High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations

Abstract A new fifth-order, semi-discrete central-upwind scheme with a Lax–Wendroff time discretization procedure for solving Hamilton–Jacobi (HJ) equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. Unlike most of the commonly used high-order upwind schemes, the new scheme is formulated as a Godunov-type method. The new scheme is based on the flux Kurganov, Noelle and Petrova (KNP flux). The spatial discretization is based on a symmetrical weighted essentially non-oscillatory reconstruction of the derivative. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. Various numerical experiments are performed to demonstrate the accuracy and stability properties of the new method. As a result, comparing with other fifth-order schemes for HJ equations, the major advantage of the new scheme is more cost effective for certain problems while the new method exhibits smaller errors without any increase in the complexity of the computations.

[1]  E. Tadmor,et al.  New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations , 2000 .

[2]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

[3]  Chi-Tien Lin,et al.  $L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions , 2001, Numerische Mathematik.

[4]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[5]  P. Souganidis.  Approximation schemes for viscosity solutions of Hamilton-Jacobi equations , 1985 .

[6]  S. Bryson,et al.  High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations , 2013 .

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

[8]  Michael Dumbser,et al.  The discontinuous Galerkin method with Lax-Wendroff type time discretizations , 2005 .

[9]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

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

[11]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[12]  Willem Hundsdorfer,et al.  High-order linear multistep methods with general monotonicity and boundedness properties , 2005 .

[13]  Alexander Kurganov,et al.  Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations , 2001, SIAM J. Sci. Comput..

[14]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

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

[16]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[17]  Steve Bryson,et al.  Semi-discrete central-upwind schemes with reduced dissipation for Hamilton–Jacobi equations , 2005 .

[18]  Alexander Kurganov,et al.  On Splitting-Based Numerical Methods for Convection-Diffusion Equations , 2008 .

[19]  S. Bryson,et al.  High-order central WENO schemes for 1D Hamilton-Jacobi equations , 2003 .

[20]  P. Souganidis,et al.  Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations , 1995 .

[21]  S. SIAMJ.,et al.  CENTRAL SCHEMES FOR MULTIDIMENSIONAL HAMILTON – JACOBI EQUATIONS , 2003 .

[22]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[23]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

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

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

[26]  Claus-Dieter Munz,et al.  ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D , 2002, J. Sci. Comput..

[27]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[28]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[29]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[30]  Steve Bryson,et al.  High-Order Central WENO Schemes for Multidimensional Hamilton-Jacobi Equations , 2013, SIAM J. Numer. Anal..

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

[32]  P. Lax,et al.  Systems of conservation laws , 1960 .

[33]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Chi-Tien Lin,et al.  High-Resolution Nonoscillatory Central Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

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

[36]  Chi-Wang Shu,et al.  Finite Difference WENO Schemes with Lax-Wendroff-Type Time Discretizations , 2002, SIAM J. Sci. Comput..

[37]  M. Minion,et al.  Performance of Under-resolved Two-Dimensional Incompressible Flow , 1995 .

[38]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[39]  Gabriella Puppo,et al.  Compact Central WENO Schemes for Multidimensional Conservation Laws , 1999, SIAM J. Sci. Comput..

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

[41]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[42]  Guy Capdeville,et al.  A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes , 2008, J. Comput. Phys..

[43]  Ashvin Gopaul,et al.  A weighted ENO-flux limiter scheme for hyperbolic conservation laws , 2010, Int. J. Comput. Math..