A CIP/multi-moment finite volume method for shallow water equations with source terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume-integrated average (VIA) for each mesh cell, the surface-integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi-Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux-based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non-oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright (c) 2007 John Wiley & Sons, Ltd.

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

[2]  Feng Xiao,et al.  A 4th-order and single-cell-based advection scheme on unstructured grids using multi-moments , 2005, Comput. Phys. Commun..

[3]  Takashi Yabe,et al.  Constructing exactly conservative scheme in a non-conservative form , 2000 .

[4]  Feng Xiao,et al.  Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow , 2004 .

[5]  M. Iskandarani,et al.  A spectral finite-volume method for the shallow water equations , 2004 .

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

[7]  Pilar García-Navarro,et al.  Efficient construction of high‐resolution TVD conservative schemes for equations with source terms: application to shallow water flows , 2001 .

[8]  Feng Xiao,et al.  Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows , 2006, J. Comput. Phys..

[9]  E. Toro.  Shock-Capturing Methods for Free-Surface Shallow Flows , 2001 .

[10]  T. Yabe,et al.  An Exactly Conservative Semi-Lagrangian Scheme (CIP–CSL) in One Dimension , 2001 .

[11]  Feng Xiao,et al.  An efficient method for capturing free boundaries in multi‐fluid simulations , 2003 .

[12]  Yongqi Wang,et al.  Comparisons of numerical methods with respect to convectively dominated problems , 2001 .

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

[14]  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..

[15]  Feng Xiao,et al.  CIP/multi-moment finite volume method for Euler equations: A semi-Lagrangian characteristic formulation , 2007, J. Comput. Phys..

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

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

[18]  Feng Xiao,et al.  Profile-modifiable Conservative Transport Schemes and a Simple Multi-integrated Moment Formulation for Hydrodynamics , 2003 .

[19]  D. Causon,et al.  The surface gradient method for the treatment of source terms in the shallow-water equations , 2001 .

[20]  Clinton N Dawson,et al.  A discontinuous Galerkin method for two-dimensional flow and transport in shallow water , 2002 .

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

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

[23]  K. S. Erduran,et al.  Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes , 2002 .

[24]  Jihn-Sung Lai,et al.  Finite-volume component-wise TVD schemes for 2D shallow water equations , 2003 .

[25]  T. Yabe,et al.  Conservative and oscillation-less atmospheric transport schemes based on rational functions , 2002 .

[26]  R. LeVeque,et al.  Balancing Source Terms and Flux Gradientsin High-Resolution Godunov Methods : The Quasi-Steady Wave-Propogation AlgorithmRandall , 1998 .

[27]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[28]  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 .

[29]  F. Xiao,et al.  Numerical simulations of free-interface fluids by a multi-integrated moment method , 2005 .

[30]  Wang Ji-wen,et al.  The composite finite volume method on unstructured meshes for the two‐dimensional shallow water equations , 2001 .

[31]  T. Yabe,et al.  Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation , 2001 .