)-u(n))/ht=(u(n+1/2,j+1)-2*u(n+1/2,j)+u(n+1/2,j-1))/hx**2); Difference scheme approximates differential equation df(u,t)=df(u,x,2) with orders of approximation: 2 hx 2 ht %*********************************************************************** %***** ***** %***** T e s t Examples --- Module C H A R P O L ***** %***** ***** %*********************************************************************** % Example C.1 coordinates t,x into i,j; grid uniform,t,x; let cos ax**2=1-sin ax**2; unfunc u,v; matrix aa(1,2),bb(2,2); aa(1,1):=(u(i+1)-u(i))/ht+(v(j+1)-v(j))/hx$ aa(1,2):=(v(i+1)-v(i))/ht+(u(j+1/2)-u(j-1/2))/hx$ bb:=ampmat aa; kx*hx ax := ------- 2 [hx 0 ] h1 := [ ] [0 hx] [ hx 2*sin(ax)*ht*( - i*cos(ax) + sin(ax))] h0 := [ ] [ - 2*i*sin(ax)*ht hx ] [ 2*sin(ax)*ht*( - i*cos(ax) + sin(ax)) ] [ 1 ---------------------------------------] [ hx ] bb := [ ] [ - 2*i*sin(ax)*ht ] [------------------- 1 ] [ hx ] bb:=denotemat bb; [ 1 ai12*i + ar12] bb := [ ] [ai21*i 1 ] factor lam; pol:=charpol bb; 2 pol := lam - 2*lam + ai12*ai21 - i*ai21*ar12 + 1 prdenot; 2 2*sin(ax) *ht ar12 := --------------- hx - 2*cos(ax)*sin(ax)*ht ai12 := ------------------------- hx - 2*sin(ax)*ht ai21 := ----------------- hx cleardenot; clear aa,bb,pol; %*********************************************************************** % Example C.2 : Reprint Vorozcov, Ganza, Mazurik: Simvolno-cislennyj % interfejs. v zadacach ..., Novosibirsk 1986, p.47. unfunc u; matrix aa(1,1),bb(1,1); aa(1,1):=(u(i+1)-u(i))/ht+a*(u(j)-u(j-1))/hx$ bb:=ampmat aa; ax := kx*hx h1 := [hx] h0 := [cos(ax)*a*ht - i*sin(ax)*a*ht - a*ht + hx] [ cos(ax)*a*ht - i*sin(ax)*a*ht - a*ht + hx ] bb := [-------------------------------------------] [ hx ] bb:=denotemat bb; bb := [ai11*i + ar11] pol:=charpol bb; pol := lam - i*ai11 - ar11 prdenot; cos(ax)*a*ht - a*ht + hx ar11 := -------------------------- hx - sin(ax)*a*ht ai11 := ----------------- hx cleardenot; clear aa,bb,pol; %*********************************************************************** % Example C.3 : Reprint Vorozcov, Ganza, Mazurik: Simvolno-cislennyj % interfejs. v zadacach ..., Novosibirsk 1986, p.52. coordinates t,x into m,j; unfunc u,r; matrix aa(1,2),bb(2,2); aa(1,1):=(r(m+1)-r(m))/ht+u0*(r(m+1,j+1)-r(m+1,j-1))/2/hx +r0*(u(m+1,j+1)-u(m+1,j-1))/2/hx$ aa(1,2):=(u(m+1)-u(m))/ht+u0*(u(m+1,j+1)-u(m+1,j-1))/2/hx +c0**2/r0*(r(m,j+1)-u(m,j-1))/2/hx$ bb:=ampmat aa; ax := kx*hx [ i*sin(ax)*ht*r0 i*sin(ax)*ht*u0 + hx] h1 := [ ] [2*r0*(i*sin(ax)*ht*u0 + hx) 0 ] h0 := mat((0,hx), 2 2 (cos(ax)*c0 *ht - i*sin(ax)*c0 *ht + 2*hx*r0, 2 c0 *ht*( - cos(ax) - i*sin(ax)))) 2 2 - i*cos(ax)*c0 *ht - sin(ax)*c0 *ht - 2*i*hx*r0 bb := mat((--------------------------------------------------, 2*r0*(sin(ax)*ht*u0 - i*hx) 2 c0 *ht*(i*cos(ax) - sin(ax)) ------------------------------), 2*r0*(sin(ax)*ht*u0 - i*hx) 2 2 sin(ax)*ht*(i*cos(ax)*c0 *ht + sin(ax)*c0 *ht + 2*i*hx*r0) (------------------------------------------------------------,( 2 2 2 2 2*(sin(ax) *ht *u0 - 2*i*sin(ax)*ht*hx*u0 - hx ) 2 2 2 2 2 - i*cos(ax)*sin(ax)*c0 *ht + sin(ax) *c0 *ht 2 - 2*i*sin(ax)*ht*hx*u0 - 2*hx )/(2 2 2 2 2 *(sin(ax) *ht *u0 - 2*i*sin(ax)*ht*hx*u0 - hx )))) bb:=denotemat bb; [ai11*i + ar11 ai12*i + ar12] bb := [ ] [ai21*i + ar21 ai22*i + ar22] pol:=charpol bb; 2 pol := lam + lam*( - i*ai11 - i*ai22 - ar11 - ar22) - ai11*ai22 + i*ai11*ar22 + ai12*ai21 - i*ai12*ar21 - i*ai21*ar12 + i*ai22*ar11 + ar11*ar22 - ar12*ar21 prdenot; 2 - c0 ar11 := --------- 2*r0*u0 2 2 - cos(ax)*c0 *ht*u0 - c0 *hx - 2*hx*r0*u0 ai11 := -------------------------------------------- 2*r0*u0*(sin(ax)*ht*u0 - hx*i) 2 - c0 ar12 := --------- 2*r0*u0 2 c0 *(cos(ax)*ht*u0 - hx) ai12 := -------------------------------- 2*r0*u0*(sin(ax)*ht*u0 - hx*i) 2 2 2 sin(ax) *c0 *ht ar21 := ---------------------------- 2 2 2 2 2*(sin(ax) *ht *u0 - hx ) 2 2 3 2 2 2 ai21 := (sin(ax)*ht*(cos(ax)*sin(ax) *c0 *ht *u0 - cos(ax)*c0 *ht*hx 2 2 2 2 2 2 3 + 2*sin(ax) *c0 *ht *hx*u0 + 2*sin(ax) *ht *hx*r0*u0 - 2*hx *r0))/ 4 4 4 3 3 3 2 2 2 2 (2*(sin(ax) *ht *u0 - 2*sin(ax) *ht *hx*i*u0 - 2*sin(ax) *ht *hx *u0 3 4 + 2*sin(ax)*ht*hx *i*u0 + hx )) 2 2 2 2 sin(ax) *c0 *ht - 2*hx ar22 := ---------------------------- 2 2 2 2 2*(sin(ax) *ht *u0 - hx ) 2 2 3 2 2 2 ai22 := (sin(ax)*ht*( - cos(ax)*sin(ax) *c0 *ht *u0 + cos(ax)*c0 *ht*hx 2 2 2 2 2 3 3 + 2*sin(ax) *c0 *ht *hx*u0 - 2*sin(ax) *ht *hx*u0 - 2*hx *u0))/(2* 4 4 4 3 3 3 2 2 2 2 (sin(ax) *ht *u0 - 2*sin(ax) *ht *hx*i*u0 - 2*sin(ax) *ht *hx *u0 3 4 + 2*sin(ax)*ht*hx *i*u0 + hx )) cleardenot; clear aa,bb,pol; %*********************************************************************** % Example C.4 : Richtmyer, Morton: Difference methods for initial value % problems, &10.3. p.262 coordinates t,x into n,j; unfunc v,w; matrix aa(1,2),bb(2,2); aa(1,1):=(v(n+1)-v(n))/ht-c*(w(j+1/2)-w(j-1/2)+ w(n+1,j+1/2)-w(n+1,j-1/2))/(2*hx)$ aa(1,2):=(w(n+1,j-1/2)-w(n,j-1/2))/ht-c*(v(n+1,j)-v(n+1,j-1)+ v(j)-v(j-1))/(2*hx)$ bb:=ampmat aa; kx*hx ax := ------- 2 [ hx - i*sin(ax)*c*ht ] h1 := [ ] [sin(ax)*c*ht*( - i*cos(ax) - sin(ax)) hx*(cos(ax) - i*sin(ax))] [ hx i*sin(ax)*c*ht ] h0 := [ ] [sin(ax)*c*ht*(i*cos(ax) + sin(ax)) hx*(cos(ax) - i*sin(ax))] [ 2 2 2 2 ] [ - sin(ax) *c *ht + hx 2*i*sin(ax)*c*ht*hx ] [-------------------------- ----------------------- ] [ 2 2 2 2 2 2 2 2 ] [ sin(ax) *c *ht + hx sin(ax) *c *ht + hx ] bb := [ ] [ 2 2 2 2 ] [ 2*i*sin(ax)*c*ht*hx - sin(ax) *c *ht + hx ] [ ----------------------- --------------------------] [ 2 2 2 2 2 2 2 2 ] [ sin(ax) *c *ht + hx sin(ax) *c *ht + hx ] bb:=denotemat bb; [ ar11 ai12*i] bb := [ ] [ai21*i ar22 ] pol:=charpol bb; 2 pol := lam - lam*(ar11 + ar22) + ai12*ai21 + ar11*ar22 prdenot; 2 2 2 2 - sin(ax) *c *ht + hx ar11 := -------------------------- 2 2 2 2 sin(ax) *c *ht + hx 2*sin(ax)*c*ht*hx ai12 := ----------------------- 2 2 2 2 sin(ax) *c *ht + hx 2*sin(ax)*c*ht*hx ai21 := ----------------------- 2 2 