Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

184 Pages·2015·13.6 MB·English

John Alexander Perez Sepulveda
Lagrangian-Eulerian approximation methods for
balance laws and hyperbolic conservation laws
Métodos de aproximação Lagrangeano-Euleriano
para leis de balanço e leis de conservação
hiperbólicas
CAMPINAS
2015

Agência de fomento: Capes
Nº processo: 0
Ficha catalográfica
Universidade Estadual de Campinas
Biblioteca do Instituto de Matemática, Estatística e Computação Científica
Maria Fabiana Bezerra Muller - CRB 8/6162

Perez Sepulveda, John Alexander, 1974-
P415L PerLagrangian-Eulerian approximation methods for balance laws and
hyperbolic conservation laws / John Alexander Perez Sepulveda. – Campinas,
SP : [s.n.], 2015.

PerOrientador: Eduardo Cardoso de Abreu.
PerTese (doutorado) – Universidade Estadual de Campinas, Instituto de
Matemática, Estatística e Computação Científica.

Per1. Equações diferenciais hiperbólicas. 2. Leis de conservação (Física). 3.
Método dos volumes finitos. 4. Mecânica dos fluídos. I. Abreu, Eduardo
Cardoso de,1974-. II. Universidade Estadual de Campinas. Instituto de
Matemática, Estatística e Computação Científica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Métodos de aproximação Lagrangeano-Euleriano para leis de
balanço e leis de conservação hiperbólicas
Palavras-chave em inglês:
Hyperbolic differential equations
Conservation laws (Physics)
Finite volume method
Fluid mechanics
Área de concentração: Matemática Aplicada
Titulação: Doutor em Matemática Aplicada
Banca examinadora:
Eduardo Cardoso de Abreu [Orientador]
Luis Felipe Feres Pereira
Sandra Mara Cardoso Malta
Fábio Antonio Dorini
Lucas Catão de Freitas Ferreira
Data de defesa: 21-07-2015
Programa de Pós-Graduação: Matemática Aplicada
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“To my wife Monica, and my sons Camila and Tómas, because all is very easy when we are
together”
Acknowledgement
Foremost, I want to thank God.
I would like to express my sincere gratitude to my wife Monica, and my sons Camila and
Tómas for their love and for helping me survive all the stress from these four years and not
letting me give up. I would also like to thank my advisor Dr. Eduardo Abreu for his un-
derstanding, wisdom, assistance and suggestions throughout my project. To my institution,
ITM-Instituto Metropolitano because gave me the space and the time to study. To the In-
stitute of Mathematics, Statistics and Scientific Computing (IMECC) - Graduate Program in
Applied Mathematics - for providing me the opportunity to be a student of this prestigious
institution. To my evaluating committee, Dr. Felipe Pereira, Dra. Sandra Malta, Dr. Favio
Dorini and Dr. Lucas Ferreira for their valuable suggestions. To the discussion group, Abel,
Arthur, Paola, Juan, Jardel, Ciro and other that they passed by the seminary, because always
their observations they brought a light. I would like to thanks Arthur, Paola and Ciro that
they help me with English. And finally, I want to give a sincere gratitude to the institutions
that they helped to get the aim. I give my thank to CAPES for the graduate fellowship (2011-
2015). In addition, I also give my thank to the financial support FAPESP through grants No.
2011/11897-6 and No. 2014/03204-9 and CNPq through grants No. MCTI/CNPQ/Universal
No 445758/2014-7.
Resumo
Neste trabalho, estudamos um volume finito de controle no espaço-tempo local, em um
marco Lagrangiano-Euleriano com o objetivo de construir um esquema localmente conservativo
que modele o delicado balanço não-linear entre as aproximações numéricas do fluxo hiperbólico
e o termo fonte, em problemas de lei de balanço ligados ao caráter puramente hiperbólico da lei
de conservação. Efetuamos a análise de estabilidade e de convergência do método para o caso
linear da lei de conservação hiperbólica e de problemas de lei de balanço em espaços discretos
convenientes. De fato, baseados nesta condição de estabilidade, nós substituímos a equação
da lei de conservação escalar dada por uma aproximação de diferenças finitas que dependente
dos parâmetros da malha, no espaço e no tempo, a fim construir uma sequência convergente
para a única solução entrópica da lei de conservação escalar, pelo menos, para o caso em que
a função de fluxo é do tipo convexo. Para o melhor de nosso conhecimento, este trabalho é o
