Consistency, convergence and stability

First, let’s clarify the expression “converge to the solution of the exact equation”, by defining the error of the numerical approximation.

Definition

The error of the numerical approximation \(u_m^n\), \(\epsilon_m^n\), is

\[ \epsilon_m^n = u_m^n - u(m\Delta x,n\Delta t), \]

where \(u_m^n\) is the numerical solution and \(u(m\Delta x,n\Delta t)\) is the exact solution at \((x_m,t^n)\).

We can now define more precisely the concepts of boundedness and convergence. Let us consider the following questions:

• What happens to \(\epsilon_m^n\) as \(\Delta t, \Delta x \to 0\) for fixed \(n\Delta t\)?

• What happens to \(\epsilon_m^n\) as \(n\to\infty\) for fixed \(\Delta t\) and \(\Delta x\)?

The first questions leads us to the concepts of consistency and convergence.

Definition

A difference equation is consistent when it approaches the corresponding PDE as \(\Delta t, \Delta x \to 0\).

The truncation error of the leapfrog scheme applied to the linear advection equation is:

\[ Tr = \frac{\Delta t^2}{3!}\frac{\partial^3 u}{\partial t^3} + c\frac{\Delta x^2}{3!}\frac{\partial^3 u}{\partial x^3} + \dots \]

or \(Tr = O(\Delta t^2)+O(\Delta x^2)\), which clearly converges to zero as \(\Delta t, \Delta x \to 0\). Thus the leapfrog discretization of the linear advection equation is consistent.

Important

Numerical schemes with truncation errors with terms \(O(\Delta t^p)\) or \(O(\Delta x^p)\) are consistent if \(p \ge 1\).

But this is not a guarantee that we will obtain meaningful results. We must also demand that, over an interval \(n\Delta t\), \(\epsilon_m^n \to 0\) as \(\Delta t, \Delta x \to 0\).

Definition

A numerical solution is convergent if, for fixed \(n\Delta t\), \(\epsilon_m^n \to 0\) as \(\Delta t, \Delta x \to 0\).

If a numerical scheme gives convergent solutions for any initial condition, the numerical scheme is said to be convergent.

The second question leads us to the concept of numerical stability. If \(u(m\Delta x,n\Delta t)\) is bounded, we can expect \(\epsilon_m^n\) to remain bounded if \(u_m^n\) remains bounded.

Boundedness

A function \(f\) defined on a set \(X\) with real or complex values is called bounded if there exists a real number \(M\) such that

\[ |f(x)|\le M \]

for all \(x\) in \(X\).

This leads us to the definition of stability of a numerical scheme:

Definition

A finite-difference scheme is stable if its solutions remain uniformly bounded functions of the initial condition, for all sufficiently small \(\Delta t\).

Uniformly bounded functions

A uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant.

Important

The concept of stability is not directly related to the underlying PDE.

Convergence is not so easily demonstrated as it involves the exact equation, whose solution may not be known. Stability is easier to demonstrate as it involved the finite-difference scheme. Luckily, we can infer convergence from stability in certain cases using the Lax Equivalence Theorem.

Lax Equivalence Theorem

Given a properly posed, linear initial value problem and a finite-difference approximation that satisfies the consistency condition, stability is the necessary and sufficient condition for convergence.

Therefore, if we can show consistency and stability, the convergence of the finite-difference scheme is guaranteed by the Lax Equivalence Theorem.