Analysis of modified Godunov type schemes for the two-dimensional linear wave equation with Coriolis source term on cartesian meshes

The study deals with colocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classical scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so, a Hodge-like decomposition is introduced. Then three different well-balanced strategies are introduced. Some properties of the associated modified equation are proven and then extended to the semi-discrete case. Stability of fully discrete schemes under a classical CFL condition is established thanks to a Von Neumann analysis. Some numerical results reinforce the purpose

Godunov type scheme for the linear wave equation with Coriolis source term

International audienceWe propose a method to explain the behavior of the Godunov finite volume scheme applied to the linear wave equation with Coriolis source term at low Froude number. In particular, we use the Hodge decomposition and we study the properties of the modified equation associated to the Godunov scheme. Based on the structure of the discrete kernel of the linear operator discretized by using the Godunov scheme, we clearly explain the inaccuracy of the classical Godunov scheme at low Froude number and we introduce the way to modify it to recover a good accuracy

Simulation of all-scale atmospheric dynamics on unstructured meshes

The advance of massively parallel computing in the nineteen nineties and beyond encouraged finer grid intervals in numerical weather-prediction models. This has improved resolution of weather systems and enhanced the accuracy of forecasts, while setting the trend for development of unified all-scale atmospheric models. This paper first outlines the historical background to a wide range of numerical methods advanced in the process. Next, the trend is illustrated with a technical review of a versatile nonoscillatory forward-in-time finite-volume (NFTFV) approach, proven effective in simulations of atmospheric flows from small-scale dynamics to global circulations and climate. The outlined approach exploits the synergy of two specific ingredients: the MPDATA methods for the simulation of fluid flows based on the sign-preserving properties of upstream differencing; and the flexible finite-volume median-dual unstructured-mesh discretisation of the spatial differential operators comprising PDEs of atmospheric dynamics. The paper consolidates the concepts leading to a family of generalised nonhydrostatic NFTFV flow solvers that include soundproof PDEs of incompressible Boussinesq, anelastic and pseudo-incompressible systems, common in large-eddy simulation of small- and meso-scale dynamics, as well as all-scale compressible Euler equations. Such a framework naturally extends predictive skills of large-eddy simulation to the global atmosphere, providing a bottom-up alternative to the reverse approach pursued in the weather-prediction models. Theoretical considerations are substantiated by calculations attesting to the versatility and efficacy of the NFTFV approach. Some prospective developments are also discussed

Numerical approximation of the shallow water equations with coriolis source term

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes

Accurate simulation of shallow flows using arbitrary order ADER schemes and overcoming numerical shockwave anomalies

En la actualidad, gracias al desarrollo de algoritmos de simulación avanzados y de tecnologías computacionales eficientes que ha tenido lugar durante las últimas décadas, es posible simular problemas de elevada complejidad que hace unos años eran inalcanzables. Parte de estos problemas se modelan mediente ecuaciones en derivadas parciales de tipo hiperbólico. Este tipo de ecuaciones reproducen con fidelidad aquellos fenómenos que involucran la propagación de ondas. En situaciones realistas, es necesario tener en cuenta efectos dinámicos adicionales más allá de los fenómenos puramente convectivos. Dichos efectos se modelan matemáticamente mediante los llamados términos fuente, que dan lugar a sistemas de ecuaciones no homogéneos y suponen un desafío computacional importante en numerosas ocasiones. Sólo unas determinadas discretizaciones del término fuente garantizan la convergencia de la solución a una solución físicamente realista; cuando se utilizan métodos numéricos sofisticados, la complejidad en el tratamiento de los términos fuentes aumenta de forma notable.Esta tesis se centra en el desarrollo de esquemas numéricos de orden arbitrario para la resolución de sistemas hiperbólicos siguiendo la metodología ADER, que permite la extensión del esquema tradicional de Godunov a orden arbitrario. Los métodos que aquí se presentan están enfocados a la resolución de las ecuaciones de aguas poco profundas, pero se formulan de forma general para su posible aplicación a otros modelos matemáticos. La particularidad fundamental de los esquemas numéricos propuestos en esta tesis reside en la manera en la que se introducen los términos fuente en la formulación discreta. A diferencia de la mayoría de métodos comunmente utilizados, aquí se propone introducir los términos fuente en la formulación de los flujos numéricos, siguiendo una metodología de discretización upwind. Esto implica considerar los términos fuente en la formulación del problema de Riemann derivativo. De este modo, es posible garantizar un equilibrio perfecto entre flujos y términos fuente a nivel discreto y reproducir con precisión aquellas situaciones de equilibrio relevantes para los problemas estudiados. Para las ecuaciones de aguas poco profundas, aquellos esquemas que satisfacen esta propiedad se denominaron tradicionalmente well-balanced, aunque dicha atribución sólo hacía referencia a la preservación de estados de reposo estático.Se muestra que sólo aquellos términos fuentes de tipo geométrico (por ejemplo, término de variación de fondo en las ecuaciones de aguas poco profundas) se deben incluir en la resolución del problema de Riemann derivativo. Otros términos fuente de distinta naturaleza se pueden integrar de forma tradicional utilizando reglas de cuadratura, o bien, se pueden reescribir como términos geométricos y pueden ser tratados del mismo modo. Siguiendo esta última aproximación, es posible garantizar la propiedad well-balanced sin perder el orden de convergencia arbitrario. Aquí se detalla la construcción de esquemas numéricos de orden arbitrario para las ecuaciones de aguas poco profundas con términos fuente de fondo, fricción y Coriolis, que satisfacen la propiedad well-balanced. Además, mediante consideraciones de conservación de energía a nivel discreto, dicha propiedad se extiende para situaciones de equilibrio unidimensionales que involucran velocidades no nulas, desde una perspectiva de un esquema ADER.Por último, en este trabajo también se estudian anomalías numéricas que pueden aparecer en la resolución de las ecuaciones de aguas poco profundas. Dichas anomalías son intrínsecas al método de volúmenes finitos y pueden dar lugar a oscilaciones severas de la solución numérica. Siguiendo estudios previos sobre anomalías numéricas en las ecuaciones de Euler, se formula un marco teórico para el estudio de dichas anomalías en las ecuaciones de aguas poco profundas. Se muestra que la presencia de resaltos hidráulicos genera oscilaciones numéricas en el caudal y se propone una corrección del flujo que lo solventa.<br /

