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Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 54 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...
ZPL: A Machine Independent Programming Language for Parallel Computers
 IEEE Transactions on Software Engineering
, 2000
"... The goal of producing architectureindependent parallel programs is complicated by the competing need for high performance. The ZPL programming language achieves both goals by building upon an abstract parallel machine and by providing programming constructs that allow the programmer to "see" thi ..."
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Cited by 25 (3 self)
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The goal of producing architectureindependent parallel programs is complicated by the competing need for high performance. The ZPL programming language achieves both goals by building upon an abstract parallel machine and by providing programming constructs that allow the programmer to "see" this underlying machine. This paper describes ZPL and provides a comprehensive evaluation of the language with respect to its goals of performance, portability, and programming convenience. In particular, we describe ZPL's machineindependent performance model, describe the programming benefits of ZPL's regionbased constructs, summarize the compilation benefits of the language's highlevel semantics, and summarize empirical evidence that ZPL has achieved both high performance and portability on diverse machines such as the IBM SP2, Cray T3E, and SGI Power Challenge. Index Terms: portable, efficient, parallel programming language. This research was supported by DARPA Grant F306029710152, a grant of HPC time from the Arctic Region Supercomputing Center, NSF Grant CCR9707056, and ONR grant N000149910402. 1 1
A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes
 J. Comput. Phys
, 2001
"... We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of appro ..."
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Cited by 23 (5 self)
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We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
Finite Volume Methods for Nonlinear Elasticity in Heterogeneous Media
, 2001
"... An approximate Riemann solver is developed for the equations of nonlinear elasticity in a heterogeneous medium, where each grid cell has an associated density and stressstrain relation. The nonlinear flux function is spatially varying and a wave decomposition of the flux difference across a cell in ..."
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Cited by 7 (3 self)
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An approximate Riemann solver is developed for the equations of nonlinear elasticity in a heterogeneous medium, where each grid cell has an associated density and stressstrain relation. The nonlinear flux function is spatially varying and a wave decomposition of the flux difference across a cell interface is used to approximate the wave structure of the Riemann solution. This solver is used in conjunction with a highresolution finitevolume method using the CLAWPACK software. As a test problem, elastic waves in a periodic layered medium are studied. Dispersive effects from the heterogeneity, combined with the nonlinearity, lead to solitary wave solutions that are well captured by the numerical method.
Noelle, Xing, Shu. Wellbalanced schemes for moving water. 1 High Order Wellbalanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water
"... A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of th ..."
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A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. In most cases, applications treated equilibria at rest, where the flow velocity vanishes. Here we present a new very high order accurate, exactly wellbalanced finite volume scheme for moving flow equilibria. Numerical experiments show excellent resolution of unperturbed as well as slightly perturbed equilibria.
3 High Order Wellbalanced Finite Volume Scheme 9
, 2007
"... A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of th ..."
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A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. In most cases, applications treated equilibria at rest, where the flow velocity vanishes. Here we present a new very high order accurate, exactly wellbalanced finite volume scheme for moving flow equilibria. Numerical experiments show excellent resolution of unperturbed as well as slightly perturbed equilibria.
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