Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients

We introduce a central-upwind scheme for one- and two-dimensional systems of shallow-water equations with horizontal temperature gradients (the Ripa system). The scheme is well-balanced, positivity preserving and does not develop spurious pressure oscillations in the neighborhood of temperature jumps, that is, near the contact waves. Such oscillations would typically appear when a conventional Godunov-type finite volume method is applied to the Ripa system, and the nature of the oscillation is similar to the ones appearing at material interfaces in compressible multifluid computations. The idea behind the proposed approach is to utilize the interface tracking method, originally developed in Chertock et al. (M2AN Math Model Numer Anal 42:991–1019, 2008) for compressible multifluids. The resulting scheme is highly accurate, preserves two types of “lake at rest” steady states, and is oscillation free across the temperature jumps, as it is illustrated in a number of numerical experiments.

[1]  Paul J. Dellar,et al.  Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields , 2003 .

[2]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

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

[4]  Giovanni Russo,et al.  Central schemes for conservation laws with application to shallow water equations , 2005 .

[5]  S. F. Davis,et al.  An interface tracking method for hyperbolic systems of conservation laws , 1992 .

[6]  D. Kröner.  Numerical Schemes for Conservation Laws , 1997 .

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

[8]  Giovanni Russo,et al.  Central Schemes for Balance Laws , 2002 .

[9]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[10]  S. Osher,et al.  Computing interface motion in compressible gas dynamics , 1992 .

[11]  Shi Jin,et al.  Two Interface-Type Numerical Methods for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations , 2005, SIAM J. Sci. Comput..

[12]  LiuYu,et al.  Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients , 2014 .

[13]  B. Perthame,et al.  A kinetic scheme for the Saint-Venant system¶with a source term , 2001 .

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

[15]  P. Ripa.  On improving a one-layer ocean model with thermodynamics , 1995, Journal of Fluid Mechanics.

[16]  Rémi Abgrall,et al.  Computations of compressible multifluids , 2001 .

[17]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[18]  P. Ripa.  Conservation laws for primitive equations models with inhomogeneous layers , 1993 .

[19]  Yulong Xing,et al.  High-order well-balanced finite volume WENO schemes for shallow water equation with moving water , 2007, J. Comput. Phys..

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

[21]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[22]  Mária Lukácová-Medvid'ová,et al.  Well-balanced finite volume evolution Galerkin methods for the shallow water equations , 2015, J. Comput. Phys..

[23]  Eitan Tadmor,et al.  Solution of two‐dimensional Riemann problems for gas dynamics without Riemann problem solvers , 2002 .

[24]  Alexander Kurganov,et al.  Interface tracking method for compressible multifluids , 2008 .

[25]  Jostein R. Natvig,et al.  Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows , 2006, J. Comput. Phys..

[26]  Yulong Xing,et al.  High order finite difference WENO schemes with the exact conservation property for the shallow water equations , 2005 .

[27]  T. Gallouët,et al.  Some approximate Godunov schemes to compute shallow-water equations with topography , 2003 .

[28]  Ning Zhao,et al.  Conservative front tracking and level set algorithms , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Shi Jin,et al.  A steady-state capturing method for hyperbolic systems with geometrical source terms , 2001 .

[30]  A. Kurganov,et al.  On the Reduction of Numerical Dissipation in Central-Upwind Schemes , 2006 .

[31]  Yulong Xing,et al.  High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms , 2006, J. Comput. Phys..

[32]  Eitan Tadmor,et al.  Strong Stability-Preserving High-Order Time Discretization , 2001 .

[33]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[34]  Doron Levy,et al.  CENTRAL-UPWIND SCHEMES FOR THE SAINT-VENANT SYSTEM , 2002 .

[35]  G. Petrova,et al.  A SECOND-ORDER WELL-BALANCED POSITIVITY PRESERVING CENTRAL-UPWIND SCHEME FOR THE SAINT-VENANT SYSTEM ∗ , 2007 .

[36]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

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