58 research outputs found
Optimized Schwarz waveform relaxation for Primitive Equations of the ocean
In this article we are interested in the derivation of efficient domain
decomposition methods for the viscous primitive equations of the ocean. We
consider the rotating 3d incompressible hydrostatic Navier-Stokes equations
with free surface. Performing an asymptotic analysis of the system with respect
to the Rossby number, we compute an approximated Dirichlet to Neumann operator
and build an optimized Schwarz waveform relaxation algorithm. We establish the
well-posedness of this algorithm and present some numerical results to
illustrate the method
A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation
The standard multilayer Saint-Venant system consists in introducing fluid
layers that are advected by the interfacial velocities. As a consequence there
is no mass exchanges between these layers and each layer is described by its
height and its average velocity. Here we introduce another multilayer system
with mass exchanges between the neighborhing layers where the unknowns are a
total height of water and an average velocity per layer. We derive it from
Navier-Stokes system with an hydrostatic pressure and prove energy and
hyperbolicity properties of the model. We also give a kinetic interpretation
leading to effective numerical schemes with positivity and energy properties.
Numerical tests show the versatility of the approach and its ability to compute
recirculation cases with wind forcing.Comment: Submitted to M2A
A Multilayer Saint-Venant Model
We introduce a new variant of the multilayer Saint-Venant system. The classical Saint-Venant system is a well-known approximation of the incompressible Navier-Stokes equations for shallow water flows with free moving boundary. Its efficiency, robustness and low computational cost make it very commonly used. Nevertheless its range of application is limited and it does not allow to access to the vertical profile of the horizontal velocity. Hence and thanks to a precise analysis of the shallow water assumption we propose here a new approximation of the Navier-Stokes equations which consists in a set of coupled Saint-Venant systems, extends the range of validity and gives a precise description of the vertical profile of the horizontal velocity while preserving the computational efficiency of the classical Saint-Venant system. We validate the model through some numerical examples
Uniqueness for a Scalar Conservation Law with Discontinuous Flux via Adapted Entropies
We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kruzkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of Kruzkov's original method. Especially, we do not need traces, interface condition, Bounded Variation assumptions (neither on the solution nor on the flux), or convex fluxes. However we use a special 'local uniform invertibility' structure of the flux which applies to cases where different interface condiftions are known to yield different solutions
Uniqueness for a Scalar Conservation Law with Discontinuous Flux via Adapted Entropies
We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kruzkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of Kruzkov's original method. Especially, we do not need traces, interface condition, Bounded Variation assumptions (neither on the solution nor on the flux), or convex fluxes. However we use a special 'local uniform invertibility' structure of the flux which applies to cases where different interface condiftions are known to yield different solutions
NUMERICAL SIMULATIONS OF THE PERIODIC INVISCID BURGERS EQUATION WITH STOCHASTIC FORCING
International audienceWe perform numerical simulations in the one-dimensional torus for the first order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various egularities in space. For the numerical tests, we apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. Our Monte-Carlo simulations are analyzed in bounded time intervals as well as in the large time limit, for various regularities in space. The empirical mean always converges to the space-average of the (deterministic) initial condition as t → ∞, just as the solution of the deterministic problem without source term, even if the stochastic source term is very rough. The empirical variance also stablizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise
Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow
In this work we are interested in numerical simulations for bedload erosion
processes. We present a relaxation solver that we apply to moving dunes test
cases in one and two dimensions. In particular we retrieve the so-called
anti-dune process that is well described in the experiments. In order to be
able to run 2D test cases with reasonable CPU time, we also describe and apply
a parallelization procedure by using domain decomposition based on the
classical MPI library.Comment: 19 page
Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system
International audienceA lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with non-flat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example Godunov, Roe, kinetic,...). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We prove also that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term
A 2d Well-balanced Positivity Preserving Second Order Scheme for Shallow Water Flows on Unstructured Meshes
We consider the solution of the Saint-Venant equations with topographic source terms on 2D unstructured meshes by a finite volume approach. We first present a stable and positivity preserving homogeneous solver issued from a kinetic representation of this hyperbolic conservation laws system. This water depth positivity property is important when dealing with wet-dry interfaces. Then we introduce a local hydrostatic reconstruction that preserves the positivity properties of the homogeneous solver and leads to a well-balanced scheme satisfying the steady state condition of still water. Finally a second order extension based on limited reconstructed values on both sides of each interface and on an enriched interpretation of the source terms satisfies the same properties and gives a noticeable accuracy improvement. Numerical examples on academic and real problems are presented
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