Mathematics Lecture Notes

Theorem8.1Lax-Richtmyer Equivalence Theorem

For a well-posed linear initial value problem and a consistent finite difference scheme, stability is equivalent to convergence. Precisely: the numerical solution $u_h^n$ converges to the exact solution $u(t_n, x_j)$ as $h, \Delta t \to 0$ (with any fixed ratio or under a CFL constraint) if and only if the scheme is stable in the sense of Lax-Richtmyer (the discrete solution operator $S_h^n$ is uniformly bounded: $\|S_h^n\| \leq C$ for $n\Delta t \leq T$ ).

Proof

Setup. Let the PDE be $u_t = Lu$ with solution operator $u(t) = E(t)u_0$ . The finite difference scheme advances as $u_h^n = S_h u_h^{n-1} = S_h^n u_h^0$ where $S_h$ is the one-step discrete operator.

Consistency means $S_h u(t_n) = E(\Delta t) u(t_n) + \Delta t \cdot \tau_n$ where $\|\tau_n\| \to 0$ as $h \to 0$ (the local truncation error vanishes).

Convergence $\Rightarrow$ Stability. If the scheme converges for all smooth initial data, then by the Uniform Boundedness Principle (Banach-Steinhaus theorem), $\|S_h^n\|$ must be uniformly bounded. If not, there would exist $u_0$ with $\|S_h^n u_0\| \to \infty$ , contradicting convergence to the bounded $E(t)u_0$ .

Stability $\Rightarrow$ Convergence. Let $e^n = S_h^n u_0 - E(n\Delta t)u_0$ be the global error. Write the telescoping decomposition: $e^n = \sum_{k=0}^{n-1} S_h^{n-1-k}[S_h E(k\Delta t)u_0 - E((k+1)\Delta t)u_0]$

Each bracketed term is the local error at step $k$ : $S_h E(k\Delta t)u_0 - E(\Delta t)E(k\Delta t)u_0 = \Delta t \cdot \tau_k$ by consistency, where $\|\tau_k\| \leq C_\tau \to 0$ as $h \to 0$ for smooth $u_0$ .

By stability: $\|S_h^{n-1-k}\| \leq M$ for all $k$ . Therefore: $\|e^n\| \leq \sum_{k=0}^{n-1} M \cdot \Delta t \cdot \|\tau_k\| \leq M \cdot n\Delta t \cdot \max_k\|\tau_k\| \leq MT \cdot \max_k\|\tau_k\| \to 0$ as $h \to 0$ , proving convergence. $\square$

■

ExampleApplication to FTCS for the Heat Equation

For the FTCS scheme applied to $u_t = u_{xx}$ : consistency holds with truncation error $O(\Delta t + (\Delta x)^2)$ . Von Neumann stability analysis gives stability iff $r = \Delta t/(\Delta x)^2 \leq 1/2$ . By the Lax equivalence theorem, the scheme converges (with order $O(\Delta t + (\Delta x)^2)$ ) under this CFL condition. Without the CFL condition, the scheme is unstable, and Lax equivalence tells us it cannot converge -- regardless of how consistent it is.

RemarkLimitations and Extensions

The Lax-Richtmyer theorem applies to linear problems. For nonlinear PDEs, convergence requires additional arguments (compactness, entropy conditions). The theorem also assumes a fixed spatial discretization refined together with time; for method-of-lines approaches where spatial and temporal discretizations are decoupled, separate stability analyses are needed for each component.

Math Notes

Lax Equivalence Theorem

Proof