This GPU based script draws 10 cross-sections ui,n/2 after every
2it weighted Jacobi iterations.
fi,j = sin(πi/n) sin(πj/n).
We consider 2D Poisson equation with fixed borders uborder = 0 Δu(x,y) = f(x,y).
On square n×n grid we get
ui-1,j + ui+1,j + ui,j-1 + ui,j+1
- 4ui = h2fi,j.
Weighted Jacobi iterations are
ui,jnew = (1 - ω)ui,j +
ω(ui-1,j + ui+1,j + ui,j-1 + ui,j+1 -
h2fi,j )/4 (*)
Transformation (*) has eigenvectors (we put fi,j = 0)
ui,j(m,k) = sin(πmi/n) sin(πkj/n) with eigenvalues
λm,k = 1 - ω (sin2πm/2n +
λn,n = 1 - 2ω,
λn/2,n/2 = λ0,n = 1 - ω,
λ0,n/2 = 1 - ω/2.
From |λn,n| = |λ0,n/2| we get
ω = 4/5 and λ0,n/2 = 3/5.
At last for small m,k we get approximately
λm,k = 1 - ωπ2(m2 +
so relaxation is very slow for long wave harmonics. As since convergence is faster
for small n we need coarse grids to accelerate relaxation. We consider
multigrid algorithm next.
Simulations on GPUupdated 6 Aug 2012