I have an iterative finite difference scheme for the Poisson equation$$\nabla^2 u=-\rho$$It's the Jacobi method, which has the form (for 1D systems)$$u^{n+1}_{i} = \frac{1}{2}(u^n_{i+1} + u^n_{i-1} + \rho_i~h^2)$$Is there a similar scheme for the nonlinear equation$$\nabla^2u = -\rho ~e^{u-v}$$I tried the following$$u^{n+1}_{i} = \frac{1}{2}(u^n_{i+1} + u^n_{i-1} + \rho_i~e^{u^n_i-v_i}~h^2)$$but couldn't get a converged answer.Above, I have given a 1D example. I am solving the 3D version, over a cube. The front and back of the cube have Von Neumann boundary conditions ( equal to 0), and the top, bottom, left, and right have Dirichlet boundary conditions ( also equal to 0 ). My initial guess is $u = 0$ everywhere.
Convergence of Jacobi's method for a semilinear elliptic PDE
finite difference;nonlinear equations;iterative method