A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations

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.

[1]  Xiangxiong Zhang,et al.  Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  Zhengfu Xu,et al.  High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes , 2014 .

[3]  Yuanyuan Liu,et al.  Maximum-principle-satisfying High Order Finite Volume Weighted Essentially Nonoscillatory Schemes for Convection-diffusion Equations , 2012, SIAM J. Sci. Comput..

[4]  Zhengfu Xu,et al.  A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows , 2013, J. Comput. Phys..

[5]  Xiangxiong Zhang,et al.  Positivity-preserving high order finite difference WENO schemes for compressible Euler equations , 2012, J. Comput. Phys..

[6]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[7]  Xiangxiong Zhang,et al.  Positivity-Preserving High Order Finite Volume HWENO Schemes for Compressible Euler Equations , 2015, Journal of Scientific Computing.

[8]  Yifan Zhang,et al.  Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes , 2013, J. Comput. Phys..

[9]  Zhengfu Xu,et al.  High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation , 2013, J. Comput. Phys..

[10]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[11]  Zhengfu Xu,et al.  High Order Maximum Principle Preserving Finite Volume Method for Convection Dominated Problems , 2016, J. Sci. Comput..

[12]  Zhengfu Xu,et al.  Positivity-Preserving Finite Difference Weighted ENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations , 2015, SIAM J. Sci. Comput..

[13]  L. Pareschi,et al.  On the stability of spectral methods for the homogeneous Boltzmann equation , 2000 .

[14]  Xiangxiong Zhang,et al.  Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms , 2011, J. Comput. Phys..

[15]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[16]  Zhengfu Xu,et al.  High Order Maximum-Principle-Preserving Discontinuous Galerkin Method for Convection-Diffusion Equations , 2014, SIAM J. Sci. Comput..

[17]  Yulong Xing,et al.  Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations , 2010 .

[18]  Chao Liang,et al.  Parametrized Maximum Principle Preserving Flux Limiters for High Order Schemes Solving Multi-Dimensional Scalar Hyperbolic Conservation Laws , 2014, J. Sci. Comput..

[19]  Yulong Xing,et al.  Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes , 2013, J. Sci. Comput..

[20]  E. Hewitt,et al.  The Gibbs-Wilbraham phenomenon: An episode in fourier analysis , 1979 .

[21]  Nikolaus A. Adams,et al.  Positivity-preserving method for high-order conservative schemes solving compressible Euler equations , 2013, J. Comput. Phys..

[22]  Xiangxiong Zhang,et al.  On maximum-principle-satisfying high order schemes for scalar conservation laws , 2010, J. Comput. Phys..

[23]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[24]  Xiangxiong Zhang,et al.  Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes , 2011, Journal of Scientific Computing.

[25]  Zhengfu Xu.  Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem , 2014, Math. Comput..

[26]  Zhengfu Xu,et al.  Parametrized Positivity Preserving Flux Limiters for the High Order Finite Difference WENO Scheme Solving Compressible Euler Equations , 2014, J. Sci. Comput..