2 2 sin(ax) *c *ht + hx 2 2 2 2 - sin(ax) *c *ht + hx ar22 := -------------------------- 2 2 2 2 sin(ax) *c *ht + hx cleardenot; clear aa,bb,pol; %*********************************************************************** % Example C.5: Mazurik: Algoritmy resenia zadaci..., Preprint no.24-85, % AN USSR SO, Inst. teor. i prikl. mechaniky, p.34 coordinates t,x,y into n,m,k; grid uniform,t,x,y; unfunc u1,u2,u3; matrix aa(1,3),bb(3,3); aa(1,1):=(u1(n+1)-u1(n))/ht+c/2*((-u1(m-1)+2*u1(m)-u1(m+1))/hx + (u2(m+1)-u2(m-1))/hx - (u1(k-1)-2*u1(k)+u1(k+1))/hy + (u3(k+1)-u3(k-1))/hy)$ aa(1,2):=(u2(n+1)-u2(n))/ht+c/2*((u1(m+1)-u1(m-1))/hx - (u2(m-1)-2*u2(m)+u2(m+1))/hx)$ aa(1,3):=(u3(n+1)-u3(n))/ht + c/2*((u1(k+1)-u1(k-1))/hy - (u3(k-1)-2*u3(k)+u3(k+1))/hy)$ off prfourmat; bb:=ampmat aa; ax := kx*hx ay := ky*hy cos(ax)*c*ht*hy + cos(ay)*c*ht*hx - c*ht*hx - c*ht*hy + hx*hy bb := mat((---------------------------------------------------------------, hx*hy - i*sin(ax)*c*ht - i*sin(ay)*c*ht -------------------,-------------------), hx hy - i*sin(ax)*c*ht cos(ax)*c*ht - c*ht + hx (-------------------,--------------------------,0), hx hx - i*sin(ay)*c*ht cos(ay)*c*ht - c*ht + hy (-------------------,0,--------------------------)) hy hy pol:=charpol bb; 3 2 2 2 pol := (lam *hx *hy + lam *hx*hy*( - 2*cos(ax)*c*ht*hy - 2*cos(ay)*c*ht*hx + 2*c*ht*hx + 2*c*ht*hy - 3*hx*hy) + lam*( 2 2 2 2 3*cos(ax)*cos(ay)*c *ht *hx*hy - 3*cos(ax)*c *ht *hx*hy 2 2 2 2 2 2 2 2 - 2*cos(ax)*c *ht *hy + 4*cos(ax)*c*ht*hx*hy + cos(ay) *c *ht *hx 2 2 2 2 2 - 2*cos(ay)*c *ht *hx - 3*cos(ay)*c *ht *hx*hy 2 2 2 2 2 2 2 2 + 4*cos(ay)*c*ht*hx *hy + sin(ay) *c *ht *hx + c *ht *hx 2 2 2 2 2 2 2 + 3*c *ht *hx*hy + 2*c *ht *hy - 4*c*ht*hx *hy - 4*c*ht*hx*hy 2 2 2 3 3 + 3*hx *hy ) - cos(ax)*cos(ay) *c *ht *hx 3 3 3 3 + 2*cos(ax)*cos(ay)*c *ht *hx + 2*cos(ax)*cos(ay)*c *ht *hy 2 2 2 3 3 - 3*cos(ax)*cos(ay)*c *ht *hx*hy - cos(ax)*sin(ay) *c *ht *hx 3 3 3 3 2 2 - cos(ax)*c *ht *hx - 2*cos(ax)*c *ht *hy + 3*cos(ax)*c *ht *hx*hy 2 2 2 2 2 3 3 + 2*cos(ax)*c *ht *hy - 2*cos(ax)*c*ht*hx*hy + cos(ay) *c *ht *hx 2 2 2 2 3 3 3 3 - cos(ay) *c *ht *hx - 2*cos(ay)*c *ht *hx - 2*cos(ay)*c *ht *hy 2 2 2 2 2 2 + 2*cos(ay)*c *ht *hx + 3*cos(ay)*c *ht *hx*hy - 2*cos(ay)*c*ht*hx *hy 2 3 3 2 2 2 2 3 3 3 3 + sin(ay) *c *ht *hx - sin(ay) *c *ht *hx + c *ht *hx + 2*c *ht *hy 2 2 2 2 2 2 2 2 2 - c *ht *hx - 3*c *ht *hx*hy - 2*c *ht *hy + 2*c*ht*hx *hy 2 2 2 2 2 + 2*c*ht*hx*hy - hx *hy )/(hx *hy ) let cos ax=cos ax2**2-sin ax2**2, cos ay=cos ay2**2-sin ay2**2, sin ax=2*sin ax2*cos ax2, sin ay=2*sin ay2*cos ay2, cos ax2**2=1-sin ax2**2, cos ay2**2=1-sin ay2**2, sin ax2=s1, sin ay2=s2, hx=c*ht/cap1, hy=c*ht/cap2; order s1,s2; pol:=pol; 3 2 2 2 2 2 pol := lam + lam *(4*s1 *cap1 + 4*s2 *cap2 - 3) + lam*(12*s1 *s2 *cap1*cap2 2 2 2 2 2 2 + 4*s1 *cap1 - 8*s1 *cap1 + 4*s2 *cap2 - 8*s2 *cap2 + 3) 2 2 2 2 2 2 2 2 + 8*s1 *s2 *cap1 *cap2 + 8*s1 *s2 *cap1*cap2 - 12*s1 *s2 *cap1*cap2 2 2 2 2 2 2 - 4*s1 *cap1 + 4*s1 *cap1 - 4*s2 *cap2 + 4*s2 *cap2 - 1 clear cos ax,cos ay,sin ax,sin ay,cos ax2**2,cos ay2**2,sin ax2,sin ay2, hx,hy; pol:=complexpol pol; 2 2 If 8*s1 *s2 *cap1*cap2*(cap1 + cap2) = 0 and 0 = 0 , a root of the polynomial is equal to 1. 3 2 2 2 2 2 pol := lam + lam *(4*s1 *cap1 + 4*s2 *cap2 - 3) + lam*(12*s1 *s2 *cap1*cap2 2 2 2 2 2 2 + 4*s1 *cap1 - 8*s1 *cap1 + 4*s2 *cap2 - 8*s2 *cap2 + 3) 2 2 2 2 2 2 2 2 + 8*s1 *s2 *cap1 *cap2 + 8*s1 *s2 *cap1*cap2 - 12*s1 *s2 *cap1*cap2 2 2 2 2 2 2 - 4*s1 *cap1 + 4*s1 *cap1 - 4*s2 *cap2 + 4*s2 *cap2 - 1 pol1:=hurw pol; 3 2 2 2 2 2 2 pol1 := 8*lam *s1 *s2 *cap1*cap2*(cap1 + cap2) + 8*lam *( - 3*s1 *s2 *cap1 *cap2 2 2 2 2 2 2 2 2 2 - 3*s1 *s2 *cap1*cap2 + 3*s1 *s2 *cap1*cap2 + s1 *cap1 + s2 *cap2 2 2 2 2 2 2 ) + 8*lam*(3*s1 *s2 *cap1 *cap2 + 3*s1 *s2 *cap1*cap2 2 2 2 2 2 2 2 - 6*s1 *s2 *cap1*cap2 - 2*s1 *cap1 + 2*s1 *cap1 - 2*s2 *cap2 2 2 2 2 2 2 2 + 2*s2 *cap2) + 8*( - s1 *s2 *cap1 *cap2 - s1 *s2 *cap1*cap2 2 2 2 2 2 2 2 + 3*s1 *s2 *cap1*cap2 + s1 *cap1 - 2*s1 *cap1 + s2 *cap2 2 - 2*s2 *cap2 + 1) denotid cp; pol:=denotepol pol; 3 2 pol := lam + lam *cpr02 + lam*cpr01 + cpr00 prdenot; 2 2 2 2 2 2 2 2 cpr00 := 8*s1 *s2 *cap1 *cap2 + 8*s1 *s2 *cap1*cap2 - 12*s1 *s2 *cap1*cap2 2 2 2 2 2 2 - 4*s1 *cap1 + 4*s1 *cap1 - 4*s2 *cap2 + 4*s2 *cap2 - 1 cpr01 := 2 2 2 2 2 2 2 2 12*s1 *s2 *cap1*cap2 + 4*s1 *cap1 - 8*s1 *cap1 + 4*s2 *cap2 - 8*s2 *cap2 + 3 2 2 cpr02 := 4*s1 *cap1 + 4*s2 *cap2 - 3 cleardenot; clear aa,bb,pol,pol1; %*********************************************************************** % Example C.6 : Lax-Wendrov (V. Ganzha) coordinates t,x,y into n,m,k; grid uniform,t,x,y; let cos ax**2=1-sin ax**2, cos ay**2=1-sin ay**2; unfunc u1,u2,u3,u4; matrix aa(1,4),bb(4,4); aa(1,1):=4*(u1(n+1)-u1(n))/ht+ (w*(u1(m+2)-u1(m-2)+u1(m+1,k+1)+u1(m+1,k-1)- u1(m-1,k+1)-u1(m-1,k-1))+p*(u2(m+2)-u2(m-2)+u2(m+1,k+1)+ u2(m+1,k-1)-u2(m-1,k+1)-u2(m-1,k-1))+ v*(u1(m+1,k+1)+u1(m-1,k+1)- u1(m+1,k-1)-u1(m-1,k-1)+u1(k+2)-u1(k-2))+p*(u3(m+1,k+1)+ u3(m-1,k+1)-u3(m+1,k-1)-u3(m-1,k-1)+u3(k+2)-u3(k-2)))/hx+ht* (2*w**2*(-u1(m+2)+2*u1(m)-u1(m-2))+4*w*p*(-u2(m+2)+2*u2(m)- u2(m-2))+2*(-u4(m+2)+2*u4(m)-u4(m-2))+2*v**2*(-u1(k+2)+ 2*u1(k)-u1(k-2))+4*v*p*(u3(k+2)+2*u3(k)-u3(k-2))+2*(-u4(k+2)+ 2*u4(k)-u4(k-2))+4*w*v*(-u1(m+1,k+1)+u1(m+1,k-1)+u1(m-1,k+1)- u1(m-1,k-1))+4*p*v*(-u2(m+1,k+1)+u2(m+1,k-1)+u2(m-1,k+1)- u2(m-1,k-1))+4*w*p*(-u3(m+1,k+1)+u3(m+1,k-1)+u3(m-1,k+1)- u3(m-1,k-1)))/hx/hx$ aa(1,2):=4*p*(u2(n+1)-u2(n))/ht+ (w*p*(u2(m+2)-u2(m-2)+u2(m+1,k+1)+ u2(m+1,k-1)-u2(m-1,k+1)-u2(m-1,k-1))+u4(m+2)-u4(m-2)+ u4(m+1,k+1)+ u4(m+1,k-1)-u4(m-1,k+1)-u4(m-1,k-1)+ p*v*(u2(m+1,k+1)+u2(m-1,k+1)+ u2(k+2)-u2(k-2)-u2(m+1,k-1)-u2(m-1,k-1)))/hx+ht*(2*w**2*p* (-u2(m+2)+2*u2(m)-u2(m-2))+2*p*c**2*(-u2(m+2)+2*u2(m)-u2(m-2)) +4*w*(-u4(m+2)+2*u4(m)-u4(m-2))+2*p*v**2*(-u2(k+2)+2*u2(k)- u2(k-2))+4*w*p*v*(-u2(m+1,k+1)+u2(m+1,k-1)+u2(m-1,k+1)- u2(m-1,k-1))+2*p*c**2*(-u3(m+1,k+1)+u3(m+1,k-1)+u3(m-1,k+1) -u3(m-1,k-1))+4*v*(-u4(m+1,k+1)+u4(m+1,k-1)+u4(m-1,k+1)- u4(m-1,k-1)))/hx/hx$ aa(1,3):=4*p*(u3(n+1)-u3(n))/ht+(w*p*(u3(m+2)-u3(m-2)+u3(m+1,k+1)+ u3(m+1,k-1)-u3(m-1,k+1)-u3(m-1,k-1))+u4(k+2)-u4(k-2)+ u4(m+1,k+1)-u4(m+1,k-1)+u4(m-1,k+1)-u4(m-1,k-1)+ p*v*(u3(m+1,k+1)+u3(m-1,k+1)+u3(k+2)-u3(k-2)-u3(m+1,k-1)- u3(m-1,k-1)))/hx+ht*(2*w**2*p*(-u3(m+2)+2*u3(m)-u3(m-2))+ 2*p*c**2*(-u3(k+2)+2*u3(k)-u3(k-2))+4*v*(-u4(k+2)+ 2*u4(k)-u4(k-2))+2*p*v**2*(-u3(k+2)+2*u3(k)-u3(k-2))+ 4*w*p*v*(-u3(m+1,k+1)+u3(m+1,k-1)+u3(m-1,k+1)- u3(m-1,k-1))+2*p*c**2*(-u2(m+1,k+1)+u2(m+1,k-1)+ u2(m-1,k+1)-u2(m-1,k-1))+4*w*(u4(m+1,k+1)+u4(m+1,k-1)+ u4(m-1,k+1)-u4(m-1,k-1)))/hx/hx$ aa(1,4):=4*(u4(n+1)-u4(n))/ht+(p*c**2*(u2(m+2)-u2(m-2)+u2(m+1,k+1)+ u2(m+1,k-1)-u2(m-1,k+1)-u2(m-1,k-1))+w*(u4(m+2)- u4(m-2)+u4(m+1,k+1)+u4(m+1,k-1)-u4(m-1,k+1)-u4(m-1,k-1))+ +p*c**2*(u3(m+1,k+1)+u3(m-1,k+1)-u3(m+1,k-1)- u3(m-1,k-1)+u3(k+2)-u3(k-2))+v*(u4(m+1,k+1)+u4(m-1,k+1)- u4(m+1,k-1)-u4(m-1,k-1)+u4(k+2)-u4(k-2)))/hx+ht* (2*w**2*(-u4(m+2)+2*u4(m)-u4(m-2))+4*w*p*c**2*(-u2(m+2)+ 2*u2(m)-u2(m-2))+2*c**2*(-u4(m+2)+2*u4(m)-u4(m-2))+ 4*p*v*c**2*(-u3(k+2)+2*u3(k)-u3(k-2))+2*c**2*(-u4(k+2)+ 2*u4(k)-u4(k-2))+2*v**2*(-u4(k+2)+2*u4(k)-u4(k-2))+ 4*p*v*c**2*(-u2(m+1,k+1)+u2(m+1,k-1)+u2(m-1,k+1)- u2(m-1,k-1))+4*w*p*c**2*(-u3(m+1,k+1)+u3(m+1,k-1)+ u3(m-1,k+1)-u3(m-1,k-1))+4*w*v*(-u4(m+1,k+1)+ u4(m+1,k-1)+u4(m-1,k+1)-u4(m-1,k-1)))/hx/hx$ bb:=ampmat aa; ax := kx*hx ay := ky*hy bb := mat((( - i*cos(ax)*sin(ax)*ht*hx*w - i*cos(ax)*sin(ay)*ht*hx*v - i*cos(ay)*sin(ax)*ht*hx*w - i*cos(ay)*sin(ay)*ht*hx*v 2 2 2 2 2 2 2 - 2*sin(ax) *ht *w - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *ht *v 2 2 + hx )/hx ,(sin(ax)*ht*p*( - i*cos(ax)*hx - i*cos(ay)*hx 2 - 4*sin(ax)*ht*w - 4*sin(ay)*ht*v))/hx ,(ht*p*( - i*cos(ax)*sin(ay)*hx - 4*i*cos(ay)*sin(ay)*ht*v 2 - i*cos(ay)*sin(ay)*hx - 4*sin(ax)*sin(ay)*ht*w - 2*ht*v))/hx 2 2 2 - 2*ht *(sin(ax) + sin(ay) ) ,--------------------------------), 2 hx (0,( - i*cos(ax)*sin(ax)*ht*hx*w - i*cos(ax)*sin(ay)*ht*hx*v - i*cos(ay)*sin(ax)*ht*hx*w - i*cos(ay)*sin(ay)*ht*hx*v 2 2 2 2 2 2 - 2*sin(ax) *c *ht - 2*sin(ax) *ht *w 2 2 2 2 2 2 - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *ht *v + hx )/hx , 2 2 - 2*sin(ax)*sin(ay)*c *ht -----------------------------,(sin(ax)*ht*( - i*cos(ax)*hx 2 hx 2 - i*cos(ay)*hx - 4*sin(ax)*ht*w - 4*sin(ay)*ht*v))/(hx *p)), 2 2 - 2*sin(ax)*sin(ay)*c *ht (0,-----------------------------,( - i*cos(ax)*sin(ax)*ht*hx*w 2 hx - i*cos(ax)*sin(ay)*ht*hx*v - i*cos(ay)*sin(ax)*ht*hx*w 2 2 2 - i*cos(ay)*sin(ay)*ht*hx*v - 2*sin(ax) *ht *w 2 2 2 2 - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *c *ht 2 2 2 2 2 - 2*sin(ay) *ht *v + hx )/hx ,(ht*( - 2*cos(ax)*cos(ay)*ht*w - 2*i*cos(ax)*sin(ay)*ht*w - i*cos(ax)*sin(ay)*hx - 2*i*cos(ay)*sin(ax)*ht*w - i*cos(ay)*sin(ay)*hx 2 2 - 2*sin(ax)*sin(ay)*ht*w - 4*sin(ay) *ht*v))/(hx *p)), 2 (0,(sin(ax)*c *ht*p*( - i*cos(ax)*hx - i*cos(ay)*hx - 4*sin(ax)*ht*w 2 2 - 4*sin(ay)*ht*v))/hx ,(sin(ay)*c *ht*p*( - i*cos(ax)*hx 2 - i*cos(ay)*hx - 4*sin(ax)*ht*w - 4*sin(ay)*ht*v))/hx ,( - i*cos(ax)*sin(ax)*ht*hx*w - i*cos(ax)*sin(ay)*ht*hx*v - i*cos(ay)*sin(ax)*ht*hx*w - i*cos(ay)*sin(ay)*ht*hx*v 2 2 2 2 2 2 - 2*sin(ax) *c *ht - 2*sin(ax) *ht *w 2 2 2 2 - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *c *ht 2 2 2 2 2 - 2*sin(ay) *ht *v + hx )/hx )) let sin(ax)=s1, cos(ax)=c1, sin(ay)=s2, cos(ay)=c2, w=k1*hx/ht, v=k2*hx/ht, c=k3*hx/ht, ht=r1*hx; denotid a; bb:=denotemat bb; [ai11*i + ar11 ai12*i + ar12 ai13*i + ar13 ar14 ] [ ] [ 0 ai22*i + ar22 ar23 ai24*i + ar24] bb := [ ] [ 0 ar32 ai33*i + ar33 ai34*i + ar34] [ ] [ 0 ai42*i + ar42 ai43*i + ar43 ai44*i + ar44] clear sin ax,cos ax,sin ay,cos ay,w,v,c,ht; pol:=charpol bb; 4 3 pol := lam + lam *( - i*ai11 - i*ai22 - i*ai33 - i*ai44 - ar11 - ar22 - ar33 - ar44) + 2 lam *( - ai11*ai22 - ai11*ai33 - ai11*ai44 + i*ai11*ar22 + i*ai11*ar33 + i*ai11*ar44 - ai22*ai33 - ai22*ai44 + i*ai22*ar11 + i*ai22*ar33 + i*ai22*ar44 + ai24*ai42 - i*ai24*ar42 - ai33*ai44 + i*ai33*ar11 + i*ai33*ar22 + i*ai33*ar44 + ai34*ai43 - i*ai34*ar43 - i*ai42*ar24 - i*ai43*ar34 + i*ai44*ar11 + i*ai44*ar22 + i*ai44*ar33 + ar11*ar22 + ar11*ar33 + ar11*ar44 + ar22*ar33 + ar22*ar44 - ar23*ar32 - ar24*ar42 + ar33*ar44 - ar34*ar43) + lam *(i*ai11*ai22*ai33 + i*ai11*ai22*ai44 + ai11*ai22*ar33 + ai11*ai22*ar44 - i*ai11*ai24*ai42 - ai11*ai24*ar42 + i*ai11*ai33*ai44 + ai11*ai33*ar22 + ai11*ai33*ar44 - i*ai11*ai34*ai43 - ai11*ai34*ar43 - ai11*ai42*ar24 - ai11*ai43*ar34 + ai11*ai44*ar22 + ai11*ai44*ar33 - i*ai11*ar22*ar33 - i*ai11*ar22*ar44 + i*ai11*ar23*ar32 + i*ai11*ar24*ar42 - i*ai11*ar33*ar44 + i*ai11*ar34*ar43 + i*ai22*ai33*ai44 + ai22*ai33*ar11 + ai22*ai33*ar44 - i*ai22*ai34*ai43 - ai22*ai34*ar43 - ai22*ai43*ar34 + ai22*ai44*ar11 + ai22*ai44*ar33 - i*ai22*ar11*ar33 - i*ai22*ar11*ar44 - i*ai22*ar33*ar44 + i*ai22*ar34*ar43 - i*ai24*ai33*ai42 - ai24*ai33*ar42 - ai24*ai42*ar11 - ai24*ai42*ar33 + ai24*ai43*ar32 + i*ai24*ar11*ar42 - i*ai24*ar32*ar43 + i*ai24*ar33*ar42 - ai33*ai42*ar24 + ai33*ai44*ar11 + ai33*ai44*ar22 - i*ai33*ar11*ar22 - i*ai33*ar11*ar44 - i*ai33*ar22*ar44 + i*ai33*ar24*ar42 + ai34*ai42*ar23 - ai34*ai43*ar11 - ai34*ai43*ar22 + i*ai34*ar11*ar43 + i*ai34*ar22*ar43 - i*ai34*ar23*ar42 + i*ai42*ar11*ar24 - i*ai42*ar23*ar34 + i*ai42*ar24*ar33 + i*ai43*ar11*ar34 + i*ai43*ar22*ar34 - i*ai43*ar24*ar32 - i*ai44*ar11*ar22 - i*ai44*ar11*ar33 - i*ai44*ar22*ar33 + i*ai44*ar23*ar32 - ar11*ar22*ar33 - ar11*ar22*ar44 + ar11*ar23*ar32 + ar11*ar24*ar42 - ar11*ar33*ar44 + ar11*ar34*ar43 - ar22*ar33*ar44 + ar22*ar34*ar43 + ar23*ar32*ar44 - ar23*ar34*ar42 - ar24*ar32*ar43 + ar24*ar33*ar42) + ai11*ai22*ai33*ai44 - i*ai11*ai22*ai33*ar44 - ai11*ai22*ai34*ai43 + i*ai11*ai22*ai34*ar43 + i*ai11*ai22*ai43*ar34 - i*ai11*ai22*ai44*ar33 - ai11*ai22*ar33*ar44 + ai11*ai22*ar34*ar43 - ai11*ai24*ai33*ai42 + i*ai11*ai24*ai33*ar42 + i*ai11*ai24*ai42*ar33 - i*ai11*ai24*ai43*ar32 - ai11*ai24*ar32*ar43 + ai11*ai24*ar33*ar42 + i*ai11*ai33*ai42*ar24 - i*ai11*ai33*ai44*ar22 - ai11*ai33*ar22*ar44 + ai11*ai33*ar24*ar42 - i*ai11*ai34*ai42*ar23 + i*ai11*ai34*ai43*ar22 + ai11*ai34*ar22*ar43 - ai11*ai34*ar23*ar42 - ai11*ai42*ar23*ar34 + ai11*ai42*ar24*ar33 + ai11*ai43*ar22*ar34 - ai11*ai43*ar24*ar32 - ai11*ai44*ar22*ar33 + ai11*ai44*ar23*ar32 + i*ai11*ar22*ar33*ar44 - i*ai11*ar22*ar34*ar43 - i*ai11*ar23*ar32*ar44 + i*ai11*ar23*ar34*ar42 + i*ai11*ar24*ar32*ar43 - i*ai11*ar24*ar33*ar42 - i*ai22*ai33*ai44*ar11 - ai22*ai33*ar11*ar44 + i*ai22*ai34*ai43*ar11 + ai22*ai34*ar11*ar43 + ai22*ai43*ar11*ar34 - ai22*ai44*ar11*ar33 + i*ai22*ar11*ar33*ar44 - i*ai22*ar11*ar34*ar43 + i*ai24*ai33*ai42*ar11 + ai24*ai33*ar11*ar42 + ai24*ai42*ar11*ar33 - ai24*ai43*ar11*ar32 + i*ai24*ar11*ar32*ar43 - i*ai24*ar11*ar33*ar42 + ai33*ai42*ar11*ar24 - ai33*ai44*ar11*ar22 + i*ai33*ar11*ar22*ar44 - i*ai33*ar11*ar24*ar42 - ai34*ai42*ar11*ar23 + ai34*ai43*ar11*ar22 - i*ai34*ar11*ar22*ar43 + i*ai34*ar11*ar23*ar42 + i*ai42*ar11*ar23*ar34 - i*ai42*ar11*ar24*ar33 - i*ai43*ar11*ar22*ar34 + i*ai43*ar11*ar24*ar32 + i*ai44*ar11*ar22*ar33 - i*ai44*ar11*ar23*ar32 + ar11*ar22*ar33*ar44 - ar11*ar22*ar34*ar43 - ar11*ar23*ar32*ar44 + ar11*ar23*ar34*ar42 + ar11*ar24*ar32*ar43 - ar11*ar24*ar33*ar42 denotid cp; pol:=denotepol pol; 4 3 2 pol := lam + lam *(cpi03*i + cpr03) + lam *(cpi02*i + cpr02) + lam*(cpi01*i + cpr01) + cpi00*i + cpr00 pol:=complexpol pol; If cpr00 + cpr01 + cpr02 + cpr03 + 1 = 0 and cpi00 + cpi01 + cpi02 + cpi03 = 0 , a root of the polynomial is equal to 1. 8 7 6 2 2 pol := lam + 2*lam *cpr03 + lam *(cpi03 + 2*cpr02 + cpr03 ) 5 + 2*lam *(cpi02*cpi03 + cpr01 + cpr02*cpr03) 4 2 2 + lam *(2*cpi01*cpi03 + cpi02 + 2*cpr00 + 2*cpr01*cpr03 + cpr02 ) 3 + 2*lam *(cpi00*cpi03 + cpi01*cpi02 + cpr00*cpr03 + cpr01*cpr02) 2 2 2 + lam *(2*cpi00*cpi02 + cpi01 + 2*cpr00*cpr02 + cpr01 ) 2 2 + 2*lam*(cpi00*cpi01 + cpr00*cpr01) + cpi00 + cpr00 denotid rp; pol:=denotepol pol; 8 7 6 5 4 3 pol := lam + lam *rpr07 + lam *rpr06 + lam *rpr05 + lam *rpr04 + lam *rpr03 2 + lam *rpr02 + lam*rpr01 + rpr00 prdenot; 2 2 2 2 ar11 := - 2*s1 *k1 - 4*s1*s2*k1*k2 - 2*s2 *k2 + 1 ai11 := - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2) ar12 := - 4*s1*p*r1*(s1*k1 + s2*k2) ai12 := - s1*p*r1*(c1 + c2) ar13 := 2*p*r1*( - 2*s1*s2*k1 - k2) ai13 := s2*p*r1*( - c1 - 4*c2*k2 - c2) 2 2 2 ar14 := - 2*r1 *(s1 + s2 ) 2 2 2 2 2 2 ar22 := - 2*s1 *k1 - 2*s1 *k3 - 4*s1*s2*k1*k2 - 2*s2 *k2 + 1 ai22 := - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2) 2 ar23 := - 2*s1*s2*k3 - 4*s1*r1*(s1*k1 + s2*k2) ar24 := ---------------------------- p - s1*r1*(c1 + c2) ai24 := -------------------- p 2 ar32 := - 2*s1*s2*k3 2 2 2 2 2 2 ar33 := - 2*s1 *k1 - 4*s1*s2*k1*k2 - 2*s2 *k2 - 2*s2 *k3 + 1 ai33 := - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2) 2 2*r1*( - s1*s2*k1 - 2*s2 *k2 - c1*c2*k1) ar34 := ------------------------------------------ p r1*( - 2*s1*c2*k1 - 2*s2*c1*k1 - s2*c1 - s2*c2) ai34 := ------------------------------------------------- p 2 - 4*s1*k3 *p*(s1*k1 + s2*k2) ar42 := ------------------------------- r1 2 - s1*k3 *p*(c1 + c2) ai42 := ----------------------- r1 2 - 4*s2*k3 *p*(s1*k1 + s2*k2) ar43 := ------------------------------- r1 2 - s2*k3 *p*(c1 + c2) ai43 := ----------------------- r1 2 2 2 2 2 2 2 2 ar44 := - 2*s1 *k1 - 2*s1 *k3 - 4*s1*s2*k1*k2 - 2*s2 *k2 - 2*s2 *k3 + 1 ai44 := - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2) cpr00 := ai11*ai22*ai33*ai44 - ai11*ai22*ai34*ai43 - ai11*ai22*ar33*ar44 + ai11*ai22*ar34*ar43 - ai11*ai24*ai33*ai42 - ai11*ai24*ar32*ar43 + ai11*ai24*ar33*ar42 - ai11*ai33*ar22*ar44 + ai11*ai33*ar24*ar42 + ai11*ai34*ar22*ar43 - ai11*ai34*ar23*ar42 - ai11*ai42*ar23*ar34 + ai11*ai42*ar24*ar33 + ai11*ai43*ar22*ar34 - ai11*ai43*ar24*ar32 - ai11*ai44*ar22*ar33 + ai11*ai44*ar23*ar32 - ai22*ai33*ar11*ar44 + ai22*ai34*ar11*ar43 + ai22*ai43*ar11*ar34 - ai22*ai44*ar11*ar33 + ai24*ai33*ar11*ar42 + ai24*ai42*ar11*ar33 - ai24*ai43*ar11*ar32 + ai33*ai42*ar11*ar24 - ai33*ai44*ar11*ar22 - ai34*ai42*ar11*ar23 + ai34*ai43*ar11*ar22 + ar11*ar22*ar33*ar44 - ar11*ar22*ar34*ar43 - ar11*ar23*ar32*ar44 + ar11*ar23*ar34*ar42 + ar11*ar24*ar32*ar43 - ar11*ar24*ar33*ar42 cpi00 := - ai11*ai22*ai33*ar44 + ai11*ai22*ai34*ar43 + ai11*ai22*ai43*ar34 - ai11*ai22*ai44*ar33 + ai11*ai24*ai33*ar42 + ai11*ai24*ai42*ar33 - ai11*ai24*ai43*ar32 + ai11*ai33*ai42*ar24 - ai11*ai33*ai44*ar22 - ai11*ai34*ai42*ar23 + ai11*ai34*ai43*ar22 + ai11*ar22*ar33*ar44 - ai11*ar22*ar34*ar43 - ai11*ar23*ar32*ar44 + ai11*ar23*ar34*ar42 + ai11*ar24*ar32*ar43 - ai11*ar24*ar33*ar42 - ai22*ai33*ai44*ar11 + ai22*ai34*ai43*ar11 + ai22*ar11*ar33*ar44 - ai22*ar11*ar34*ar43 + ai24*ai33*ai42*ar11 + ai24*ar11*ar32*ar43 - ai24*ar11*ar33*ar42 + ai33*ar11*ar22*ar44 - ai33*ar11*ar24*ar42 - ai34*ar11*ar22*ar43 + ai34*ar11*ar23*ar42 + ai42*ar11*ar23*ar34 - ai42*ar11*ar24*ar33 - ai43*ar11*ar22*ar34 + ai43*ar11*ar24*ar32 + ai44*ar11*ar22*ar33 - ai44*ar11*ar23*ar32 cpr01 := ai11*ai22*ar33 + ai11*ai22*ar44 - ai11*ai24*ar42 + ai11*ai33*ar22 + ai11*ai33*ar44 - ai11*ai34*ar43 - ai11*ai42*ar24 - ai11*ai43*ar34 + ai11*ai44*ar22 + ai11*ai44*ar33 + ai22*ai33*ar11 + ai22*ai33*ar44 - ai22*ai34*ar43 - ai22*ai43*ar34 + ai22*ai44*ar11 + ai22*ai44*ar33 - ai24*ai33*ar42 - ai24*ai42*ar11 - ai24*ai42*ar33 + ai24*ai43*ar32 - ai33*ai42*ar24 + ai33*ai44*ar11 + ai33*ai44*ar22 + ai34*ai42*ar23 - ai34*ai43*ar11 - ai34*ai43*ar22 - ar11*ar22*ar33 - ar11*ar22*ar44 + ar11*ar23*ar32 + ar11*ar24*ar42 - ar11*ar33*ar44 + ar11*ar34*ar43 - ar22*ar33*ar44 + ar22*ar34*ar43 + ar23*ar32*ar44 - ar23*ar34*ar42 - ar24*ar32*ar43 + ar24*ar33*ar42 cpi01 := ai11*ai22*ai33 + ai11*ai22*ai44 - ai11*ai24*ai42 + ai11*ai33*ai44 - ai11*ai34*ai43 - ai11*ar22*ar33 - ai11*ar22*ar44 + ai11*ar23*ar32 + ai11*ar24*ar42 - ai11*ar33*ar44 + ai11*ar34*ar43 + ai22*ai33*ai44 - ai22*ai34*ai43 - ai22*ar11*ar33 - ai22*ar11*ar44 - ai22*ar33*ar44 + ai22*ar34*ar43 - ai24*ai33*ai42 + ai24*ar11*ar42 - ai24*ar32*ar43 + ai24*ar33*ar42 - ai33*ar11*ar22 - ai33*ar11*ar44 - ai33*ar22*ar44 + ai33*ar24*ar42 + ai34*ar11*ar43 + ai34*ar22*ar43 - ai34*ar23*ar42 + ai42*ar11*ar24 - ai42*ar23*ar34 + ai42*ar24*ar33 + ai43*ar11*ar34 + ai43*ar22*ar34 - ai43*ar24*ar32 - ai44*ar11*ar22 - ai44*ar11*ar33 - ai44*ar22*ar33 + ai44*ar23*ar32 cpr02 := - ai11*ai22 - ai11*ai33 - ai11*ai44 - ai22*ai33 - ai22*ai44 + ai24*ai42 - ai33*ai44 + ai34*ai43 + ar11*ar22 + ar11*ar33 + ar11*ar44 + ar22*ar33 + ar22*ar44 - ar23*ar32 - ar24*ar42 + ar33*ar44 - ar34*ar43 cpi02 := ai11*ar22 + ai11*ar33 + ai11*ar44 + ai22*ar11 + ai22*ar33 + ai22*ar44 - ai24*ar42 + ai33*ar11 + ai33*ar22 + ai33*ar44 - ai34*ar43 - ai42*ar24 - ai43*ar34 + ai44*ar11 + ai44*ar22 + ai44*ar33 cpr03 := - (ar11 + ar22 + ar33 + ar44) cpi03 := - (ai11 + ai22 + ai33 + ai44) 2 2 rpr00 := cpi00 + cpr00 rpr01 := 2*(cpi00*cpi01 + cpr00*cpr01) 2 2 rpr02 := 2*cpi00*cpi02 + cpi01 + 2*cpr00*cpr02 + cpr01 rpr03 := 2*(cpi00*cpi03 + cpi01*cpi02 + cpr00*cpr03 + cpr01*cpr02) 2 2 rpr04 := 2*cpi01*cpi03 + cpi02 + 2*cpr00 + 2*cpr01*cpr03 + cpr02 rpr05 := 2*(cpi02*cpi03 + cpr01 + cpr02*cpr03) 2 2 rpr06 := cpi03 + 2*cpr02 + cpr03 rpr07 := 2*cpr03 cleardenot; clear aa,bb,pol; %*********************************************************************** %***** ***** %***** T e s t Examples --- Module H U R W P ***** %***** ***** %*********************************************************************** % Example H.1 x0:=lam-1; x0 := lam - 1 x1:=lam-(ar+i*ai); x1 := lam - (ai*i + ar) x2:=lam-(br+i*bi); x2 := lam - (bi*i + br) x3:=lam-(cr+i*ci); x3 := lam - (ci*i + cr) hurwitzp x1; Necessary and sufficient conditions are: - ar > 0 cond % Example H.2 x:=hurw(x0*x1); x := 2*lam*( - ai*i - ar + 1) + 2*(ai*i + ar + 1) hurwitzp x; Necessary and sufficient conditions are: 2 2 4*( - ai - ar + 1) > 0 cond % Example H.3 x:=(x1*x2); 2 x := lam - lam*(ai*i + ar + bi*i + br) - ai*bi + ai*br*i + ar*bi*i + ar*br hurwitzp x; Necessary and sufficient conditions are: - (ar + br) > 0 2 2 2 2 ar*br*(ai - 2*ai*bi + ar + 2*ar*br + bi + br ) > 0 cond % Example H.4 x:=(x1*x2*x3); 3 2 x := lam - lam *(ai*i + ar + bi*i + br + ci*i + cr) + lam*( - ai*bi + ai*br*i - ai*ci + ai*cr*i + ar*bi*i + ar*br + ar*ci*i + ar*cr - bi*ci + bi*cr*i + br*ci*i + br*cr) + ai*bi*ci*i + ai*bi*cr + ai*br*ci - ai*br*cr*i + ar*bi*ci - ar*bi*cr*i - ar*br*ci*i - ar*br*cr hurwitzp x; Necessary and sufficient conditions are: - (ar + br + cr) > 0 2 2 3 3 ai *ar*br + ai *ar*cr - 2*ai*ar*bi*br - 2*ai*ar*ci*cr + ar *br + ar *cr 2 2 2 2 2 2 3 2 + 2*ar *br + 4*ar *br*cr + 2*ar *cr + ar*bi *br + ar*br + 4*ar*br *cr 2 2 3 2 3 + 4*ar*br*cr + ar*ci *cr + ar*cr + bi *br*cr - 2*bi*br*ci*cr + br *cr 2 2 2 3 + 2*br *cr + br*ci *cr + br*cr > 0 4 2 4 4 2 4 4 2 4 2 ar*br*cr*( - ai *bi + 2*ai *bi*ci - ai *br - 2*ai *br*cr - ai *ci - ai *cr 3 3 3 2 3 2 3 + 2*ai *bi - 2*ai *bi *ci + 2*ai *bi*br + 4*ai *bi*br*cr 3 2 3 2 3 2 3 - 2*ai *bi*ci + 2*ai *bi*cr + 2*ai *br *ci + 4*ai *br*ci*cr 3 3 3 2 2 2 2 2 2 + 2*ai *ci + 2*ai *ci*cr - 2*ai *ar *bi + 4*ai *ar *bi*ci 2 2 2 2 2 2 2 2 2 2 2 - 2*ai *ar *br - 4*ai *ar *br*cr - 2*ai *ar *ci - 2*ai *ar *cr 2 2 2 2 2 - 2*ai *ar*bi *br - 2*ai *ar*bi *cr + 4*ai *ar*bi*br*ci 2 2 3 2 2 + 4*ai *ar*bi*ci*cr - 2*ai *ar*br - 6*ai *ar*br *cr 2 2 2 2 2 2 2 3 - 2*ai *ar*br*ci - 6*ai *ar*br*cr - 2*ai *ar*ci *cr - 2*ai *ar*cr 2 4 2 3 2 2 2 2 2 - ai *bi - 2*ai *bi *ci - 2*ai *bi *br - 2*ai *bi *br*cr 2 2 2 2 2 2 2 2 2 + 6*ai *bi *ci - 2*ai *bi *cr - 2*ai *bi*br *ci - 8*ai *bi*br*ci*cr 2 3 2 2 2 4 2 3 - 2*ai *bi*ci - 2*ai *bi*ci*cr - ai *br - 2*ai *br *cr 2 2 2 2 2 2 2 2 2 3 - 2*ai *br *ci - 2*ai *br *cr - 2*ai *br*ci *cr - 2*ai *br*cr 2 4 2 2 2 2 4 2 3 2 2 - ai *ci - 2*ai *ci *cr - ai *cr + 2*ai*ar *bi - 2*ai*ar *bi *ci 2 2 2 2 2 + 2*ai*ar *bi*br + 4*ai*ar *bi*br*cr - 2*ai*ar *bi*ci 2 2 2 2 2 + 2*ai*ar *bi*cr + 2*ai*ar *br *ci + 4*ai*ar *br*ci*cr 2 3 2 2 3 2 + 2*ai*ar *ci + 2*ai*ar *ci*cr + 4*ai*ar*bi *cr + 4*ai*ar*bi *br*ci 2 2 2 - 8*ai*ar*bi *ci*cr + 4*ai*ar*bi*br *cr - 8*ai*ar*bi*br*ci 2 2 3 + 8*ai*ar*bi*br*cr + 4*ai*ar*bi*ci *cr + 4*ai*ar*bi*cr 3 2 3 + 4*ai*ar*br *ci + 8*ai*ar*br *ci*cr + 4*ai*ar*br*ci 2 4 3 2 3 2 + 4*ai*ar*br*ci*cr + 2*ai*bi *ci - 2*ai*bi *ci + 2*ai*bi *cr 2 2 2 2 3 + 4*ai*bi *br *ci + 4*ai*bi *br*ci*cr - 2*ai*bi *ci 2 2 2 2 2 2 - 2*ai*bi *ci*cr - 2*ai*bi*br *ci + 2*ai*bi*br *cr 2 3 4 2 2 + 4*ai*bi*br*ci *cr + 4*ai*bi*br*cr + 2*ai*bi*ci + 4*ai*bi*ci *cr 4 4 3 2 3 + 2*ai*bi*cr + 2*ai*br *ci + 4*ai*br *ci*cr + 2*ai*br *ci 2 2 4 2 4 4 2 4 + 2*ai*br *ci*cr - ar *bi + 2*ar *bi*ci - ar *br - 2*ar *br*cr 4 2 4 2 3 2 3 2 3 - ar *ci - ar *cr - 2*ar *bi *br - 2*ar *bi *cr + 4*ar *bi*br*ci 3 3 3 3 2 3 2 + 4*ar *bi*ci*cr - 2*ar *br - 6*ar *br *cr - 2*ar *br*ci 3 2 3 2 3 3 2 4 2 3 - 6*ar *br*cr - 2*ar *ci *cr - 2*ar *cr - ar *bi + 2*ar *bi *ci 2 2 2 2 2 2 2 2 2 2 2 - 2*ar *bi *br - 6*ar *bi *br*cr - 2*ar *bi *ci - 2*ar *bi *cr 2 2 2 2 3 + 2*ar *bi*br *ci + 8*ar *bi*br*ci*cr + 2*ar *bi*ci 2 2 2 4 2 3 2 2 2 + 2*ar *bi*ci*cr - ar *br - 6*ar *br *cr - 2*ar *br *ci 2 2 2 2 2 2 3 2 4 - 10*ar *br *cr - 6*ar *br*ci *cr - 6*ar *br*cr - ar *ci 2 2 2 2 4 4 3 - 2*ar *ci *cr - ar *cr - 2*ar*bi *cr + 4*ar*bi *ci*cr 2 2 2 2 2 2 - 4*ar*bi *br *cr - 2*ar*bi *br*ci - 6*ar*bi *br*cr 2 2 2 3 2 3 - 2*ar*bi *ci *cr - 2*ar*bi *cr + 4*ar*bi*br *ci*cr + 4*ar*bi*br*ci 2 4 3 2 3 2 + 4*ar*bi*br*ci*cr - 2*ar*br *cr - 2*ar*br *ci - 6*ar*br *cr 2 2 2 3 4 2 2 - 6*ar*br *ci *cr - 6*ar*br *cr - 2*ar*br*ci - 4*ar*br*ci *cr 4 4 2 4 2 3 3 3 2 - 2*ar*br*cr - bi *ci - bi *cr + 2*bi *ci + 2*bi *ci*cr 2 2 2 2 2 2 2 2 2 3 - 2*bi *br *ci - 2*bi *br *cr - 2*bi *br*ci *cr - 2*bi *br*cr 2 4 2 2 2 2 4 2 3 2 2 - bi *ci - 2*bi *ci *cr - bi *cr + 2*bi*br *ci + 2*bi*br *ci*cr 4 2 4 2 3 2 3 3 2 4 - br *ci - br *cr - 2*br *ci *cr - 2*br *cr - br *ci 2 2 2 2 4 - 2*br *ci *cr - br *cr ) > 0 cond clear x,x0,x1,x2,x3; %*********************************************************************** %***** ***** %***** T e s t Examples --- Module L I N B A N D ***** %***** ***** %*********************************************************************** on evallhseqp; % So both sides of equations evaluate. % Example L.1 operator v; off echo; dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3) dx=5.0e-2 x=0.1 do 25001 i=1,101 v(i)=x**2/2.0 x=x+dx 25001 continue iacof=200 iarhs=200 n=1 ad(n)=1.0 aucd(n)=0.0e+000 arhs(n)=v(1) n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n)=1.0 arhs(n)=v(3)-(2.0*v(2))+v(1) do 25002 k=3,99,1 n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n)=1.0 arhs(n)=v(k-1)+v(k+1)-(2.0*v(k)) 25002 continue n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n)=1.0 arhs(n)=v(101)-(2.0*v(100))+v(99) n=n+1 alcd(n)=0.0e+000 ad(n)=1.0 arhs(n)=v(101) call dgtsl(n,alcd,ad,aucd,arhs,ier) c n is number of equations c alcd,ad,aucd,arhs are arrays of dimension at least (n) c if (ier.ne.0) matrix acof is algorithmically singular if(ier.ne.0) call errout n=1 u(1)=arhs(n) n=n+1 u(2)=arhs(n) do 25003 k=3,99,1 n=n+1 u(k)=arhs(n) 25003 continue n=n+1 u(100)=arhs(n) n=n+1 u(101)=arhs(n) amer=0.0 arer=0.0 do 25004 i=1,101 am=abs(real(u(i)-v(i))) ar=am/v(i) if(am.gt.amer) amer=am if(ar.gt.arer) arer=ar 25004 continue write(*,100)amer,arer stop 100 format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2) end %*********************************************************************** % Example L.2 on nag; off echo; dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3) dx=5.0e-2 x=0.1 do 25005 i=1,101 v(i)=x**2/2.0 x=x+dx 25005 continue iacof=200 iarhs=200 n=1 ad(n)=1.0 aucd(n+1)=0.0e+000 arhs(n)=v(1) n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n+1)=1.0 arhs(n)=v(3)-(2.0*v(2))+v(1) do 25006 k=3,99,1 n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n+1)=1.0 arhs(n)=v(k-1)+v(k+1)-(2.0*v(k)) 25006 continue n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n+1)=1.0 arhs(n)=v(101)-(2.0*v(100))+v(99) n=n+1 alcd(n)=0.0e+000 ad(n)=1.0 arhs(n)=v(101) ier=0 call f01lef(n,ad,0.,aucd,alcd,1.e-10,au2cd,in,ier) c n is number of equations c alcd,ad,aucd,au2cd,arhs are arrays of dimension at least (n) c in is integer array of dimension at least (n) if(ier.ne.0 .or. in(n).ne.0) call errout call f04lef(1,n,ad,aucd,alcd,au2cd,in,arhs,0.,ier) if(ier.ne.0) call errout n=1 u(1)=arhs(n) n=n+1 u(2)=arhs(n) do 25007 k=3,99,1 n=n+1 u(k)=arhs(n) 25007 continue n=n+1 u(100)=arhs(n) n=n+1 u(101)=arhs(n) amer=0.0 arer=0.0 do 25008 i=1,101 am=abs(real(u(i)-v(i))) ar=am/v(i) if(am.gt.amer) amer=am if(ar.gt.arer) arer=ar 25008 continue write(*,100)amer,arer stop 100 format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2) end %*********************************************************************** % Example L.3 on imsl; off echo,nag; dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3) dx=5.0e-2 x=0.1 do 25009 i=1,101 v(i)=x**2/2.0 x=x+dx 25009 continue iacof=200 iarhs=200 n=1 acof(n,1)=0.0e+000 acof(n,2)=1.0 acof(n,3)=0.0e+000 arhs(n)=v(1) n=n+1 acof(n,1)=1.0 acof(n,2)=-2.0 acof(n,3)=1.0 arhs(n)=v(3)-(2.0*v(2))+v(1) do 25010 k=3,99,1 n=n+1 acof(n,1)=1.0 acof(n,2)=-2.0 acof(n,3)=1.0 arhs(n)=v(k-1)+v(k+1)-(2.0*v(k)) 25010 continue n=n+1 acof(n,1)=1.0 acof(n,2)=-2.0 acof(n,3)=1.0 arhs(n)=v(101)-(2.0*v(100))+v(99) n=n+1 acof(n,1)=0.0e+000 acof(n,2)=1.0 acof(n,3)=0.0e+000 arhs(n)=v(101) call leqt1b(acof,n,1,1,iacof,arhs,1,iarhs,0,xl,ier) c iacof is actual 1-st dimension of the acof array c iarhs is actual 1-st dimension of the arhs array c xl is working array with size n*(nlc+1) c where n is number of equations nlc number of lower c codiagonals c if ier=129( .ne.0) matrix acof is algorithmically singular if(ier.ne.0) call errout n=1 u(1)=arhs(n) n=n+1 u(2)=arhs(n) do 25011 k=3,99,1 n=n+1 u(k)=arhs(n) 25011 continue n=n+1 u(100)=arhs(n) n=n+1 u(101)=arhs(n) amer=0.0 arer=0.0 do 25012 i=1,101 am=abs(real(u(i)-v(i))) ar=am/v(i) if(am.gt.amer) amer=am if(ar.gt.arer) arer=ar 25012 continue write(*,100)amer,arer stop 100 format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2) end %*********************************************************************** % Example L.4 on essl; off echo,imsl; dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3) dx=5.0e-2 x=0.1 do 25013 i=1,101 v(i)=x**2/2.0 x=x+dx 25013 continue iacof=200 iarhs=200 n=1 ad(n)=1.0 aucd(n)=0.0e+000 arhs(n)=v(1) n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n)=1.0 arhs(n)=v(3)-(2.0*v(2))+v(1) do 25014 k=3,99,1 n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n)=1.0 arhs(n)=v(k-1)+v(k+1)-(2.0*v(k)) 25014 continue n=n+1 alcd(n)=1.0 ad(n)=-2.0 aucd(n)=1.0 arhs(n)=v(101)-(2.0*v(100))+v(99) n=n+1 alcd(n)=0.0e+000 ad(n)=1.0 arhs(n)=v(101) call dgtf(n,alcd,ad,aucd,af,ipvt) c n is number of equations c alcd,ad,aucd,af,arhs are arrays of dimension at least (n) c these arrays are double precision type c for single precision change dgtf to sgtf and dgts to sgts c ipvt is integer array of dimension at least (n+3)/8 call dgts(n,alcd,ad,aucd,af,ipvt,arhs) n=1 u(1)=arhs(n) n=n+1 u(2)=arhs(n) do 25015 k=3,99,1 n=n+1 u(k)=arhs(n) 25015 continue n=n+1 u(100)=arhs(n) n=n+1 u(101)=arhs(n) amer=0.0 arer=0.0 do 25016 i=1,101 am=abs(real(u(i)-v(i))) ar=am/v(i) if(am.gt.amer) amer=am if(ar.gt.arer) arer=ar 25016 continue write(*,100)amer,arer stop 100 format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2) end off essl; %*********************************************************************** %***** ***** %***** T e s t Complex Examples --- More Modules ***** %***** ***** %*********************************************************************** % Example M.1 off exp; coordinates t,x into n,j; grid uniform,x,t; dependence v(t,x),w(t,x); isgrid v(x..one),w(x..half); iim