On curl-preserving finite volume discretizations for shallow water equations

The preservation of intrinsic or inherent constraints, like divergence-conditions, has gained increasing interest in numerical simulations of various physical evolution equations. In Torrilhon and Fey, SIAM J. Numer. Anal. (42/4) 2004, a general framework is presented how to incorporate the preservation of a discrete constraint into upwind finite volume methods. This paper applies this framework to the wave equation system and the system of shallow water equations. For the wave equation a curl-preservation for the momentum variable is present and easily identified. The preservation in case of the shallow water system is more involved due to the presence of convection. It leads to the vorticity evolution as generalized curl-constraint. The mechanisms of vorticity generation are discussed.For the numerical discretization special curl-preserving flux distributions are discussed and their incorporation into a finite volume scheme described. This leads to numerical discretizations which are exactly curl-preserving for a specific class of discrete curl-operators.The numerical experiments for the wave equation show a significant improvement of the new method against classical schemes. The extension of the curl-free numerical discretization to the shallow water case is possible after isolating the pressure flux. Simulation examples demonstrate the influence of the modification. The vortex structure is more clearly resolved.

[1]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[2]  Manuel Torrilhon,et al.  Constraint-Preserving Upwind Methods for Multidimensional Advection Equations , 2004, SIAM J. Numer. Anal..

[3]  Eun Heui Kim,et al.  A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks , 2002 .

[4]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

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

[6]  J. Brackbill,et al.  The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations☆ , 1980 .

[7]  G. Tóth.  The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes , 2000 .

[8]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 2: ARTIFICIAL DIFFUSION AND DISCRETE SHOCK STRUCTURE , 1994 .

[9]  J. Scalo,et al.  The Galactic Shock Pump: A Source of Supersonic Internal Motions in the Cool Interstellar Medium , 1997, astro-ph/9708131.

[10]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[11]  Manuel Torrilhon,et al.  Locally Divergence-preserving Upwind Finite Volume Schemes for Magnetohydrodynamic Equations , 2005, SIAM J. Sci. Comput..

[12]  M. Shashkov,et al.  Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆ , 1997 .

[13]  C. Truesdell.  The Kinematics Of Vorticity , 1954 .

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

[15]  Manuel Torrilhon,et al.  A Constrained Transport Upwind Scheme for Divergence-free Advection , 2003 .

[16]  Shian-Jiann Lin,et al.  An explicit flux‐form semi‐lagrangian shallow‐water model on the sphere , 1997 .

[17]  M. Torrilhon,et al.  Vorticity Preserving Finite Volume Method for the Wave Equation System , 2022 .

[18]  Quasilinear Hyperbolic Systems , 1990 .

[19]  K. W. Morton,et al.  Vorticity-Preserving Lax-Wendroff-Type Schemes for the System Wave Equation , 2001, SIAM J. Sci. Comput..