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Fully adaptive multiresolution finite volume schemes for conservation laws. (English) Zbl 1010.65035
This paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes for the numerical solution of the Cauchy problem for hyperbolic systems of conservation laws of the form \( \partial_t u+\text{Div}_x(f(u(t,x)))=0\), \(u\in\mathbb R^m\), \(x\in\mathbb R^d\), \(t>0\), with initial value \(u(t=0,x)=u_0(x)\). The authors present an alternative strategy for adaptive discretizations, based on coupling of multiresolution representation and finite volume schemes, which allows a relatively simple implementation and a rigorous error analysis.
The solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions of this problem are proposed, analyzed, and compared in terms of accuracy and complexity. Numerical tests for 1D and 2D problems are presented together with a discussion concerning the practical relevance of the refinement strategy and error estimate.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65T60 Numerical methods for wavelets
65M15 Error bounds 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
Full Text: DOI
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