numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

, the Lax equivalence theorem is a fundamental theorem in the analysis of linear

finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial doma ...

s for the numerical solution of linear

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s. It states that for a linear

consistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...

finite difference method for a well-posed linear

initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ...

, the method is convergent if and only if it is

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

. The importance of the theorem is that while the convergence of the solution of the linear finite difference method to the solution of the linear partial differential equation is what is desired, it is ordinarily difficult to establish because the numerical method is defined by a

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

while the differential equation involves a

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

function. However, consistency—the requirement that the linear finite difference method approximates the correct linear 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 In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Roun ...

will not destroy the computation). Hence convergence is usually shown via the Lax equivalence theorem. Stability in this context means that a

matrix norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...

of the matrix used in the iteration is at most unity, called (practical) Lax–Richtmyer stability. Often a

von Neumann stability analysis In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations. The analysis is based on ...

is substituted for convenience, although von Neumann stability only implies Lax–Richtmyer stability in certain cases. This theorem is due to

Peter Lax Peter David Lax (1 May 1926 – 16 May 2025) was a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax made important contributions to integrable systems, fluid dynamics an ...

. It is sometimes called the Lax–Richtmyer theorem, after Peter Lax and Robert D. Richtmyer.

References

Numerical differential equations Theorems in mathematical analysis {{mathapplied-stub