A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, called PVM (polynomial viscosity matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of a Roe matrix is needed. As a consequence, they are faster than the Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [C. R. Acad. Sci. Paris Ser. l Math., 328 (1999), pp. 479--483] for balance laws and nonconservative systems. The first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems. In this work, some well known solvers, such as Rusanov, Lax--Friedrichs, FORCE (see [E. F. Toro and S. J. Billett, IMA J. Num...

[1]  Carlos Parés,et al.  On the well-balance property of Roe?s method for nonconservative hyperbolic systems , 2004 .

[2]  Marica Pelanti,et al.  A Roe-type scheme for two-phase shallow granular flows over variable topography , 2008 .

[3]  Manuel Jesús Castro Díaz,et al.  High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems , 2009, J. Sci. Comput..

[4]  Randall J. LeVeque,et al.  Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods , 1998 .

[5]  I. Toumi.  A weak formulation of roe's approximate riemann solver , 1992 .

[6]  Eleuterio F. Toro,et al.  On some fast well-balanced first order solvers for nonconservative systems , 2009, Math. Comput..

[7]  Rémi Abgrall,et al.  A comment on the computation of non-conservative products , 2010, J. Comput. Phys..

[8]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[9]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[10]  T. Hou,et al.  Why nonconservative schemes converge to wrong solutions: error analysis , 1994 .

[11]  M. J. Castro,et al.  FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems , 2010 .

[12]  Pierre Degond,et al.  Polynomial upwind schemes for hyperbolic systems , 1999 .

[13]  T. Morales de Luna,et al.  On a shallow water model for the simulation of turbidity currents , 2009 .

[14]  Long Le,et al.  A two-fluid model for avalanche and debris flows , 2005, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  Eleuterio F. Toro,et al.  MUSTA fluxes for systems of conservation laws , 2006, J. Comput. Phys..

[16]  Carlos Parés,et al.  Godunov method for nonconservative hyperbolic systems , 2007 .

[17]  V. Guinot.  Approximate Riemann Solvers , 2010 .

[18]  R. Courant,et al.  On the solution of nonlinear hyperbolic differential equations by finite differences , 1952 .

[19]  李幼升,et al.  Ph , 1989 .

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

[21]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

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

[23]  Eleuterio F. Toro,et al.  Centred TVD schemes for hyperbolic conservation laws , 2000 .

[24]  Carlos Parés Madroñal,et al.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework , 2006, SIAM J. Numer. Anal..

[25]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[26]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[27]  Carlos Parés,et al.  A Q-SCHEME FOR A CLASS OF SYSTEMS OF COUPLED CONSERVATION LAWS WITH SOURCE TERM. APPLICATION TO A TWO-LAYER 1-D SHALLOW WATER SYSTEM , 2001 .

[28]  M. J. Castro,et al.  ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows , 2009 .

[29]  Manuel Jesús Castro Díaz,et al.  High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems , 2006, Math. Comput..

[30]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .

[31]  Carlos Parés Madroñal,et al.  On some difficulties of the numerical approximation of nonconservative hyperbolic systems , 2009 .

[32]  Carlos Parés Madroñal,et al.  On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws , 2011, J. Sci. Comput..

[33]  Manuel Jesús Castro Díaz,et al.  Available Online at Www.sciencedirect.com Mathematical and So,snos ~__d,~ot" Computer Modelling the Numerical Treatment of Wet/dry Fronts in Shallow Flows: Application to One-layer and Two-layer Systems , 2022 .

[34]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..