Finite difference methods approximate partial derivatives on grids. Their analysis combines Taylor expansion, stability, and discrete operator theory.
Finite difference operator
Definition. A finite difference operator replaces derivatives by differences of nearby grid values. For example, the centered second difference \((u_{i-1}-2u_i+u_{i+1})/h^2\) approximates \(\partial^2 u/\partial x^2\).
Elliptic model problem
Proposition. The standard finite difference discretization of the Dirichlet Poisson problem produces a sparse linear system whose matrix is a discrete Laplacian. Maximum principles and energy estimates give stability.
Heat equation stability
Theorem. For the explicit heat scheme, stability requires a parabolic step restriction of the form \(\Delta t\le C h^2\). Implicit schemes relax this restriction but require linear solves.
Hyperbolic CFL condition
Theorem. For wave or transport equations, the time step must respect a Courant-Friedrichs-Lewy condition: numerical information must propagate at least as fast as physical information. Otherwise convergence fails even if the formal truncation error is small.
Discrete conservation
Remark. For conservation laws, the discretization should respect flux balance and entropy conditions. Smooth Taylor analysis alone does not capture shocks or weak solutions.
Elliptic finite differences
For \(-\Delta u=f\) on a rectangular grid, the five-point stencil is \[ -\Delta_hu_{ij} =\frac{4u_{ij}-u_{i+1,j}-u_{i-1,j}-u_{i,j+1}-u_{i,j-1}}{h^2}. \] Taylor expansion gives second-order consistency for \(u\in C^4\). The matrix is symmetric positive definite under Dirichlet conditions and an M-matrix, giving a discrete maximum principle.
Theorem. On a rectangle with smooth solution and Dirichlet data, \[ \|u_h-R_hu\|_\infty\le Ch^2\|u\|_{C^4}. \]
Proof. The error solves \(-\Delta_he=\tau_h\), with \(\|\tau_h\|_\infty=O(h^2)\). A discrete barrier such as \(x(L-x)+y(H-y)\) and the maximum principle bounds \(\|(-\Delta_h)^{-1}\|_{\infty\to\infty}\) independently of \(h\). Apply it to the error equation.
Heat equation and the CFL condition
Forward Euler with centered space differences gives \[ u_j^{n+1} =u_j^n+\mu(u_{j-1}^n-2u_j^n+u_{j+1}^n), \qquad \mu=\frac{\Delta t}{h^2}. \] Fourier mode \(e^{ij\xi}\) has amplification \[ G(\xi)=1-4\mu\sin^2(\xi/2). \] Stability requires \(|G(\xi)|\le1\) for all \(\xi\), hence \[ 0\le\mu\le\frac12. \] This parabolic CFL restriction makes explicit time steps scale like \(h^2\). Backward Euler is unconditionally stable; Crank--Nicolson is A-stable but can retain oscillatory stiff modes.
Advection and numerical diffusion
For \(u_t+au_x=0\), \(a>0\), upwind discretization is \[ u_j^{n+1}=u_j^n-\nu(u_j^n-u_{j-1}^n), \qquad \nu=a\Delta t/h. \] It is stable and monotone for \(0\le\nu\le1\). Its modified equation is \[ u_t+au_x =\frac{ah}{2}(1-\nu)u_{xx}+O(h^2), \] revealing artificial diffusion. Centered explicit Euler has no damping and is unstable for pure advection.
Von Neumann analysis and its limits
Fourier analysis exactly diagnoses constant-coefficient periodic linear schemes in \(L_2\). Variable coefficients, boundaries, nonlinearities, and nonnormal matrices require energy estimates, matrix stability, or monotonicity. A favorable amplification factor on the infinite grid is not by itself a boundary-stability theorem.
Conservation laws
A conservative update has \[ u_j^{n+1} =u_j^n-\frac{\Delta t}{h} \{\widehat f_{j+1/2}-\widehat f_{j-1/2}\}. \] Fluxes cancel when summed, preserving total mass up to boundary terms. Godunov's theorem says a linear monotone scheme cannot exceed first order. High-resolution methods use nonlinear limiters to obtain higher accuracy away from shocks while preserving entropy-compatible nonoscillatory behavior.