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Finite difference schemes for one-dimensional systems

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Numerical Approximation of Hyperbolic Systems of Conservation Laws

Part of the book series: Applied Mathematical Sciences ((AMS,volume 118))

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Abstract

Let us consider again the Cauchy problem for a general system of conservation laws

$$ \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}f\left( u \right) = 0,\quad x \in \mathbb{R},t > 0,} \\ {u\left( {x,0} \right) = {u_0}\left( x \right),} \end{array}} \right. $$
((1.1))

, where u = (u 1,..., u p )T is a p-vector. As usual, we assume that this system is hyperbolic, i.e., the Jacobian matrix A(u) = f′(u) of f(u) has p real eigenvalues ranked in increasing order,

, and a complete set of eigenvectors. Moreover, we shall assume that for all 1 ≤ k ≤ p, the kth characteristic field is either genuinely nonlinear or linearly degenerate. For the notations concerning the difference schemes, we refer to Chapter III of G.R..

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Authors and Affiliations

  1. Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Edwige Godlewski

  2. Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Pierre-Arnaud Raviart

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  1. Edwige Godlewski
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  2. Pierre-Arnaud Raviart
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Godlewski, E., Raviart, PA. (1996). Finite difference schemes for one-dimensional systems. In: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol 118. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0713-9_4

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  • DOI: https://doi.org/10.1007/978-1-4612-0713-9_4

  • Publisher Name: Springer, New York, NY

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