Finite difference method

Remarks

The local truncation error is the residual left when the exact solution is put into the scheme. The scheme is consistent of order $p$ if this error is $O(h^p)$; for example, Taylor expansion gives $\frac{u(x+h) - 2u(x) + u(x-h)}{h^2} = u''(x) + \frac{h^2}{12} u^{(4)}(\xi)$ for some $\xi$ in $(x - h, x + h)$ when $u \in C^4$. Example: for the heat equation $u_t = u_{xx}$, the explicit scheme is $u_j^{n+1} = u_j^n + r (u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ with $r = k/h^2$. It is consistent of order $1$ in $k$ and $2$ in $h$. It is stable in the maximum norm when $r \le \tfrac12$, since then $u_j^{n+1}$ is a convex combination of $u_{j-1}^n, u_j^n, u_{j+1}^n$. Non-example: the finite element method also yields algebraic equations on a mesh, but it does not satisfy clause 2. It imposes the equation in weak form against basis functions instead of replacing derivatives by difference quotients.