primeiro a estabelecer uma prova rigorosa para convergência para da única solução entrópica
construída por tubos integrais em um procedimento Lagrangiano-Euleriano, para leis de conser-
vação hiperbólicas em uma dimensão espacial com fluxo convexo. Ressaltamos a importância e
a utilidade de identificar equações modificadas, no escopo das diferencias finitas, associada ao
método Lagrangiano-Euleriano o qual usamos para dar uma explicação sobre a possibilidade
de instabilidade na construção dos tubos integrais longos; tal construção tem sido chamado a
atenção de muitos autores na literatura disponível. O esquema construido é livre de Riemann
solvers, mas se soluções locais de Riemann estão disponíveis para um determinado problema,
estes podem ser incorporados naturalmente no esquema. Um grande conjunto não-trivial, dis-
tinto e bem conhecido de experimentos numéricos tanto para problemas unidimensionais, como
para problemas bidimensionais não-lineares estão disponíveis na literatura especializada, que
incluileisdeconservaçãoescalareshiperbólicasesistemasescalaresdeleisdeconservaçãohiper-
bólicas e de leis de balanço, são discutidos para ilustrar o desempenho do novo método, onde
incluímos experimentos numéricos com fluxos convexo, não-convexo e funções de fluxo descon-
tínuos. Os resultados numéricos são comparados com soluções exatas sempre que possível ou
soluções aproximadas com malha fina em outros casos.
Palavras-chave: Enfoque Lagrangiano-Euleriano, Volume Finito, Leis de balanço, Leis de
conservação hiperbólicas, Mecânica dos fluidos, Problemas de fluxo em meios porosos.
Abstract
In this work, we study a local space-time finite control volume in a Lagrangian-Eulerian
framework in order to design a locally conservative scheme to account the delicate nonlinear
balance between the numerical approximations of the hyperbolic flux and the source term for
balance law problems linked to the purely hyperbolic character of conservation laws. We have
performedstabilityandconvergenceanalysisofthemethodforlineardifferentialhyperbolicand
balance law problems in convenient discrete spaces. Indeed, based on this stability condition,
we replace the given scalar conservation law equation by a finite difference approximation
depending on mesh parameters in space and in time in order construct a sequence that is
convergent to the unique entropy solution to the scalar conservation law, at least for the case
where flux function is of convex type. To the best of our knowledge, this work is the first to
establish a rigorous convergence proof for the uniqueness of the entropy solution constructed
integral tubes by a Lagrangian-Eulerian procedure for hyperbolic conservation laws in one-
space dimension with convex flux. We also add some meaningful comments on the usefulness
of identifying modified equations, which models the behavior of the analogue difference scheme
associated to the Lagrangian-Eulerian method and use it to give a possible explanation on
the possibility of instability in construction of long integral tubes; such construction has been
called the attention of many authors in the available literature. The designed scheme is also
free of Riemann solvers, but if local Riemann solutions are available for a particular problem
it is natural to incorporate such feature into the scheme. Furthermore, by combining ideas
of the new approach, we give a formal construction of a new algorithm for solving several
nonlinear hyperbolic conservation laws in two space dimensions. A set of nontrivial and distinct
well-known one-dimensional as well as two-dimensional numerical experiments for nonlinear
problems available in the specialized literature - scalar and system - of hyperbolic conservation
law and balance law types are discussed to illustrate the performance of the new method,
including numerical experiments with convex, non-convex and discontinuous flux functions.
The numerical results are compared with accurate approximate solutions or exact solutions
whenever possible.
Keywords: Lagrangian-Eulerian approach, Finite volume, Balance laws, Hyperbolic con-
servation laws, fluid mechanics, porous media flow problems
Contents
Dedicatória 5
Acknowledgement 6
1 Introduction 11
1.1 Motivation and significance of the research work . . . . . . . . . . . . . . . . . . 11
