A large time step Godunov scheme for free-surface shallow water equations

AbstractWe present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

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On large time step Godunov scheme for hyperbolic conservation laws

Communications in Mathematical Sciences ◽

10.4310/cms.2004.v2.n3.a7 ◽

2004 ◽

Vol 2 (3) ◽

pp. 477-495 ◽

Cited By ~ 2

Author(s):

Jinghua Wang ◽

Hairui Wen ◽

Tie Zhou

Keyword(s):

Conservation Laws ◽

Hyperbolic Conservation Laws ◽

Large Time ◽

Godunov Scheme ◽

Time Step ◽

Hyperbolic Conservation ◽

Large Time Step

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Large Time Step Godunov Method for 2-D Conservation Laws

Acta Analysis Functionalis Applicata ◽

10.3724/sp.j.1160.2010.00097 ◽

2010 ◽

Vol 12 (2) ◽

pp. 97-109

Author(s):

Hairui WEN ◽

Jinghua WANG

Keyword(s):

Conservation Laws ◽

Large Time ◽

Godunov Method ◽

Time Step ◽

Large Time Step

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WAVETRISK-2.1: an adaptive dynamical core for ocean modelling

10.5194/gmd-2021-365 ◽

2021 ◽

Author(s):

Nicholas Keville-Reynolds Kevlahan ◽

Florian Lemarié

Keyword(s):

Free Surface ◽

Shallow Water ◽

Time Integration ◽

Ocean Model ◽

Mesh Adaptivity ◽

Time Step ◽

Vertical Coordinate ◽

Slip Boundary ◽

Modelling Framework ◽

Atmosphere Model

Abstract. This paper introduces WAVETRISK-2.1 (i.e. WAVETRISK-OCEAN), an incompressible version of the atmosphere model wavetrisk-1.x with free-surface. This new model is built on the same wavelet-based dynamically adaptive core as wavetrisk, which itself uses DYNANICO's mimetic vector-invariant multilayer rotating shallow water formulation. Both codes use a Lagrangian vertical coordinate with conservative remapping. The ocean variant solves the incompressible multilayer shallow water equations with inhomogeneous density layers. Time integration uses barotropic--baroclinic mode splitting via an semi-implicit free surface formulation, which is about 34–44 times faster than an unsplit explicit time-stepping. The barotropic and baroclinic estimates of the free surface are reconciled at each time step using layer dilation. No slip boundary conditions at coastlines are approximated using volume penalization. The vertical eddy viscosity and diffusivity coefficients are computed from a closure model based on turbulent kinetic energy (TKE). Results are presented for a standard set of ocean model test cases adapted to the sphere (seamount, upwelling and baroclinic turbulence). An innovative feature of wavetrisk-ocean is that it could be coupled easily to the wavetrisk atmosphere model, thus providing a first building block toward an integrated Earth-system model using a consistent modelling framework with dynamic mesh adaptivity and mimetic properties.

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