| 14 | | The concept of LTE is illustrated with the following example. Suppose we replace the 2nd derivative of the following ODE |
| | 14 | The concept of LTE is illustrated with the following example. Suppose we replace the 2nd derivative of the 1D Poisson equation, |
| | 15 | |
| | 16 | [[latex($\frac{d^2 u}{dx^2} = f(x)$)]] |
| | 17 | |
| | 18 | with the centered (2nd order, as can be verified by above discussion) finite different approximation, |
| | 19 | |
| | 20 | [[latex($\frac{1}{h}[\frac{u(x+h)-u(x)}{h} - \frac{u(x-h)-u(x)}{-h}] = \frac{1}{h^2}[u(x+h)+u(x-h)-2u(x)]$)]] |
| | 21 | |
| | 22 | (which is easily interpreted as the finite difference version of the derivative of the derivative). With these, and replacing the functions with discrete versions of the functions, we have |
| | 23 | |
| | 24 | [[latex($\frac{d^2 u}{dx^2} = f(x)$)]] |