1.2 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Preliminary results and ongoing work . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 The revisited Lagrangian-Eulerian scheme for linear hyperbolic conservation
laws 19
2.1 Formal construction: statement of the Lagrangian-Eulerian formulation and no-
tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The local Lagrangian-Eulerian conservation relation . . . . . . . . . . . . 20
2.1.2 The Lagrangian-Eulerian scheme for linear hyperbolic conservation laws
and its mathematical properties . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 A multiple time-step formulation for the Lagrangian-Eulerian scheme . . . . . . 29
3 The conservative finite difference Lagrangian-Eulerian scheme for nonlinear
scalar conservation laws 33
3.1 Preliminary concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 On the construction of a unique entropy solution based on the Lagrangian-
Eulerian scheme for convex scalar conservation law . . . . . . . . . . . . . . . . 38
3.3 The finite difference Lagrangian-Eulerian scheme in conservative form . . . . . . 48
3.4 Numerical experiments for hyperbolic problems with convex and nonconvex fluxes 49
3.5 Numericalexperimentswithdiscontinuousfluxfunctions,modelproblemsAdimurthi,
J. Jaffré and V. Gowda §3.5.1 and R. Burger, K. H. Karlsen and J. D. Towers
§3.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 The Lagrangian-Eulerian approach to nonlinear hyperbolic conservation laws 59
4.1 The Lagrangian-Eulerian scheme in conservative form . . . . . . . . . . . . . . . 59
4.2 A Lipschitz condition to the Lagrangian-Eulerian numerical flux function . . . . 61
4.3 Monotonicity and TVD properties for the Lagrangian-Eulerian scheme . . . . . 64
4.4 Equivalence between finite difference and finite volume Lagrangian-Eulerian for-
mulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 Numerical experiments for hyperbolic problems with convex and non-convex fluxes 66
5 The Lagrangian-Eulerian scheme for hyperbolic balance laws 70
5.1 Linear case for balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 Midpoint rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.2 The trapezoidal rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.3 Stability and convergence for Lagrangian-Eulerian scheme for linear bal-
ance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Nonlinear balance law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.1 Predictor-corrector method . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.2 Midpoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.3 Trapezoidal method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Stability of the Lagrangian Eulerian scheme to nonlinear balance law . . . . . . 82
5.3.1 Predictor corrector approximation . . . . . . . . . . . . . . . . . . . . . . 83
5.3.2 An approximation of the source term by the Midpoint quadrature rule . 84
5.3.3 An approximation of the source term by the Trapezoidal quadrature rule 84
5.4 Numerical experiments for nonlinear scalar balance laws . . . . . . . . . . . . . 85
6 The Lagrangian-Eulerian scheme for systems of hyperbolic conservation laws
and balance laws 98
6.1 Extension to systems of hyperbolic conservation laws and balance laws . . . . . 98
6.2 Numerical experiments for systems of nonlinear balance laws . . . . . . . . . . . 101
6.3 Numerical experiments for systems of nonlinear hyperbolic conservation laws . . 107
7 The extension of the Lagragian-Eulerian scheme for hyperbolic conservation
laws in two-space dimensions 112
7.1 The Lagrangian-Eulerian relation for hyperbolic conservation laws linked to bal-
ance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.1.1 The locally conservative Lagrangian-Eulerian relation for scalar multidi-
mensional hyperbolic conservation laws linked to one-dimensional system
of balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Numerical experiments for nonlinear hyperbolic law em 2D . . . . . . . . . . . . 125
8 Concluding remarks and perspectives for the future 160
8.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Perspectives for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.3 Final Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography 165
A Uniqueness of the Entropy Solution 177
B Licença 184
B.1 Sobre a licença dessa obra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184