A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context recent implicit-explicit schemes developed in the framework of compressibleflows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases

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

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.Comment: accepted versio

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

FullSWOF_Paral: Comparison of two parallelization strategies (MPI and SKELGIS) on a software designed for hydrology applications

In this paper, we perform a comparison of two approaches for the parallelization of an existing, free software, FullSWOF 2D (http://www. univ-orleans.fr/mapmo/soft/FullSWOF/ that solves shallow water equations for applications in hydrology) based on a domain decomposition strategy. The first approach is based on the classical MPI library while the second approach uses Parallel Algorithmic Skeletons and more precisely a library named SkelGIS (Skeletons for Geographical Information Systems). The first results presented in this article show that the two approaches are similar in terms of performance and scalability. The two implementation strategies are however very different and we discuss the advantages of each one.Comment: 27 page

FATODE: A Library for Forward, Adjoint, and Tangent Linear Integration of ODEs

FATODE is a FORTRAN library for the integration of ordinary differential equations with direct and adjoint sensitivity analysis capabilities. The paper describes the capabilities, implementation, code organization, and usage of this package. FATODE implements four families of methods -- explicit Runge-Kutta for nonstiff problems and fully implicit Runge-Kutta, singly diagonally implicit Runge-Kutta, and Rosenbrock for stiff problems. Each family contains several methods with different orders of accuracy; users can add new methods by simply providing their coefficients. For each family the forward, adjoint, and tangent linear models are implemented. General purpose solvers for dense and sparse linear algebra are used; users can easily incorporate problem-tailored linear algebra routines. The performance of the package is demonstrated on several test problems. To the best of our knowledge FATODE is the first publicly available general purpose package that offers forward and adjoint sensitivity analysis capabilities in the context of Runge Kutta methods. A wide range of applications are expected to benefit from its use; examples include parameter estimation, data assimilation, optimal control, and uncertainty quantification