Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

[1]  Eitan Tadmor,et al.  Non-Oscillatory Central Schemes for the Incompressible 2-D Euler Equations , 1997 .

[2]  Gabriella Puppo,et al.  High-Order Central Schemes for Hyperbolic Systems of Conservation Laws , 1999, SIAM J. Sci. Comput..

[3]  P. Arminjon,et al.  Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace , 1995 .

[4]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[5]  G. Russo,et al.  NUMERICAL SOLUTION FOR HYDRODYNAMICAL MODELS OF SEMICONDUCTORS , 2000 .

[6]  Vittorio Romano,et al.  Extended Hydrodynamical Model of Carrier Transport in Semiconductors , 2000, SIAM J. Appl. Math..

[7]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

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

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

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

[11]  Sebastian Noelle,et al.  A Comparison of Third and Second Order Accurate Finite Volume Schemes for the Two-dimensional Compressible Euler Equations , 1999 .

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

[13]  Raz Kupferman,et al.  Simulation of Viscoelastic Fluids , 1998 .

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

[15]  Raz Kupferman,et al.  A Numerical Study of the Axisymmetric Couette-Taylor Problem Using a Fast High-Resolution Second-Order Central Scheme , 1998, SIAM J. Sci. Comput..

[16]  Stanley Osher,et al.  Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I , 1996 .

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

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

[19]  Gabriella Puppo,et al.  A third order central WENO scheme for 2D conservation laws , 2000 .

[20]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[21]  S. Osher,et al.  On the convergence of difference approximations to scalar conservation laws , 1988 .

[22]  Doron Levy,et al.  A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations , 2000, SIAM J. Sci. Comput..

[23]  B. Engquist,et al.  Multi-phase computations in geometrical optics , 1996 .

[24]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[25]  Chi-Wang Shu.  Numerical experiments on the accuracy of ENO and modified ENO schemes , 1990 .

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

[27]  M. Falcone,et al.  Numerical schemes for conservation laws via Hamilton-Jacobi equations , 1995 .

[28]  E. Tadmor,et al.  Third order nonoscillatory central scheme for hyperbolic conservation laws , 1998 .

[29]  Eitan Tadmor,et al.  New High-Resolution Semi-discrete Central Schemes for Hamilton—Jacobi Equations , 2000 .

[30]  Knut-Andreas Lie,et al.  Remarks On High-Resolution Non-Oscillatory Central Schemes For Multi-Dimensional Systems Of Conserva , 2000 .

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

[32]  Paul Arminjon,et al.  A Finite Volume Extension of the Lax-Friedrichs and Nessyahu-Tadmor Schemes for Conservation Laws on Unstructured Grids , 1998 .

[33]  G. Russo,et al.  Central WENO schemes for hyperbolic systems of conservation laws , 1999 .

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

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

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

[37]  Paul Arminjon,et al.  Convergence of a Finite Volume Extension of the Nessyahu--Tadmor Scheme on Unstructured Grids for a Two-Dimensional Linear Hyperbolic Equation , 1999 .

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

[39]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[40]  E. Tadmor,et al.  A fast, high resolution, second-order central scheme for incompressible flows. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

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

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

[43]  Kolumban Hutter,et al.  Shock-capturing and front-tracking methods for granular avalanches , 2015, 1501.04756.

[44]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[45]  Eitan Tadmor,et al.  Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws , 1998, SIAM J. Sci. Comput..

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

[47]  B. Wendroff,et al.  Approximate Riemann Solvers, Godunov Schemes and Contact Discontinuities , 2001 .

[48]  IXu-Dong Liu,et al.  Nonoscillatory High Order Accurate Self-similar Maximum Principle Satisfying Shock Capturing Schemes I , 1996 .

[49]  Alexander Kurganov,et al.  A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems , 2001, Numerische Mathematik.

[50]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .