Lax equivalence theorem
In the numerical analysis of the equivalence theorem of Lax is the fundamental theorem in the analysis of finite difference method for the numerical solution of partial differential equations. The claim is that a consistent method is convergent if and only for a properly excepted linear initial value problem, if it is stable.
The sentence means that although the desired convergence of the solution of the finite difference method for the solution of the partial differential equation is very difficult to detect because the numerical solution is defined recursively. However, the consistency of the method, i.e., that the numerical method approximates the differential equation, easy to check and stability is usually much easier to show than the convergence (this would be evidence in any case, to show that rounding errors the solution does not distort ). Therefore, convergence is usually shown on the equivalence set.
Stability in this context means that a matrix norm of the matrix which is used in the iteration, a maximum of unity. This is (practical ) Lax- Richtmyer called stability. Often, a Von Neumann - stability analysis is carried out, although a Von Neumann stability implies Lax Richtmyer stability in certain cases, instead.
The set is named after Peter Lax. Sometimes it is also referred to by Peter Lax and Richtmyer as Robert Lax- Richtmyer theorem.