Object-oriented hyperbolic solver on 2D-unstructured meshes applied to the shallow water equations

Fluid dynamics, like other physical sciences, is divided into theoretical and experimental branches. However, computational fluid dynamics (CFD) is third branch of Fluid dynamics, which has aspects of both the previous two branches. CFD is a supplement rather than a replacement to the experiment or theory. It turns a computer into a virtual laboratory, providing insight, foresight, return on investment and cost savings1. This work is a step toward an approach that realise a new and effective way of developing these CFD models

2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

In this work, an arbitrary order augmented WENO-ADER scheme for the resolution of the 2D Shallow Water Equations (SWE) with geometric source term is presented and its application to other shallow water models involving non-geometric sources is explored. This scheme is based in the 1D Augmented Roe Linearized-ADER (ARL-ADER) scheme, presented by the authors in a previous work and motivated by a suitable compromise between accuracy and computational cost. It can be regarded as an arbitrary order version of the Augmented Roe solver, which accounts for the contribution of continuous and discontinuous geometric source terms at cell interfaces in the resolution of the Derivative Riemann Problem (DRP). The main novelty of this work is the extension of the ARL-ADER scheme to 2 dimensions, which involves the design of a particular procedure for the integration of the source term with arbitrary order that ensures an exact balance between flux fluctuations and sources. This procedure makes the scheme preserve equilibrium solutions with machine precision and capture the transient waves accurately. The scheme is applied to the SWE with bed variation and is extended to handle non-geometric source terms such as the Coriolis source term. When considering the SWE with bed variation and Coriolis, the most relevant equilibrium states are the still water at rest and the geostrophic equilibrium. The traditional well-balanced property is extended to satisfy the geostrophic equilibrium. This is achieved by means of a geometric reinterpretation of the Coriolis source term. By doing this, the formulation of the source terms is unified leading to a single geometric source regarded as an apparent topography. The numerical scheme is tested for a broad variety of situations, including some cases where the first order scheme ruins the solution

Stratified shallow flow modelling

Environmental hydraulics covers a very wide range of applications including free surface flows in rivers. estuaries and lakes. To find engineering solutions to environmental hydraulics problems. 3D numerical modelling is nowadays widely used. However. the computation of sharp spatial gradients (such as found in stratified estuaries and lakes. around plumes near outfalls along rivers and coasts or in exchange areas of high shear). and the modelling of these processes along steep bathymetric slopes (such as found at the edge of dredged channels or of the continental shelf) remains a challenge. In addition. crude assumptions (such as the hydrostatic assumption) are often made to the primary differential equations in order to simplify the problem and enable long term prediction of environmental hydraulic changes. In this thesis. a robust adaptive mesh displacement (AMD) method is implemented and validated against the lock exchange case in particular. The AMD method aims at vertically focusing nodes within each water column to capture sharp gradients. while reducing the number of nodes or requiring prior knowledge of the flow structure. Second. a direct computation of dynamic pressure is introduced based on the equation of vertical momentum and validated against the analytical potential flow theory solution of a source-sink pair. Dynamic pressure is necessary to model destratification recirculation devices. or flow over dredge channel. or solitary waves. for instance. This direct computation method makes the hydrostatic assumption redundant. Third. a new advection scheme is implemented. whose main advantage is simplicity averaging over Riemann problems without solving them. while excessive numerical viscosity is compensated for by using high-resolution MUSCL type reconstruction. Recommendations are made in this thesis to extend the advection scheme developed herein for tracer advection to the non-linear shallow water equations. to the diffusion terms and to turbulence closure laws within the same finite element framework

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com