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Front tracking for scalar balance equations
 J. Hyperbolic Differ. Equ
"... Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the f ..."
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Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and nonstiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.
Asymptotic Highorder schemes for integrodifferential problems arising in markets with jumps
, 2006
"... In this paper we deal with the numerical approximation of integrodifferential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are highorder accurate for large time regimes. Therefore ..."
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In this paper we deal with the numerical approximation of integrodifferential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are highorder accurate for large time regimes. Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic highorder schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.
A Numerical Scheme for Axisymmetric Elastic Waves in Solids
"... This paper is dedicated to Professor Alan Jeffrey, University of Newcastle Upon Tyne, on the occasion of his 65th birthday. 1 Some analytical solutions were obtained, e.g. by Laturelle [1,2] for a half space using Laplace and Hankel transformations, and by Miklowitz [3] for a rod using the approxim ..."
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This paper is dedicated to Professor Alan Jeffrey, University of Newcastle Upon Tyne, on the occasion of his 65th birthday. 1 Some analytical solutions were obtained, e.g. by Laturelle [1,2] for a half space using Laplace and Hankel transformations, and by Miklowitz [3] for a rod using the approximate MindlinHerrmann theory. These show that the source term complicates the analytical solving of the system in a way that does not occur in the plane problem. Therefore a numerical method giving a good approximate solution will be very useful for practical applications. Hirose and Achenbach [4] developed a timedomain boundary element method to study the elastic wave interaction in an axisymmetric body. For plane problems without a source term, Lin and Ballmann [58] have extended the Godunovtype characteristicbased finite difference methods of gasdynamics for stress waves in elasticplastic solids and have obtained a great number of results. Problems with cylindrical symmetry can be treated in a similar way. Nevertheless, the existing numerical schemes dealing with hyperbolic systems with a source term seem not yet as welldeveloped as those for the systems without a source term. A widely used method is the time splitting technique which alternately solves a system of conservation laws without any source term and a system of ordinary differential equations modeling the source effect. However, it seems that this technique can produce misleading results, see Westenberger and Ballmann [9]. For onedimensional problems, some promising efforts were made by Glimm et al [10], Glaz and Liu [11] and Roe [12]. In this paper, we first propose an explicit finite difference scheme for the numerical integration of hyperbolic PDEs with a source term. Then, this scheme is applied to so...
ON WELLBALANCED FINITE VOLUME METHODS FOR NONCONSERVATIVE NONHOMOGENEOUS HYPERBOLIC SYSTEMS∗
"... Abstract. In this work we introduce a general family of finite volume methods for nonhomogeneous hyperbolic systems with nonconservative terms. We prove that all of them are “asymptotically wellbalanced”: they preserve all smooth stationary solutions in all the domain except for a set whose measur ..."
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Abstract. In this work we introduce a general family of finite volume methods for nonhomogeneous hyperbolic systems with nonconservative terms. We prove that all of them are “asymptotically wellbalanced”: they preserve all smooth stationary solutions in all the domain except for a set whose measure tends to zero as Δx tends to zero. This theory is applied to solve the bilayer shallowwater equations with arbitrary crosssection. Finally, some numerical tests are presented for simplified but meaningful geometries, comparing the computed solution with approximated asymptotic analytical solutions.
Fluxgradient and source term balancing for certain high resolution shockcapturing schemes
, 2006
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Numerical Techniques for Conservation Laws with Source Terms, MSc Dissertation
, 1998
"... In this dissertation we will discuss the finite difference method for approximating conservation laws with a source term present which is considered to be a known function of x, t and u. Finite difference schemes for approximating conservation laws without a source term present are discussed and are ..."
