Computer Algebra Recipes

6.3. NUMERICAL SIMULATION OF PDES 283
> f:=(i,j)->2*(1+r)*V[i,j]-V[i+1,j]-V[i-1,j]-r*(V[i,j+1]
+V[i,j-1])+h^2*x[i]*exp(y[j])=0;
f := (i; j) ! 2(1+r) Vi; j ¡ Vi+1;j ¡ Vi¡1;j ¡ r (Vi; j+1 + Vi; j¡1)+h 2 xi e yj
The mesh equations are determined for all the internal grid points using two
nested sequences running from i =1tom ¡ 1andfromj =1ton ¡ 1.
> eqs:=fseq(seq(f(i,j),i=1..m-1),j=1..n-1)g:
The unknown potentials at the internal grid points are entered as the variables.
> vars:=fseq(seq(V[i,j],i=1..m-1),j=1..n-1)g:
The mesh equations are solved (to 6 digits) for the variables, and the solution
is assigned.
> sol:=evalf(fsolve(eqs,vars),6): assign(sol):
Three-dimensional plotting points are formed for all the grid points, including
those on the boundaries.
> pts:=seq(seq([x[i],y[j],V[i,j]],i=0..m),j=0..n):
Using the pointplot3d command, the points representing the potential at each
grid point are plotted as size-6 blue circles, the result being shown in Figure 6.4.
> pointplot3d([pts],symbol=circle,symbolsize=6,color=blue,
axes=boxed,orientation=[135,60],tickmarks=[3,3,3],
labels=["x","y","V"]);
4
V
2
0
2
1
x
0
0.5 y
Figure 6.4: Numerical solution of the Poisson equation.
The CPU time to execute the entire recipe is about 6 seconds.
> cpu:=time()-begin;
cpu := 5:978
0