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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 78 (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...
A wavepropagation method for conservation laws and balance laws with spatially varying flux functions
 SIAM J. Sci. Comput
, 2002
"... Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a ge ..."
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Cited by 42 (5 self)
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Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A highresolution wavepropagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)− fi−1(Qi−1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be secondorder accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws qt + f(q, x)x = ψ(q, x) are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasisteady problems close to steady state. Key words. finitevolume methods, highresolution methods, conservation laws, source terms, discontinuous flux functions AMS subject classifications. 65M06, 35L65 PII. S106482750139738X
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 27 (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.
Wave Propagation Methods for Conservation Laws with Source Terms
 In preparation
, 1998
"... . An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm ..."
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Cited by 14 (3 self)
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. An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm was developed and used to compute nearsteady shallow water flow over variable topography. In this paper we extend this algorithm to nearsteady flow of an ideal gas subject to a static gravitational field. The method is implemented in the software package clawpack. The ability of this method to capture perturbed quasisteady solutions is demonstrated with numerical examples. 1. Introduction We consider the Euler equations in conservation form @ t q +r \Delta f (q) = / (q) (1) where q 2 R m is a vector of conserved quantities, f : R m ! R m is the flux, and / is a source term due to a static gravitational field. It is well known that if f is a nonlinear function of q as for the Eule...
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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Cited by 4 (2 self)
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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.
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
HIGHRESOLUTION FINITE VOLUME METHODS FOR THE SHALLOW WATER EQUATIONS WITH
"... We give a brief review of the wavepropagation algorithm, a highresolution finite volume method for solving hyperbolic systems of conservation laws. These methods require a Riemann solver to resolve the jump in variables at each cell interface into waves. We present a Riemann solver for the shallow ..."
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We give a brief review of the wavepropagation algorithm, a highresolution finite volume method for solving hyperbolic systems of conservation laws. These methods require a Riemann solver to resolve the jump in variables at each cell interface into waves. We present a Riemann solver for the shallow water equations that works robustly with bathymetry and dry states. This method is implemented in clawpack and applied to benchmark problems from the Third International Workshop on LongWave Runup Models, including a twodimensional simulation of runup during the 1993 tsunami event on Okushiri Island. Comparison is made with wave tank experimental data provided for the workshop. Some preliminary results using adaptive mesh refinement on the 26 December 2004 Sumatra event are also presented. 1.