aa, v, diff(v,t)=c*diff(w,x), w, diff(w,t)=c*diff(v,x); ***************************** ***** Program ***** IIMET Ver 1.1.2 ***************************** Partial Differential Equations ============================== diff(v,t) - diff(w,x)*c = 0 - diff(v,x)*c + diff(w,t) = 0 0 interpolations are needed in t coordinate Equation for v variable is integrated in half grid point Equation for w variable is integrated in half grid point 0 interpolations are needed in x coordinate Equation for v variable is integrated in one grid point Equation for w variable is integrated in half grid point Equations after Discretization Using IIM : ========================================== 2*j - 1 2*j + 1 2*j + 1 2*j - 1 ((w(n,---------) - w(n,---------) - w(n + 1,---------) + w(n + 1,---------))*c 2 2 2 2 *ht + 2*(v(n + 1,j) - v(n,j))*hx)/(2*ht*hx) = 0 ( - (v(n,j + 1) - v(n,j) - v(n + 1,j) + v(n + 1,j + 1))*c*ht 2*j + 1 2*j + 1 + 2*(w(n + 1,---------) - w(n,---------))*hx)/(2*ht*hx) = 0 2 2 on exp; center t=1/2; functions v,w; approx( aa(0,0)=aa(0,1)); Difference scheme approximates differential equation df(v,t) - df(w,x)*c=0 with orders of approximation: 2 ht 2 hx center x=1/2; approx( aa(1,0)=aa(1,1)); Difference scheme approximates differential equation - df(v,x)*c + df(w,t)=0 with orders of approximation: 2 ht 2 hx let cos ax**2=1-sin ax**2; unfunc v,w; matrix a(1,2),b(2,2),bt(2,2); a(1,1):=aa(0,0); 2*j - 1 a(1,1) := (2*v(n + 1,j)*hx - 2*v(n,j)*hx + w(n + 1,---------)*c*ht 2 2*j + 1 2*j - 1 - w(n + 1,---------)*c*ht + w(n,---------)*c*ht 2 2 2*j + 1 - w(n,---------)*c*ht)/(2*ht*hx) 2 a(1,2):=aa(1,0); a(1,2) := ( - v(n + 1,j + 1)*c*ht + v(n + 1,j)*c*ht - v(n,j + 1)*c*ht 2*j + 1 2*j + 1 + v(n,j)*c*ht + 2*w(n + 1,---------)*hx - 2*w(n,---------)*hx)/(2*ht 2 2 *hx) off prfourmat; b:=ampmat a; kx*hx ax := ------- 2 [ 2 2 2 2 ] [ - sin(ax) *c *ht + hx 2*i*sin(ax)*c*ht*hx ] [-------------------------- ----------------------- ] [ 2 2 2 2 2 2 2 2 ] [ sin(ax) *c *ht + hx sin(ax) *c *ht + hx ] b := [ ] [ 2 2 2 2 ] [ 2*i*sin(ax)*c*ht*hx - sin(ax) *c *ht + hx ] [ ----------------------- --------------------------] [ 2 2 2 2 2 2 2 2 ] [ sin(ax) *c *ht + hx sin(ax) *c *ht + hx ] clear a,aa; factor lam; pol:=charpol b; 2 4 4 4 2 2 2 2 4 pol := (lam *(sin(ax) *c *ht + 2*sin(ax) *c *ht *hx + hx ) 4 4 4 4 4 4 4 + 2*lam*(sin(ax) *c *ht - hx ) + sin(ax) *c *ht 2 2 2 2 4 4 4 4 2 2 2 2 + 2*sin(ax) *c *ht *hx + hx )/(sin(ax) *c *ht + 2*sin(ax) *c *ht *hx 4 + hx ) pol:=troot1 pol; 2 2 2 4*sin(ax) *c *ht If ----------------------- = 0 and 0 2 2 2 2 sin(ax) *c *ht + hx = 0 , a root of the polynomial is equal to 1. 2 4 4 4 2 2 2 2 4 pol := (lam *(sin(ax) *c *ht + 2*sin(ax) *c *ht *hx + hx ) 4 4 4 4 4 4 4 + 2*lam*(sin(ax) *c *ht - hx ) + sin(ax) *c *ht 2 2 2 2 4 4 4 4 2 2 2 2 + 2*sin(ax) *c *ht *hx + hx )/(sin(ax) *c *ht + 2*sin(ax) *c *ht *hx 4 + hx ) pol:=hurw num pol; pol := 2 2 2 2 2 2 2 2 2 2 2 2 2 4*lam *sin(ax) *c *ht *(sin(ax) *c *ht + hx ) + 4*hx *(sin(ax) *c *ht + hx ) hurwitzp pol; Necessary and sufficient conditions are: 2 hx ----------------- > 0 2 2 2 sin(ax) *c *ht cond bt:=tcon b; [ 2 2 2 2 ] [ - sin(ax) *c *ht + hx - 2*sin(ax)*c*ht*hx*i ] [-------------------------- ------------------------ ] [ 2 2 2 2 2 2 2 2 ] [ sin(ax) *c *ht + hx sin(ax) *c *ht + hx ] bt := [ ] [ 2 2 2 2 ] [ - 2*sin(ax)*c*ht*hx*i - sin(ax) *c *ht + hx ] [ ------------------------ --------------------------] [ 2 2 2 2 2 2 2 2 ] [ sin(ax) *c *ht + hx sin(ax) *c *ht + hx ] bt*b; [1 0] [ ] [0 1] bt*b-b*bt; [0 0] [ ] [0 0] clear aa,a,b,bt; %*********************************************************************** % Example M.2 : Richtmyer, Morton: Difference methods for initial value % problems, &10.2. p.261 coordinates t,x into n,j; grid uniform,t,x; let cos ax**2=1-sin ax**2; unfunc v,w; matrix a(1,2),b(2,2),bt(2,2); a(1,1):=(v(n+1)-v(n))/ht-c*(w(j+1/2)-w(j-1/2))/hx$ a(1,2):=(w(n+1,j-1/2)-w(n,j-1/2))/ht-c*(v(n+1,j)-v(n+1,j-1))/hx$ off prfourmat; b:=ampmat a; kx*hx ax := ------- 2 [ 2*i*sin(ax)*c*ht ] [ 1 ------------------ ] [ hx ] [ ] b := [ 2 2 2 2 ] [ 2*i*sin(ax)*c*ht - 4*sin(ax) *c *ht + hx ] [------------------ ----------------------------] [ hx 2 ] [ hx ] clear a; factor lam; pol:=charpol b; 2 2 2 2 2 2 2 lam *hx + 2*lam*(2*sin(ax) *c *ht - hx ) + hx pol := -------------------------------------------------- 2 hx pol:=hurw num pol; 2 2 2 2 2 2 2 2 pol := 4*lam *sin(ax) *c *ht + 4*( - sin(ax) *c *ht + hx ) hurwitzp pol; Necessary and sufficient conditions are: 2 2 2 2 - sin(ax) *c *ht + hx -------------------------- > 0 2 2 2 sin(ax) *c *ht cond bt:=tcon b; [ - 2*sin(ax)*c*ht*i ] [ 1 --------------------- ] [ hx ] [ ] bt := [ 2 2 2 2 ] [ - 2*sin(ax)*c*ht*i - 4*sin(ax) *c *ht + hx ] [--------------------- ----------------------------] [ hx 2 ] [ hx ] bt*b; [ 2 2 2 2 3 3 3 ] [ 4*sin(ax) *c *ht + hx 8*sin(ax) *c *ht *i ] [------------------------- --------------------- ] [ 2 3 ] [ hx hx ] [ ] [ 3 3 3 4 4 4 2 2 2 2 4 ] [ - 8*sin(ax) *c *ht *i 16*sin(ax) *c *ht - 4*sin(ax) *c *ht *hx + hx ] [------------------------ --------------------------------------------------] [ 3 4 ] [ hx hx ] bt*b-b*bt; [ 3 3 3 ] [ 16*sin(ax) *c *ht *i ] [ 0 ----------------------] [ 3 ] [ hx ] [ ] [ 3 3 3 ] [ - 16*sin(ax) *c *ht *i ] [------------------------- 0 ] [ 3 ] [ hx ] clear a,b,bt; %*********************************************************************** % Example M.3: Mazurik: Algoritmy resenia zadaci..., preprint no.24-85, % AN USSR SO, Inst. teor. i prikl. mechaniky, p.34 operator v1,v2; matrix a(1,3),b(3,3),bt(3,3); a(1,1):=(p(n+1)-p(n))/ht+c/2*((-p(m-1)+2*p(m)-p(m+1))/hx + (v1(m+1)-v1(m-1))/hx - (p(k-1)-2*p(k)+p(k+1))/hy + (v2(k+1)-v2(k-1))/hy)$ a(1,2):=(v1(n+1)-v1(n))/ht+c/2*((p(m+1)-p(m-1))/hx - (v1(m-1)-2*v1(m)+v1(m+1))/hx)$ a(1,3):=(v2(n+1)-v2(n))/ht + c/2*((p(k+1)-p(k-1))/hy - (v2(k-1)-2*v2(k)+v2(k+1))/hy)$ coordinates t,x,y into n,m,k; functions p,v1,v2; for