Esquemas 2D de alto orden basados en reconstrucciones de estado, para sistemas hiperbólicos no conservativos. Aplicación a problemas de transporte de sedimentos

En este trabajo se aborda la aproximación numérica del problema de arrastre de sedimentos causada por la evolución del agua. Para la componente hidrodinámica se consideran las ecuaciones de aguas poco profundas. La componente morfodinámica se define mediante una ecuación de continuidad, dada en función del caudal sólido. Ambas componentes constituyen un sistema acoplado que puede reescribirse como un sistema hiperbólico no conservativo (ver [4] A.M. Ferreiro Ferreiro. Desarrollo de técnicas de post-proceso de flujos hidrodinámicos, modelización de problemas de transporte de sedimentos y simulación numérica mediante técnicas de volúmenes finitos. Tesis Doctoral. Universidad de Sevilla. 2006). Se propone un esquema 2D generalizado de Roe con reconstrucciones de estado para sistemas hiperbólicos no conservativos (ver [4]), mediante esquemas de volúmenes finitos y el método de líneas (ver [5] J.A. García Rodríguez. Paralelización de esquemas de vol´umenes finitos: aplicación a la resolución de sistemas de tipo aguas someras. Tesis Doctoral. Universidad de Málaga. 2005), extendiendo los esquemas de alto orden para el caso 1D propuesto en [2] M.J. Castro, J.M. Gallardo and C. Parés. High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow water systems. Math. Comp., 75: 1103-1134. 2006. La reconstrucción de estado empleada es de tipo MUSCL (ver [1] B. Van Leer. MUSCL. A new approach to numerical gas dynamics. Computing in plasma physics and astrophysics, Max-Planck-Institut fur plama physik.. Carchung, Germany, April 1976), que proporciona orden dos para mallas no estructuradas de volúmenes finitos de tipo arista. Finalmente se presenta un test num´erico en el que se estudia la evolución del ángulo de expansi´on de una monta˜na de arena (ver [4])

Solving shallow-water systems in 2D domains using Finite Volume methods and multimedia SSE instructions

AbstractThe goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. García, J.M. González, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques.In this paper, in order to optimize the computations, a new parallelization of SIMD type is performed at each MPI thread, by means of SSE (“Streaming SIMD Extensions”), which are present in common processors. More specifically, as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations, we use the Intel© Integrated Performance Primitives small matrix library. To make easy the use of this library, which is implemented using assembler and SSE instructions, we have developed a C++ wrapper of this library in an efficient way. Some numerical tests were carried out to validate the performance of the C++ small matrix wrapper. The specific application of the scheme to one-layer Shallow-Water systems has been implemented on a PC’s cluster. The correct behavior of the one-layer model is assessed using laboratory data

A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies

We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties and the severe time step restrictions arising from higher order terms. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an energy conservation law from which the energy conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in "C. Escalante, M. Dumbser and M.J. Castro. An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes, Journal of Computational Physics 2018"

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around $N=7$). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings