High-order well-balanced finite volume WENO schemes for shallow water equation with moving water

A characteristic feature of hyperbolic systems of balance laws is the existence of non-trivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of so-called well-balanced schemes were developed which satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. In most cases, applications treated equilibria at rest, where the flow velocity vanishes. Here we present a new very high-order accurate, exactly well-balanced finite volume scheme for moving flow equilibria. Numerical experiments show excellent resolution of unperturbed as well as slightly perturbed equilibria.

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

[2]  L. Gosse A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms☆ , 2000 .

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

[4]  Xin Wen,et al.  A steady state capturing and preserving method for computing hyperbolic systems with geometrical source terms having concentrations , 2006, J. Comput. Phys..

[5]  Shi Jin,et al.  A STEADY-STATE CAPTURING METHOD FOR HYPERBOLIC SYSTEMS WITH GEOMETRICAL SOURCE , 2022 .

[6]  Rupert Klein,et al.  Well balanced finite volume methods for nearly hydrostatic flows , 2004 .

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

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

[9]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[10]  Centro internazionale matematico estivo. Session,et al.  Advanced Numerical Approximation of Nonlinear Hyperbolic Equations , 1998 .

[11]  Shi Jin,et al.  AN EFFICIENT METHOD FOR COMPUTING HYPERBOLIC SYSTEMS WITH GEOMETRICAL SOURCE TERMS HAVING CONCENTRATIONS ∗1) , 2004 .

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

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

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

[15]  Luka Sopta,et al.  ENO and WENO Schemes with the Exact Conservation Property for One-Dimensional Shallow Water Equations , 2002 .

[16]  V. T. Chow Open-channel hydraulics , 1959 .

[17]  Randall J. LeVeque,et al.  A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions , 2002, SIAM J. Sci. Comput..

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

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

[20]  Shi Jin,et al.  An efficient method for computing hyperbolic systems with geometrical source terms having concentrations ∗ , 2004 .

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

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

[23]  Chi-Wang Shu Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[24]  Chi-Wang Shu TVB uniformly high-order schemes for conservation laws , 1987 .

[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]  A. Leroux,et al.  A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon , 2004 .

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

[28]  Yulong Xing,et al.  High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms , 2006, J. Sci. Comput..

[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]  M. Vázquez-Cendón Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

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

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

[33]  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..

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

[35]  Yulong Xing,et al.  A New Approach of High OrderWell-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms† , 2005 .