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Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed. [ABSTRACT FROM AUTHOR]

The computational simulation of shallow stratified fluids is a very active research topic because these types of systems are very common in a variety of natural environments. The simulation of such systems can be modeled using multilayer shallow-water equations but do impose important computational requirements, especially when applied to large domains. General Purpose Computing on Graphics Processing Units (GPGPU) has become a vivid research field due to the arrival of massively parallel hardware platforms (based on graphics cards) and adequate programming frameworks which have allowed important speed-up factors with respect to not only sequential but also parallel CPU based simulation systems. In this work we present simulation of shallow stratified fluids with an arbitrary number of layers using GPUs. The designed system does fully adapt to the many-core architecture of modern GPUs and several experiments have been carried out to illustrate its scalability and behavior on different GPU models. We propose a new multilayer computational scheme for an underlying 2D mathematical model. This scheme is capable of handling an arbitrary number of layers. The system adds no overhead when used for two-layer scenarios, compared to an existing 2D system specifically designed for just two layers. Our proposal is aimed at creating a GPU-based computational scheme suitable for the simulation of multilayer large-scale real-world scenarios. [ABSTRACT FROM AUTHOR]

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed. [ABSTRACT FROM AUTHOR]

We present an adaptation of the MOOD method, initially introduced in [1,2], for the two-dimensional shallow-water system with varying bathymetry, where the major novelty of the study is the non-conservative term discretization in the framework of the MOOD strategy. We derive a robust sixth-order well-balanced scheme and propose a large panel of numerical tests to assess the accuracy of the method and show that numerical solutions are free of oscillations in the vicinity of discontinuities. We also demonstrate that the MOOD method guarantees the height positivity as long as the first-order scheme does. [ABSTRACT FROM AUTHOR]

Highlights: [•] The Multiple-grid FEM is detailed to accelerate the convergence of the SWE’s solution. [•] This approach is applied for subcritical and supercritical flows. [•] Solution is free of numerical oscillations. [•] Smooth solutions are obtained near shock fronts. [•] The CPU time is competitive compared with the existent shock capturing methods. [Copyright &y& Elsevier]

Abstract: A fully conservative semi-Lagrangian (SL) scheme is presented to solve for the shallow-water equations. Existing inherently conservative SL schemes only ensure the conservation of mass while momentum is not fully conserved. The gravity terms, which are mainly responsible for the wave structure in dam break flows, are then discretized by using traditional non-conservative Eulerian schemes. In the presence of large variations in water surface (e.g., dam-break type flows), such an approach leads to incorrect shock speed and highly oscillatory results. Indeed, if the conservation of the gravity terms is forced, the use of existing schemes will be restricted to small time steps. In this paper we present a fully conservative scheme which can accurately simulate the shallow flows with a large time step. In our approach, both convective and gravitational terms are treated in a conservative manner, which ensures an accurate shock speed. The fully conservation property improves considerably the performance of common SL schemes for a wide range of practical applications. [Copyright &y& Elsevier]