On maximum-principle-satisfying high order schemes for scalar conservation laws

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)...

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Bibliographic Details
Published in:Journal of computational physics 2010-05, Vol.229 (9), p.3091-3120
Main Authors: Zhang, Xiangxiong, Shu, Chi-Wang
Format: Article
Language:English
Subjects:
Computational techniques
Conservation laws
Construction
Discontinuous Galerkin method
Discretization
Essentially non-oscillatory scheme
Exact sciences and technology
Finite volume scheme
High order accuracy
Hyperbolic conservation laws
Incompressible flow
Mathematical analysis
Mathematical methods in physics
Mathematical models
Maximum principle
Passive convection equation
Physics
Preserves
Scalars
Strong stability preserving time discretization
Weighted essentially non-oscillatory scheme
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Summary:We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2009.12.030