A well‐balanced stable generalized Riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes

SUMMARY We propose a well-balanced stable generalized Riemann problem (GRP) scheme for the shallow water equations with irregular bottom topography based on moving, adaptive, unstructured, triangular meshes. In order to stabilize the computations near equilibria, we use the Rankine–Hugoniot condition to remove a singularity from the GRP solver. Moreover, we develop a remapping onto the new mesh (after grid movement) based on equilibrium variables. This, together with the already established techniques, guarantees the well-balancing. Numerical tests show the accuracy, efficiency, and robustness of the GRP moving mesh method: lake at rest solutions are preserved even when the underlying mesh is moving (e.g., mesh points are moved to regions of steep gradients), and various comparisons with fixed coarse and fine meshes demonstrate high resolution at relatively low cost. Copyright © 2013 John Wiley & Sons, Ltd.

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

[2]  Sudi Mungkasi,et al.  A finite volume method for shallow water flows on triangular computational grids , 2011, 2011 International Conference on Advanced Computer Science and Information Systems.

[3]  Emmanuel Audusse,et al.  A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes , 2005 .

[4]  Joseph Falcovitz,et al.  Application of the GRP scheme to open channel flow equations , 2007, J. Comput. Phys..

[5]  Huazhong Tang.  Solution of the shallow‐water equations using an adaptive moving mesh method , 2004 .

[6]  Michael Dumbser,et al.  Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed , 2009 .

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

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

[9]  Gerald Warnecke,et al.  A direct Eulerian GRP scheme for compressible fluid flows , 2006, J. Comput. Phys..

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

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

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

[13]  A. Dvinsky.  Adaptive grid generation from harmonic maps on Reimannian manifolds , 1991 .

[14]  Jianqiang Han,et al.  An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics , 2007, J. Comput. Phys..

[15]  Roger Alexander Falconer,et al.  Modelling estuarine and coastal flows using an unstructured triangular finite volume algorithm , 2004 .

[16]  Jiequan Li,et al.  Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem , 2007, Numerische Mathematik.

[17]  Pingwen Zhang,et al.  Moving mesh methods in multiple dimensions based on harmonic maps , 2001 .

[18]  Jiequan Li,et al.  An adaptive GRP scheme for compressible fluid flows , 2010, J. Comput. Phys..

[19]  Matania Ben-Artzi,et al.  The generalized Riemann problem for reactive flows , 1989 .

[20]  Ee Han,et al.  Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations , 2011 .

[21]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[22]  Jiequan Li,et al.  The generalized Riemann problem method for the shallow water equations with bottom topography , 2006 .

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

[24]  Pingwen Zhang,et al.  Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes , 2008, J. Sci. Comput..

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

[26]  Robert D. Russell,et al.  Adaptive mesh movement — the MMPDE approach and its applications , 2001 .

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

[28]  Giorgio Rosatti,et al.  The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations , 2010, J. Comput. Phys..

[29]  The gas-kinetic scheme for shallow water equations * * Project supported by the Hong Kong Research G , 2006 .

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

[31]  Keith Miller,et al.  Moving Finite Elements. I , 1981 .

[32]  Huazhong Tang,et al.  A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case , 2012, J. Comput. Phys..

[33]  Yulong Xing,et al.  Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations , 2010 .

[34]  Weiqing Ren,et al.  An Iterative Grid Redistribution Method for Singular Problems in Multiple Dimensions , 2000 .

[35]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .

[36]  Mario Ricchiuto,et al.  Stabilized residual distribution for shallow water simulations , 2009, J. Comput. Phys..

[37]  Christophe Berthon,et al.  Efficient well-balanced hydrostatic upwind schemes for shallow-water equations , 2012, J. Comput. Phys..

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

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

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