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Convergence of Runge-Kutta methods for nonlinear parabolic equations. (English) Zbl 1004.65093

The aim of the authors is to study fully nonlinear parabolic initial-boundary value problems. As these problems can be cast in the form of an abstract initial value problem, the authors derive existence and convergence results for Runge-Kutta time discretizations for this last problem. The main idea is to linearize the problem along the exact solution, because Runge-Kutta methods are invariant under this linearization. Among others, an interesting example from detonation theory is studied by means of the suggested approach.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations

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References:

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