Directly solving the Hamilton-Jacobi equations by Hermite WENO Schemes

In this paper, we present a class of new Hermite weighted essentially non-oscillatory (HWENO) schemes based on finite volume framework to directly solve the Hamilton-Jacobi (HJ) equations. For HWENO reconstruction, both the cell average and the first moment of the solution are evolved, and for two dimensional case, HWENO reconstruction is based on a dimension-by-dimension strategy which is the first used in HWENO reconstruction. For spatial discretization, one of key points for directly solving HJ equation is the reconstruction of numerical fluxes. We follow the idea put forward by Cheng and Wang (2014) 3 to reconstruct the values of solution at Gauss-Lobatto quadrature points and numerical fluxes at the interfaces of cells, and for neither the convex nor concave Hamiltonian case, the monotone modification of numerical fluxes is added, which can guarantee the precision in the smooth region and converge to the entropy solution when derivative discontinuities come up. The third order TVD Runge-Kutta method is used for the time discretization. Extensive numerical experiments in one dimensional and two dimensional cases are performed to verify the efficiency of the methods.

[1]  Chi-Wang Shu,et al.  A technique of treating negative weights in WENO schemes , 2000 .

[2]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[3]  Chi-Wang Shu,et al.  High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes , 2003, SIAM J. Sci. Comput..

[4]  Stanley Osher,et al.  A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations , 2011, J. Comput. Phys..

[5]  R. Abgrall.  Numerical discretization of the first‐order Hamilton‐Jacobi equation on triangular meshes , 1996 .

[6]  P. Lions,et al.  Two approximations of solutions of Hamilton-Jacobi equations , 1984 .

[7]  Wang Hai-bing,et al.  High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations , 2006 .

[8]  Chi-Wang Shu,et al.  Hermite WENO schemes for Hamilton-Jacobi equations , 2005 .

[9]  Chi-Wang,et al.  ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT , 2006 .

[10]  B. A. Bernstein.  Postulates for abelian groups and fields in terms of non-associative operations , 1938 .

[11]  Chi-Wang Shu.  HIGH ORDER NUMERICAL METHODS FOR TIME DEPENDENT HAMILTON-JACOBI EQUATIONS , 2007 .

[12]  S. Osher,et al.  High Order Two Dimensional Nonoscillatory Methods for Solving Hamilton-Jacobi Scalar Equations , 1996 .

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

[14]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[15]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[16]  Chi-Wang Shu,et al.  A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations , 2007, Journal of Computational Physics.

[17]  Chi-Wang Shu,et al.  Convergence of High Order Finite Volume Weighted Essentially Nonoscillatory Scheme and Discontinuous Galerkin Method for Nonconvex Conservation Laws , 2008, SIAM J. Sci. Comput..

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

[19]  J. Qiu.  WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations , 2007 .

[20]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case , 2004 .

[21]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[22]  Chi-Wang Shu,et al.  A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[23]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[24]  Chi-Wang Shu,et al.  Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations , 2005, Appl. Math. Lett..

[25]  Yingda Cheng,et al.  A new discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations , 2013, J. Comput. Phys..

[26]  J. Qiu.  HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS , 2007 .