A conservative fully-discrete numerical method for the regularised shallow water wave equations

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods

Twenty-eight years with “Hyperbolic Conservation Laws with Relaxation”

This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves (specifically, rarefaction and shock waves). The aim is twofold: firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the (near) future

Finite element approximation of a non-local problem in non-fickian polymer diffusion

This is the post-print version of the Article. Copyright @ 2011 Institute for Scientific Computing and InformationThe problem of non-local nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with a nonlinearly coupled boundary value problem for a viscoelastic ‘pseudostress’ is considered (see, for example, DA Edwards in Z. angew. Math. Phys., 52, 2001, pp. 254—288). We present two numerical schemes using the implicit Euler method and also the Crank-Nicolson method. Each scheme uses a Galerkin finite element method for the spatial discretisation. Special attention is paid to linearising the discrete equations by extrapolating the value of the nonlinear terms from previous time steps. A priori error estimates are given, based on the usual assumptions that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates. We demonstrate by example that although both schemes converge at their optimal rates the Euler method may be more robust than the Crank-Nicolson method for problems of practical relevance

FISH: A 3D parallel MHD code for astrophysical applications

FISH is a fast and simple ideal magneto-hydrodynamics code that scales to ~10 000 processes for a Cartesian computational domain of ~1000^3 cells. The simplicity of FISH has been achieved by the rigorous application of the operator splitting technique, while second order accuracy is maintained by the symmetric ordering of the operators. Between directional sweeps, the three-dimensional data is rotated in memory so that the sweep is always performed in a cache-efficient way along the direction of contiguous memory. Hence, the code only requires a one-dimensional description of the conservation equations to be solved. This approach also enable an elegant novel parallelisation of the code that is based on persistent communications with MPI for cubic domain decomposition on machines with distributed memory. This scheme is then combined with an additional OpenMP parallelisation of different sweeps that can take advantage of clusters of shared memory. We document the detailed implementation of a second order TVD advection scheme based on flux reconstruction. The magnetic fields are evolved by a constrained transport scheme. We show that the subtraction of a simple estimate of the hydrostatic gradient from the total gradients can significantly reduce the dissipation of the advection scheme in simulations of gravitationally bound hydrostatic objects. Through its simplicity and efficiency, FISH is as well-suited for hydrodynamics classes as for large-scale astrophysical simulations on high-performance computer clusters. In preparation for the release of a public version, we demonstrate the performance of FISH in a suite of astrophysically orientated test cases.Comment: 27 pages, 11 figure

Sistema de leis de balanço em problemas de dinâmicas de fluidos : modelagem matemática e aproximação numérica

