In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of finite difference methods for the numerical solution of partial differential equations. 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.[1]
The importance of the theorem is that while the 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 approximates 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.[2] 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.[3]
^Strikwerda, John C. (1989). Finite Difference Schemes and Partial Differential Equations (1st ed.). Chapman & Hall. pp. 26, 222. ISBN 0-534-09984-X.
^
Smith, G. D. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods (3rd ed.). Oxford University Press. pp. 67–68. ISBN 0-19-859641-3.
^Lax, P. D.; Richtmyer, R. D. (1956). "Survey of the Stability of Linear Finite Difference Equations". Comm. Pure Appl. Math.9 (2): 267–293. doi:10.1002/cpa.3160090206. MR 0079204.
and 24 Related for: Lax equivalence theorem information
In numerical analysis, the Laxequivalencetheorem is a fundamental theorem in the analysis of finite difference methods for the numerical solution of...
Compass equivalencetheorem, a theorem in straightedge and compass construction Equivalence principle, in general relativity Laxequivalencetheorem, a theorem...
Babuška–Lax–Milgram theorem Lions–Lax–Milgram theorem The Martians (scientists) Lewis, Adrian S.; Parrilo, Pablo A.; Ramana, Motakuri V. (2005). "The Lax conjecture...
necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the Laxequivalencetheorem): The PDE and the finite difference scheme models...
fixed time remains bounded as the step size goes to zero. The Laxequivalencetheorem states that an algorithm converges if it is consistent and stable...
as the Dahlquist equivalencetheorem, named after Germund Dahlquist; this theorem is similar in spirit to the Laxequivalencetheorem for finite difference...
vertices, line segments, triangles, tetrahedra, ..., making up a mesh Laxequivalencetheorem — a consistent method is convergent if and only if it is stable...
Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered...
ensures unique solvability for a large class of problems is the Lax–Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite...
possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone...
Wuytack 2001, p. 216. Gustafsson 2018, pp. 112–113. Lax, Peter D. (2005). "Interview with Peter D. Lax" (PDF) (Interview). Interviewed by Martin Raussen;...
the index set. While we can make the requirements on the ordering quite lax in order to allow for great generality, the most natural and common prerequisite...
The measure class of μ in the sense of the Radon–Nikodym theorem is unique. Davies 1989 Lax & Phillips 1976 Helgason 1984, p. 38 Flensted-Jensen 1978...
D(\mathbf {P} ^{\vee ,d}).} The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves...
uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of existence and uniqueness theorems for ODE...
c\|u\|^{2}} for some constant c > 0. {\displaystyle c>0.} By the Lax-Milgram theorem (see weak formulation), these two conditions imply well-posedness...
arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems. Like any system implementing the Peano axioms, TNT is capable of referring...
Fano varieties Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory Nagata's conjecture...
applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields...
Poincaré recurrence theorem to imply that the action of the pentagram map on X is recurrent: The orbit of almost any equivalence class of convex polygon...
2-monad). Q Q-category. Quillen Quillen’s theorem A provides a criterion for a functor to be a weak equivalence. reflect 1. A functor is said to reflect...
the role of mathematics in human history, beginning with the Pythagorean theorem (Pythagorean equation) to the equation that transformed the twenty-first...
Global Information