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TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework

Authors: Bernardo Cockburn and Chi-Wang Shu
Journal: Math. Comp. 52 (1989), 411-435
MSC: Primary 65M60; Secondary 35L65, 65N30
MathSciNet review: 983311
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Abstract: This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws $ {u_t} + \sum\nolimits_{i = 1}^d {{{({f_i}(u))}_{{x_i}}} = 0} $. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

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Keywords: TVD, TVB, Runge-Kutta, discontinuous finite elements, conservation law
Article copyright: © Copyright 1989 American Mathematical Society