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Nesta tese, estamos preocupados com o comportamento limite de sistemas hiperbólicos de leis de conservação com termos de relaxamento {\it stiff} para os sistemas locais de leis de conservação, com particular interesse na questão da estabilidade e limites singulares dessas soluções no tempo zero de relaxão. O relaxamento é importante em muitas situações físicas, tais como, em teoria cinética, dinâmica de gases fora do equilíbrio termodinâmico local, em elasticidade com memória (histerese), transição de fase em fluxo multifásico e problemas lineares e não lineares de progação de ondas. Embora a teoria matemática para modelos não lineares de leis equilíbrio com relaxamento tem apresentado algum significativo progresso na boa colocação no contexto de modelos em termodinâmica e teoria cinética, uma compreensão completa sobre o comportamento assintótico para sistemas maiores que $2\times2$, sobre os quais soluções evoluem a partir de um determinado dado inicial com regularidade, permanece indefinida, notadamente para soluções fracas de sistemas hiperbólicos. Assim, devido à complexidade inerente a esta classe de modelos, existem poucas soluções para tais leis de equilíbrio de relaxamento por meio de métodos analíticos. Então, uma análise abstrata, bem como a computação numérica prática por meio de algoritmos de aproximação, constituem ferramentas importantes para estudar tal classe de modelos, bem como para obter novas perspectivas para ampliar o conhecimento geral de sistemas de leis de balanço, ou de leis de equilíbrio. Portanto, foi também desenvolvido um novo método de volumes finitos de tipo {\it unsplitting}, localmente conservativo, via construção formal. Este método foi capaz de computar para sistemas de Euler tanto novas soluções não monótonas como também de reproduzir soluções qualitativamente corretas em regime de fricção alta e gravidade, recentemente publicados na literatura. De fato, os novos algoritmos de apro\-xima\-ção {\it unsplitting} também foram usados para ajudar a compreender um problema de injeção de nitrogênio e de vapor em meios poroso. Outro ponto de vista fundamental perseguido nesta tese é a comparação entre duas metodologias para abordar a questão da resolução de leis equilíbrio com termos fonte de relaxamento: uma metodologia baseia-se do pressuposto que o fenômeno físico está sob equilíbrio termodinâmico (equilíbrio instantâneo), que é modelado por sistemas de leis de conservação hiperbólicas, e a outra metodologia é baseada no relaxamento de tal equilíbrio, que por sua vez dá origem à utilização dos sistemas de leis de equilíbrio na modelagem do processo de relaxamento, como por exemplo, em modelos de transição de fase. Neste momento, uma série de perguntas naturais surgem: quão diferentes são essas soluções de ambas as soluções obtidas por meio destas duas abordagens? A este respeito, uma pergunta mais rigorosa - e mais fundamental - é: como é o comportamento de tais soluções durante o processo de relaxamento e qual é o seu limite? A fim de entender melhor essas metodologias, vamos considerar dois formalismos matemáticos distintos. Nesta tese, nós damos um exemplo de modelagem utilizando esta nova metodologia para a injeção de nitrogênio e de vapor de água em meios porosos. Nós não fomos capazes de dar uma resposta assertiva a todas as perguntas acima, mas um sólido ponto de partida é um estudo aprofundado do caso unidimensional para um problema concreto, que é feito nesta tese. Acreditamos que temos um campo muito interessante (e promissor) de trabalho pela frente, que temos a intenção de continuar a estudar, a fim de entender melhor, via análises abstrata e numérica, tais perguntas importantes e que permanecem indefinidas. Esta tese é uma pequena tentativa de obter uma nova compreensão sobre tais modelos de leis de balançoAbstract: In this thesis, we are concerned with the limit behaviour of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems of conservation laws, particularly the question of stability and singular limits of such solutions to the zero relaxation time. Relaxation is important in many physical situations, as such, in kinetic theory, gases not in local thermodynamic equilibrium, elasticity with memory (hysteresis), multiphase and phase transition and linear and nonlinear waves. Although the mathematical theory of nonlinear balance law with relaxation has presented significant progress on well-posedness linked to extended thermodynamics and kinetic theory, a complete understanding for systems larger than $2\times2$ about how solutions evolve from a given initial data and their regularity and asymptotic behaviour remains elusive, mainly for weak solutions of hyperbolic systems. Thus, due to the complexity inherent to this class of models, there are few solutions for such relaxation balance laws by means of analytical methods. Then, abstract analysis as well as practical computing via approximation algorithms are both significant mathematical tools to tackle as well as to get further insights to enlarge the knowledge for systems of balance laws. Therefore, it was also developed a new unsplitting finite volume methods, which in turn is locally conservative by formal construction. This method was able to corroborate the new solutions for Euler systems with a non-monotonic character as well as to reproduce correct qualitatively solutions of the Euler models with high friction regime and gravity, recently published in the literature. Indeed, the novel unsplitting approximation algorithms were also used to address injection problems of nitrogen and steam in porous media. Another crucial viewpoint pursued in this thesis is the comparison between two methodologies to tackle the issue of solving balance laws with relaxation source terms: one methodology is based by assuming that the physical phenomenon is under thermodynamic equilibrium (instantaneous equilibrium), which is modelled by systems of conservation laws, and the other methodology is based in the relaxation of such equi\-li\-brium, which in turn gives rise to the use of systems of balance laws in the modelling of the relaxation process, for instance, in the modelling of phase transition. At this moment a natural questions is: how different are these both solutions obtained by means of two approaches? In this regard, a more stringent -- and more fundamental -- question is: how is the behaviour of such solutions during the relaxation process and how is its limit? In order to better understand these methodologies we will consider two distinct mathematical formalisms. In thesis, we give an example of modelling using this novel methodology for the injection of nitrogen and steam in porous media. We were not able to give assertive answers to the above questions, but a solid starting point is a thorough study of the one-dimensional case for a concrete problem, which is done in this thesis. We believe we have a very interesting (and promising) field of work ahead of us, which we intend to continue studying in order to better understand abstract and numerical analysis for these important questions that remains elusive. This thesis is a small attempt to get new insights in this directionDoutoradoMatematica AplicadaDoutor em Matemática Aplicada2011/23628-0FAPES

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp. energy estimates of Section 7, corrected bad forward references, expanded Remark 1.17, end of introductio