Lax equivalence theorem
E87776
The Lax equivalence theorem is a fundamental result in numerical analysis stating that for a well-posed linear initial value problem, consistency and stability of a finite difference scheme together imply its convergence.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Lax–Richtmyer equivalence theorem | 1 |
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in numerical analysis ⓘ |
| alsoKnownAs |
Lax equivalence theorem
ⓘ
surface form:
Lax–Richtmyer equivalence theorem
|
| appliesTo |
discretizations on nonuniform grids
ⓘ
discretizations on uniform grids ⓘ well-posed linear initial value problems ⓘ |
| assumes |
finite difference discretization
ⓘ
linear problem ⓘ well-posed problem ⓘ |
| clarifies | relationship between stability and convergence ⓘ |
| concerns |
consistency of numerical schemes
ⓘ
convergence of numerical schemes ⓘ finite difference methods ⓘ linear initial value problems ⓘ stability of numerical schemes ⓘ |
| domain |
linear partial differential equations
ⓘ
time-dependent problems ⓘ |
| field |
numerical analysis
ⓘ
partial differential equations ⓘ |
| formalizes | equivalence of stability plus consistency with convergence for linear well-posed problems ⓘ |
| hasConsequence |
for linear well-posed problems, convergence analysis can focus on stability and consistency
ⓘ
stability is necessary for convergence of consistent schemes ⓘ |
| historicalContext | 20th-century numerical analysis ⓘ |
| implies | convergence of a finite difference scheme under stability and consistency ⓘ |
| mathematicalNature | equivalence theorem ⓘ |
| namedAfter | Peter Lax ⓘ |
| provenBy | Peter Lax ⓘ |
| relatedTo |
Cauchy problem
ⓘ
Lax stability condition ⓘ finite difference scheme ⓘ von Neumann stability analysis ⓘ |
| relatesConcept |
consistency
ⓘ
convergence ⓘ stability ⓘ well-posedness ⓘ |
| states | for a well-posed linear initial value problem, stability and consistency of a finite difference scheme imply convergence ⓘ |
| typicalApplication |
numerical solution of hyperbolic PDEs
ⓘ
numerical solution of parabolic PDEs ⓘ |
| usedIn |
analysis of numerical schemes for PDEs
ⓘ
design of stable numerical methods ⓘ verification of convergence of discretization schemes ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lax–Richtmyer equivalence theorem