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Fully discrete, entropy conservative schemes of arbitrary order. (English) Zbl 1033.65073
The authors consider the numerical approximation of discontinuous solutions of general systems of conservation laws of the form $$\partial_t u+\partial_x f(u)=0$$, $$u(x,t)\in\mathbb R^N$$, $$x\in\mathbb R$$, $$t>0$$, endowed with a smooth entropy-entropy flux pair $$(U,F):\mathbb R^N \rightarrow \mathbb R^2$$ and the flux-function $$f:\mathbb R^N\rightarrow\mathbb R^N$$ is a given smooth mapping. Solutions are sought which satisfy the entropy inequality $$\partial_t U(u)+\partial_x F(u)\leq 0$$ in the sense of distribution.
The first aim is to construct finite difference schemes that are fully discrete in space and time, conservative in the sense of Lax and Wendroff, entropy conservative in the sense of Tadmor, and at least third-order accurate. Semidiscrete entropy conservative schemes of arbitrary high order are investigated.
Numerical experiments underline the good performance of such schemes with appropriate dissipative terms. These techniques also apply to other types of evolution equations for which an energy conservation or dissipation is available.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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