An efficient scheme on wet/dry transitions for shallow water equations with friction

The present work concerns the derivation of a suitable discretization to approximate the friction source terms in the shallow-water model. Such additional source terms are known to be very stiff as soon as the water height is vanishing. The proposed numerical procedure comes from a relevant correction of a Godunov-type scheme that approximates the solutions of hyperbolic systems of conservation laws. The adopted correction gives a discretization of the source term which preserves the robustness and does not change the CFL condition. The scheme is shown to be particularly efficient for wet/dry transition simulations. In addition, this numerical procedure can be used together with any robust and well-balanced discretization of the topography source term. Second order extension is also investigated. Extensive numerical validations illustrate the interest of this new approach.

[1]  Christophe Berthon,et al.  Asymptotic preserving HLL schemes , 2011 .

[2]  Alan K. Easton,et al.  Moving boundary shallow water flow in a region with quadratic bathymetry , 2008 .

[3]  P. Raviart,et al.  GODUNOV-TYPE SCHEMES FOR HYPERBOLIC SYSTEMS WITH PARAMETER-DEPENDENT SOURCE: THE CASE OF EULER SYSTEM WITH FRICTION , 2010 .

[4]  Fabien Marche,et al.  A Positive Preserving High Order VFRoe Scheme for Shallow Water Equations: A Class of Relaxation Schemes , 2008, SIAM J. Sci. Comput..

[5]  Manmohan Singh,et al.  Moving boundary shallow water flow above parabolic bottom topography , 2006 .

[6]  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..

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

[8]  C. Berthon,et al.  A Free Streaming Contact Preserving Scheme for the M 1 Model , 2010 .

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

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

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

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

[13]  Rou-Fei Chen,et al.  Simulation of Tsaoling landslide, Taiwan, based on Saint Venant equations over general topography , 2009 .

[14]  M M Hafez,et al.  Innovative Methods for Numerical Solution of Partial Differential Equations , 2001 .

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

[16]  Q. Liang,et al.  Numerical resolution of well-balanced shallow water equations with complex source terms , 2009 .

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

[18]  Olivier Delestre.  Simulation du ruissellement d'eau de pluie sur des surfaces agricoles. (Rain water overland flow on agricultural fields simulation) , 2010 .

[19]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[20]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

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

[22]  Javier Murillo,et al.  Friction term discretization and limitation to preserve stability and conservation in the 1D shallow‐water model: Application to unsteady irrigation and river flow , 2008 .

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

[24]  Olivier Delestre,et al.  A Numerical Scheme for a Viscous Shallow Water Model with Friction , 2011, J. Sci. Comput..

[25]  Randall J. LeVeque,et al.  A class of approximate Riemann solvers and their relation to relaxation schemes , 2001 .

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

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

[28]  Nancy Nichols,et al.  Analytic Benchmark Solutions for Open-Channel Flows , 1997 .

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

[30]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[31]  Derek M. Causon,et al.  On the Choice of Wavespeeds for the HLLC Riemann Solver , 1997, SIAM J. Sci. Comput..