Consider the 2D heat equation with a source term in the domain −1 ≤ x ≤ 1 −1 ≤ y ≤ 1:
,
where α is the thermal conductivity and assumed to be 1. The equation is subject to homogeneous initial and boundary conditions, namely, ϕ(x,y,0) = 0, ϕ(±1,y,t) = 0, and ϕ(x,±1,t) = 0.
Complete the following tasks:
1. Determine the exact steady-state solution of ϕ when the source term is given by S(x,y) = 2(2−x2−y2).
2. Employ the Crank-Nicolson method for time stepping and a second-order central difference schemefor the spatial derivative to solve the equation up to steady state on a uniform grid. Afterwards, plot both the exact and numerical steady-state solutions, considering parameters like time step ∆t and the number of grid points in the x and y directions, N and M respectively.
3. Based on your numerical findings, provide a discussion about the order of accuracy in both time andspace.
