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61
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...
An immersed interface method for incompressible navierstokes equations
 SIAM J. Sci. Comput
, 2003
"... Abstract. The method developed in this paper is motivated by Peskin’s immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discr ..."
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Cited by 48 (0 self)
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Abstract. The method developed in this paper is motivated by Peskin’s immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the twodimensional incompressible Navier–Stokes equations using a highresolution finitevolume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included.
Adaptive mesh refinement using wavepropagation algorithms for hyperbolic systems
 SIAM J. Numer. Anal
, 1998
"... Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ highresolution wavepropagation algorithms in a more general framework. This allows its use on a ..."
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Cited by 44 (6 self)
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Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ highresolution wavepropagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available.
Approximate Solutions of Nonlinear Conservation Laws and Related Equations
, 1997
"... During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical ..."
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Cited by 34 (11 self)
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During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^1compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools  the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finitedifference schemes; error estimates derived from the onesided stability of Godunovtype methods for convex conservation laws (and their multidimensional analogue  viscosity solutions of demiconcave HamiltonJacobi equations); we outline, in the onedimensional case, the convergence proof of finiteelement streamlinediffusion and spectral viscosity schemes based on the divcurl lemma; we also address the questions of convergence and error estimates for multidimensional finitevolume schemes on nonrectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finitevolume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.
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 28 (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 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.
A Wave Propagation Method for Three Dimensional Hyperbolic Problems
, 1996
"... A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. ..."
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Cited by 23 (5 self)
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A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse direction to model crossderivative terms. Due to proper upwinding, the method is stable for Courant numbers up to one. Several examples using the Euler equations are included.
Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations I. The conformal field equations
, 1997
"... This is the first in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to physical problems and why there is good hope that this mig ..."
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Cited by 20 (6 self)
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This is the first in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to physical problems and why there is good hope that this might even be a good idea from the numerical point of view. We describe in detail the derivation of the conformal field equations in the spinor formalism which we use for the implementation of the equations, and present all the equations as a reference for future work. Finally, we discuss the implications of the assumptions of a continuous symmetry. 1 Introduction Much of the current work in numerical and experimental relativity is devoted to obtaining information about the gravitational radiation that is emitted by astrophysical processes which are taking place in our universe. The goal is to obtain wave forms, i.e., the "finger prints" by which different processes can be identified when...
Construction and Application of an AMR Algorithm for Distributed Memory Computers
 PROC. OF CHICAGO WORKSHOP ON ADAPTIVE MESH REFINEMENT METHODS
, 2003
"... ... In this paper, a localitypreserving domain decomposition is proposed that partitions the entire AMR hierarchy from the base level on. It is shown that the approach reduces the communication costs and simplifies the implementation. Emphasis is put on the effective parallelization of the flux cor ..."
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Cited by 20 (11 self)
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... In this paper, a localitypreserving domain decomposition is proposed that partitions the entire AMR hierarchy from the base level on. It is shown that the approach reduces the communication costs and simplifies the implementation. Emphasis is put on the effective parallelization of the flux correction procedure at coarsefine boundaries, which is indispensable for conservative finite volume schemes. An easily reproducible standard benchmark and a highly resolved parallel AMR simulation of a diffrracting hydrogenoxygen detonation demonstrate the proposed strategy in practice.
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 12 (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...