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In this dissertation we will discuss the finite difference method for approximating conservation laws with a source term present which is considered to be a known function of x, t and u. Finite difference schemes for approximating conservation laws without a source term present are discussed and are adapted to approximate conservation laws with a source term present. First we consider the source term to be a function of x and t only and then we consider the source term to be a function of u also. Some numerical results of the different approaches are discussed throughout the dissertation and an overall comparison of
New NAG Library Software Partial Differential Equations for FirstOrder
"... New NAG Fortran Library routines are described for the solution of systems of nonlinear, firstorder, timedependent partial differential equations in one space dimension, with scope for coupled ordinary differential or algebraic equations. The methodoflines is used with spatial discretization by ..."
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New NAG Fortran Library routines are described for the solution of systems of nonlinear, firstorder, timedependent partial differential equations in one space dimension, with scope for coupled ordinary differential or algebraic equations. The methodoflines is used with spatial discretization by either the centraldifference Keller box scheme or an upwind scheme for hyperbolic systems of conservation laws. The new routines have the same structure as existing library routines for the solution of secondorder partial differential equations, and much of the existing library software is reused. Results are presented for several computational examples to show that the software provides physically realistic numerical solutions to a challenging class of problems.
An Eulerian Finite Volume solver for multimaterial fluid flows with cylindrical symmetry
"... In this paper, we adapt a preexisting 2D cartesian cell centered finite volume solver to treat the compressible 3D Euler equations with cylindrical symmetry. We then extend it to multimaterial flows. Assuming cylindrical symmetry with respect to the z axis (i.e. all the functions do not depend ex ..."
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In this paper, we adapt a preexisting 2D cartesian cell centered finite volume solver to treat the compressible 3D Euler equations with cylindrical symmetry. We then extend it to multimaterial flows. Assuming cylindrical symmetry with respect to the z axis (i.e. all the functions do not depend explicitly on the angular variable θ), we obtain a set of 5 conservation laws with source terms that can be decoupled in 2 systems solved on a 2D orthogonal mesh in which a cell as a torus geometry. A specific upwinding treatment of the source term is required and implemented for the stationary case. Test cases will be presented for vanishing and nonvanishing azimuthal velocity uθ.
Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts
"... We study ordinary nonlinear differential equations which arise from steady nonlinear conservation laws with source terms. Two examples of conservation laws which lead to these equations are the SaintVenant and the Euler equations. In each case there is a reduction to a scalar equation and we use th ..."
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We study ordinary nonlinear differential equations which arise from steady nonlinear conservation laws with source terms. Two examples of conservation laws which lead to these equations are the SaintVenant and the Euler equations. In each case there is a reduction to a scalar equation and we use the ideas of upwinding and discretisation of source terms to devise methods for the solution. Numerical results are presented with both the EngquistOsher and the Roe scheme with different strategies for discretising the source terms based on balance ideas. Acknowledgements Firstly, I would like to express my gratitude to Professor Mike Baines. His supervision, support and patience were constant throughout this work and encouraged me to go on. My thanks go also to Professor Nancy Nichols. Her supervision and advice were very helpful. As a team, their supervision complemented each other and I benefited from their knowledge and teaching. I am grateful to my sponsors in Portugal, Funda ção para a Ciência e a Tecnologia (grant PRAXIS XXI/BD/15905/98 from the Subprograma Ciência e Tecnologia do 2o Quadro Comunitário de Apoio, andtheEscola Superior de Tecnologia e Gestão from the Instituto Politécnico de Leiria, who made this project viable. I wish to thank the help of staff and colleagues in the Mathematics Department in Reading who always made me feel welcome. Studying in Department of Mathematics of the University of Reading afforded the opportunity to learn with very good teachers and to meet colleagues and fellow researchers. In Reading, I met new friends and their friendship and support were very important in making me feel less lonely. I would like to thank especially Jessica, Ana Teresa, Hussain and Giovanni. We shared very happy moments that I will cherish forever. I extend my thanks to Helena, who made my stay in the University of Reading possible, and to Fernando, Cacilda, Teresa Mota and Cristine and other members of the Brazilian and Portuguese Speakers Society. Among the friends I met in in the Mathematics Department, I will remember with