Lax equivalence theorem
Encyclopedia
In numerical analysis
, the Lax equivalence theorem is the fundamental theorem in the analysis of finite difference method
s for the numerical solution of partial differential equation
s. It states that for a consistent finite difference method for a well-posed linear initial value problem
, the method is convergent
if and only if it is stable
.
The importance of the theorem is that while convergence of the solution of the finite difference method to the solution of the partial differential equation is what is desired, it is ordinarily difficult to establish because the numerical method is defined by a recurrence relation
while the differential equation
involves a differentiable function. However, consistency—the requirement that the finite difference method approximate the correct partial differential equation—is straightforward to verify, and stability is typically much easier to show than convergence (and would be needed in any event to show that round-off error
will not destroy the computation). Hence convergence is usually shown via the Lax equivalence theorem.
Stability in this context means that a matrix norm of the matrix used in the iteration is at most unity, called (practical) Lax-Richtmyer stability. Often a von Neumann stability analysis
is substituted for convenience, although von Neumann stability only implies Lax-Richtmyer stability in certain cases.
This theorem is due to Peter Lax
. It is sometimes called the Lax–Richtmyer theorem, after Peter Lax and Robert D. Richtmyer
.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, the Lax equivalence theorem is the fundamental theorem in the analysis of finite difference method
Finite difference method
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...
s for the numerical solution of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s. It states that for a consistent finite difference method for a well-posed linear initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
, the method is convergent
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
if and only if it is stable
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
.
The importance of the theorem is that while convergence of the solution of the finite difference method to the solution of the partial differential equation is what is desired, it is ordinarily difficult to establish because the numerical method is defined by a recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
while the differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
involves a differentiable function. However, consistency—the requirement that the finite difference method approximate the correct partial differential equation—is straightforward to verify, and stability is typically much easier to show than convergence (and would be needed in any event to show that round-off error
Round-off error
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...
will not destroy the computation). Hence convergence is usually shown via the Lax equivalence theorem.
Stability in this context means that a matrix norm of the matrix used in the iteration is at most unity, called (practical) Lax-Richtmyer stability. Often a von Neumann stability analysis
Von Neumann stability analysis
In numerical analysis, von Neumann stability analysis is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations...
is substituted for convenience, although von Neumann stability only implies Lax-Richtmyer stability in certain cases.
This theorem is due to Peter Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
. It is sometimes called the Lax–Richtmyer theorem, after Peter Lax and Robert D. Richtmyer
Robert D. Richtmyer
Robert Davis Richtmyer was an American physicist, mathematician, educator, author, and musician.-Biography:Richtmyer was born on October 10, 1910 in Ithaca, New York.His father was physicist Floyd K...
.