| 35 | | The Poisson equation can be thought of as being the steady state limit of the diffusion equation. Can be thought of the steady state, static form of the diffusion equaiton - begin with equaiotn, expect relaxation to steady state (i.e. as t-> inf, d/dt -> 0), get the following Poisson equation: |
| | 35 | By replacing the 2nd-order derivatives of Poisson's equation with 2nd-order central finite differences, we have the discretized version of Poisson's equation in 1D (See Leveque, |
| | 36 | Finite Difference Methods for Ordinary and Partial Differential Equations), |
| | 37 | |
| | 38 | [[latex($\frac{1}{h}(u_{i+1} + u_{i-1} - 2 u_i) = f$)]] |
| | 39 | |
| | 40 | |
| | 41 | which can be re-arranged for |
| | 42 | |
| | 43 | [[latex($u_i = \frac{1}{2}(u_{i-1}+ u_{i+1} - h f_i) $)]] |
| | 44 | |
| | 45 | Here, h is the internode size, which for a 1D line of mesh points is h=L/m+1, where L is domain length, and m is the number of grid points. |