k:=1:3 do approx(a(1,k)=0); Difference scheme approximates differential equation df(p,t) + df(v1,x)*c + df(v2,y)*c=0 with orders of approximation: 2 ht hx hy Difference scheme approximates differential equation df(p,x)*c + df(v1,t)=0 with orders of approximation: 2 ht hx 1 Difference scheme approximates differential equation df(p,y)*c + df(v2,t)=0 with orders of approximation: 2 ht 1 hy grid uniform,t,x,y; unfunc p,v1,v2; hy:=hx; hy := hx off prfourmat; b:=ampmat a; ax := kx*hx ay := ky*hy cos(ax)*c*ht + cos(ay)*c*ht - 2*c*ht + hx - i*sin(ax)*c*ht b := mat((-------------------------------------------,-------------------, hx hx - i*sin(ay)*c*ht -------------------), hx - i*sin(ax)*c*ht cos(ax)*c*ht - c*ht + hx (-------------------,--------------------------,0), hx hx - i*sin(ay)*c*ht cos(ay)*c*ht - c*ht + hx (-------------------,0,--------------------------)) hx hx pol:=charpol b; 3 3 2 2 pol := (lam *hx + lam *hx *( - 2*cos(ax)*c*ht - 2*cos(ay)*c*ht + 4*c*ht - 3*hx) 2 2 2 2 + lam*hx*(3*cos(ax)*cos(ay)*c *ht - 5*cos(ax)*c *ht 2 2 + 4*cos(ax)*c*ht*hx - 5*cos(ay)*c *ht + 4*cos(ay)*c*ht*hx 2 2 2 3 3 + 7*c *ht - 8*c*ht*hx + 3*hx ) + 4*cos(ax)*cos(ay)*c *ht 2 2 3 3 2 2 - 3*cos(ax)*cos(ay)*c *ht *hx - 4*cos(ax)*c *ht + 5*cos(ax)*c *ht *hx 2 3 3 2 2 - 2*cos(ax)*c*ht*hx - 4*cos(ay)*c *ht + 5*cos(ay)*c *ht *hx 2 3 3 2 2 2 3 3 - 2*cos(ay)*c*ht*hx + 4*c *ht - 7*c *ht *hx + 4*c*ht*hx - hx )/hx let cos ax=cos ax2**2-sin ax2**2, cos ay=cos ay2**2-sin ay2**2, sin ax=2*sin ax2*cos ax2, sin ay=2*sin ay2*cos ay2, cos ax2**2=1-sin ax2**2, cos ay2**2=1-sin ay2**2, sin ax2=s1, sin ay2=s2, hx=c*ht/cap; factor lam; order s1,s2; pol:=troot1 pol; 2 2 3 If 16*s1 *s2 *cap = 0 and 0 = 0 , a root of the polynomial is equal to 1. 3 2 2 2 pol := lam + lam *(4*s1 *cap + 4*s2 *cap - 3) + lam 2 2 2 2 2 2 2 2 2 *(12*s1 *s2 *cap + 4*s1 *cap - 8*s1 *cap + 4*s2 *cap - 8*s2 *cap + 3) 2 2 3 2 2 2 2 2 2 + 16*s1 *s2 *cap - 12*s1 *s2 *cap - 4*s1 *cap + 4*s1 *cap 2 2 2 - 4*s2 *cap + 4*s2 *cap - 1 clear cos ax,cos ay,sin ax,sin ay,cos ax2**2,cos ay2**2,sin ax2,sin ay2, hx,hy; pol:=hurw num pol; 3 2 2 3 pol := 16*lam *s1 *s2 *cap 2 2 2 2 2 2 2 2 + 8*lam *cap *( - 6*s1 *s2 *cap + 3*s1 *s2 + s1 + s2 ) + 16*lam*cap 2 2 2 2 2 2 2 2 2 *(3*s1 *s2 *cap - 3*s1 *s2 *cap - s1 *cap + s1 - s2 *cap + s2 ) + 8*( 2 2 3 2 2 2 2 2 2 2 2 - 2*s1 *s2 *cap + 3*s1 *s2 *cap + s1 *cap - 2*s1 *cap + s2 *cap 2 - 2*s2 *cap + 1) hurwitzp pol; If we denote: 2 2 3 (1) 16*s1 *s2 *cap > 0 2 2 2 2 2 2 2 (2) 8*cap *( - 6*s1 *s2 *cap + 3*s1 *s2 + s1 + s2 ) > 0 2 2 2 2 2 2 2 2 2 (3) 16*cap*(3*s1 *s2 *cap - 3*s1 *s2 *cap - s1 *cap + s1 - s2 *cap + s2 ) > 0 2 2 3 2 2 2 2 2 2 2 2 (4) 8*( - 2*s1 *s2 *cap + 3*s1 *s2 *cap + s1 *cap - 2*s1 *cap + s2 *cap 2 - 2*s2 *cap + 1) > 0 2 2 2 2 2 2 2 (5) 8*cap *( - 6*s1 *s2 *cap + 3*s1 *s2 + s1 + s2 ) > 0 3 4 4 3 4 4 2 4 4 (6) 128*cap *( - 16*s1 *s2 *cap + 24*s1 *s2 *cap - 9*s1 *s2 *cap 4 2 2 4 2 4 2 4 4 + 8*s1 *s2 *cap - 10*s1 *s2 *cap + 3*s1 *s2 - s1 *cap + s1 2 4 2 2 4 2 4 2 2 + 8*s1 *s2 *cap - 10*s1 *s2 *cap + 3*s1 *s2 - 2*s1 *s2 *cap 2 2 4 4 + s1 *s2 - s2 *cap + s2 ) > 0 3 6 6 6 6 6 5 6 6 4 (7) 1024*cap *(32*s1 *s2 *cap - 96*s1 *s2 *cap + 90*s1 *s2 *cap 6 6 3 6 4 5 6 4 4 - 27*s1 *s2 *cap - 32*s1 *s2 *cap + 100*s1 *s2 *cap 6 4 3 6 4 2 6 2 4 - 93*s1 *s2 *cap + 27*s1 *s2 *cap + 10*s1 *s2 *cap 6 2 3 6 2 2 6 2 6 3 - 31*s1 *s2 *cap + 26*s1 *s2 *cap - 6*s1 *s2 *cap - s1 *cap 6 2 6 4 6 5 4 6 4 + 3*s1 *cap - 2*s1 *cap - 32*s1 *s2 *cap + 100*s1 *s2 *cap 4 6 3 4 6 2 4 4 4 - 93*s1 *s2 *cap + 27*s1 *s2 *cap + 20*s1 *s2 *cap 4 4 3 4 4 2 4 4 - 76*s1 *s2 *cap + 73*s1 *s2 *cap - 21*s1 *s2 *cap 4 2 3 4 2 2 4 2 - 3*s1 *s2 *cap + 16*s1 *s2 *cap - 14*s1 *s2 *cap 4 2 4 4 2 6 4 + 3*s1 *s2 - s1 *cap + s1 + 10*s1 *s2 *cap 2 6 3 2 6 2 2 6 - 31*s1 *s2 *cap + 26*s1 *s2 *cap - 6*s1 *s2 *cap 2 4 3 2 4 2 2 4 - 3*s1 *s2 *cap + 16*s1 *s2 *cap - 14*s1 *s2 *cap 2 4 2 2 2 2 6 3 6 2 + 3*s1 *s2 - 2*s1 *s2 *cap + s1 *s2 - s2 *cap + 3*s2 *cap 6 4 4 - 2*s2 *cap - s2 *cap + s2 ) > 0 (c1) (1) AND (2) AND (4) (c2) (1) AND (3) AND (4) (d1) (5) AND (7) (d2) (6) Necessary and sufficient conditions are: ( (C1) OR (C2) ) AND ( (D1) OR (D2) ) cond bt:=tcon b; cos(ax)*c*ht + cos(ay)*c*ht - 2*c*ht + hx sin(ax)*c*ht*i bt := mat((-------------------------------------------,----------------, hx hx sin(ay)*c*ht*i ----------------), hx sin(ax)*c*ht*i cos(ax)*c*ht - c*ht + hx (----------------,--------------------------,0), hx hx sin(ay)*c*ht*i cos(ay)*c*ht - c*ht + hx (----------------,0,--------------------------)) hx hx bt*b; 2 2 2 2 mat(((2*cos(ax)*cos(ay)*c *ht - 4*cos(ax)*c *ht + 2*cos(ax)*c*ht*hx 2 2 2 2 2 2 - 4*cos(ay)*c *ht + 2*cos(ay)*c*ht*hx + 6*c *ht - 4*c*ht*hx + hx )/hx , 2 2 2 2 sin(ax)*c *ht *i*( - cos(ay) + 1) sin(ay)*c *ht *i*( - cos(ax) + 1) -----------------------------------,-----------------------------------), 2 2 hx hx 2 2 sin(ax)*c *ht *i*(cos(ay) - 1) (--------------------------------, 2 hx 2 2 2 2 2 - 2*cos(ax)*c *ht + 2*cos(ax)*c*ht*hx + 2*c *ht - 2*c*ht*hx + hx ----------------------------------------------------------------------, 2 hx 2 2 sin(ax)*sin(ay)*c *ht ------------------------), 2 hx 2 2 2 2 sin(ay)*c *ht *i*(cos(ax) - 1) sin(ax)*sin(ay)*c *ht (--------------------------------,------------------------, 2 2 hx hx 2 2 2 2 2 - 2*cos(ay)*c *ht + 2*cos(ay)*c*ht*hx + 2*c *ht - 2*c*ht*hx + hx ----------------------------------------------------------------------)) 2 hx bt*b-b*bt; 2 2 2*sin(ax)*c *ht *i*( - cos(ay) + 1) mat((0,-------------------------------------, 2 hx 2 2 2*sin(ay)*c *ht *i*( - cos(ax) + 1) -------------------------------------), 2 hx 2 2 2*sin(ax)*c *ht *i*(cos(ay) - 1) (----------------------------------,0,0), 2 hx 2 2 2*sin(ay)*c *ht *i*(cos(ax) - 1) (----------------------------------,0,0)) 2 hx clear a,b,bt,pol; %*********************************************************************** end; (TIME: fide 15030 15900)