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Liu, Yuan; Cheng, Yingda; Shu, Chi-Wang

A simple bound-preserving sweeping technique for conservative numerical approximations. (English) Zbl 1381.65087

J. Sci. Comput. 73, No. 2-3, 1028-1071 (2017).
Summary: In this paper, we propose a simple bound-preserving sweeping procedure for conservative numerical approximations. Conservative schemes are of importance in many applications, yet for high order methods, the numerical solutions do not necessarily satisfy maximum principle. This paper constructs a simple sweeping algorithm to enforce the bound of the solutions. It has a very general framework acting as a postprocessing step accommodating many point-based or cell average-based discretizations. The method is proven to preserve the bounds of the numerical solution while conserving a prescribed quantity designated as a weighted average of values from all points. The technique is demonstrated to work well with a spectral method, high order finite difference and finite volume methods for scalar conservation laws and incompressible flows. Extensive numerical tests in 1D and 2D are provided to verify the accuracy of the sweeping procedure.

Cited in 3 Documents

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics

Keywords:

bound-preserving; conservative schemes; Fourier spectral methods; high-order finite difference and finite volume methods; numerical example; bound-preserving sweeping; algorithm; incompressible flows
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