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Total variation diminishing Runge-Kutta schemes. (English) Zbl 0897.65058
Suppose that a hyperbolic conservation law \(u_t = - f(u)_x\) is discretized in space by a certain total variation diminishing (TVD) finite difference or finite element approximation, and that the resulting ordinary differential equation is solved by an explicit Runge-Kutta method. This article investigates methods which, under a suitable restriction on the stepsize, the total variation does not increase in time. First, a numerical example is presented which illustrates the importance of this property. Then, the authors construct optimal TVD Runge-Kutta methods of second, third and fourth order. Details are very technical and postponed to an appendix.
Reviewer: E.Hairer (Genève)

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
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