Finite Difference Methods for PDEs

Finite difference methods approximate partial derivatives on a discrete grid. They are the most direct approach to numerical PDEs and provide the clearest illustration of stability, consistency, and convergence concepts.

Discretization of Differential Operators

Definition8.1Finite Difference Approximations

On a grid with spacing $h$ , standard finite difference approximations include:

Forward: $u'(x) \approx \frac{u(x+h) - u(x)}{h}$ ( $O(h)$ )
Backward: $u'(x) \approx \frac{u(x) - u(x-h)}{h}$ ( $O(h)$ )
Central: $u'(x) \approx \frac{u(x+h) - u(x-h)}{2h}$ ( $O(h^2)$ )
Second derivative: $u''(x) \approx \frac{u(x+h) - 2u(x) + u(x-h)}{h^2}$ ( $O(h^2)$ )

The five-point Laplacian on a 2D grid: $\nabla^2 u \approx \frac{u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4u_{i,j}}{h^2}$ .

ExampleSolving the Poisson Equation

The Poisson equation $-\nabla^2 u = f$ on the unit square with Dirichlet boundary conditions, discretized on an $N \times N$ grid, gives a system $Au = f$ where $A$ is the $(N-1)^2 \times (N-1)^2$ block tridiagonal matrix with the 5-point stencil. The matrix $A$ is SPD, sparse (at most 5 nonzeros per row), and has condition number $\kappa(A) = O(1/h^2) = O(N^2)$ .

Heat Equation: Explicit and Implicit Schemes

Definition8.2Forward Euler (FTCS) for Heat Equation

For $u_t = \alpha u_{xx}$ , the Forward Time Central Space (FTCS) scheme is $\frac{u_j^{n+1} - u_j^n}{\Delta t} = \alpha \frac{u_{j+1}^n - 2u_j^n + u_{j-1}^n}{(\Delta x)^2}$ , or $u_j^{n+1} = u_j^n + r(u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ where $r = \alpha\Delta t/(\Delta x)^2$ . The scheme is stable if and only if $r \leq 1/2$ (CFL condition), giving $\Delta t \leq (\Delta x)^2/(2\alpha)$ . This severe restriction makes FTCS impractical for fine grids.

Definition8.3Crank-Nicolson Scheme

The Crank-Nicolson scheme averages the explicit and implicit discretizations: $\frac{u_j^{n+1} - u_j^n}{\Delta t} = \frac{\alpha}{2}\left[\frac{\delta^2 u_j^{n+1}}{(\Delta x)^2} + \frac{\delta^2 u_j^n}{(\Delta x)^2}\right]$ where $\delta^2 u_j = u_{j+1} - 2u_j + u_{j-1}$ . Crank-Nicolson is unconditionally stable (no CFL restriction) and second-order in both time and space: $O((\Delta t)^2 + (\Delta x)^2)$ . It requires solving a tridiagonal system at each time step, costing $O(N)$ .

Wave Equation

RemarkCFL Condition for the Wave Equation

For $u_{tt} = c^2 u_{xx}$ , the standard explicit scheme $u_j^{n+1} = 2u_j^n - u_j^{n-1} + \lambda^2(u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ where $\lambda = c\Delta t/\Delta x$ is stable iff $\lambda \leq 1$ (CFL condition). At $\lambda = 1$ the scheme is exact (it reproduces d'Alembert's solution), a remarkable property not shared by other PDEs. The CFL condition has a physical interpretation: the numerical domain of dependence must contain the physical domain of dependence.