Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound. (English) Zbl 1262.65126
Summary: We consider 3-point numerical schemes, that resolve scalar conservation laws, that are oscillatory either to their dispersive or anti-diffusive nature. The spatial discretization is performed over nonuniform adaptively redefined meshes. We provide a model for studying the evolution of the extremes of the oscillations. We prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the total variation (TV) of the numerical solution. We, moreover, prove under more strict assumptions that the increase of the TV, due to the oscillatory behavior of the numerical schemes, decreases with time; hence proving that the overall scheme is TV increase-decreasing. We finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique.
MSC:
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65M08 | Finite volume 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 |
| 35L05 | Wave equation |
Keywords:
adaptive mesh reconstruction; hyperbolic conservation laws; total variation bound; oscillation; stabilization; numerical examples; finite volume methodReferences:
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