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Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws. (English) Zbl 0805.65088

The author constructs a local third-order accurate shock capturing method for hyperbolic scalar conservation laws, based on numerical fluxes with a total variation diminishing (TVD) Runge-Kutta evolution in time, using the idea introduced by C. W. Shu and S. J. Osher [J. Comput. Phys. 77, No. 2, 439-471 (1988; Zbl 0653.65072) and ibid. 83, No. 1, 32- 78 (1989; Zbl 0674.65061)] for essentially nonoscillatory methods. The method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell. It is third-order accurate in smooth regions of the solution, except at local extrema where it may become \(O(h^{3/2})\), thus giving better accuracy than TVD methods at local extrema. The presented method is efficient since it is low cost and is not sensitive to the Courant-Friedrichs-Lewy number.

MSC:

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
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