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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 56 (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...
Numerical Schemes For Hyperbolic Conservation Laws With Stiff Relaxation Terms
 J. Comput. Phys
, 1996
"... Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution meth ..."
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Cited by 56 (11 self)
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Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid. We introduce a modification of higher order Godunov methods that possesses the correct asymptotic behavior, allowing the use of coarse grids (large cell Peclet numbers). The idea is to build into the numerical scheme the asymptotic balances that lead to this behavior. Numerical experiments on 2 \Theta 2 systems verify our analysis. 1 Email address: jin@math.gatech.edu 2 Email address: lvrmr@math.arizona.edu Typeset by A M ST E X 2 1. Introduction Hyperbolic systems of partial differential equations that arise in applications ofter have re...
Global Solutions to the Compressible Euler Equations with Geometrical Structure
, 1995
"... We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especia ..."
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Cited by 21 (7 self)
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We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, j~xj 1, and to transonic nozzle flow. Arbitrary data with L 1 bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework. 1. Introduction We develop new mathematical existence theory and numerical schemes for global discontinuous solutions to the Euler equations of compressible isentropic gas dynamics with large initial data and with geometrical structure. The compressib...
Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary crosssections
, 2003
"... Abstract. We establish the existence and stability of multidimensional transonic flows with transonic shocks through an infinite nozzle of arbitrary crosssections, including a slowly varying de Lavel nozzle. The transonic flow is governed by the inviscid steady potential flow equation with superson ..."
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Cited by 12 (7 self)
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Abstract. We establish the existence and stability of multidimensional transonic flows with transonic shocks through an infinite nozzle of arbitrary crosssections, including a slowly varying de Lavel nozzle. The transonic flow is governed by the inviscid steady potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the infinite exit, and the slip boundary condition on the nozzle boundary. The multidimensional transonic nozzle problem is reformulated into a free boundary problem, for which the free boundary is a transonic shock dividing two phases of C1,α flow in the infinite nozzle, and the equation is hyperbolic in the supersonic upstream phase and elliptic in the subsonic downstream phase. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain and to solve the multidimensional transonic nozzle problem in a direct fashion. Our results indicate that, for the transonic nozzle problem, there exists a transonic flow such that the flow is divided into a C1,α subsonic flow up to the nozzle boundary in the unbounded downstream region from the supersonic upstream flow by a C1,α multidimensional transonic shock that is orthogonal to the nozzle boundary at every intersection point, and the uniform velocity state at the infinite exit in the downstream direction is uniquely determined by the supersonic upstream flow at the entrance which is sufficiently close to a uniform flow. The uniform velocity state at the exit can not be apriori prescribed from the corresponding pressure for such a flow to exist. We further prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance. 1.
Explicit CharacteristicBased HighResolution Algorithms For Hyperbolic Conservation Laws With Stiff Source Terms
, 1996
"... o TA a class while simultaneously taking it for credit. More importantly, for being an extremely valuable mentor, taking special care to introduce me to his colleagues. Further, it was he who provided the initial impetus for the work in Chapter V. 1 Whatever you do, do well. Even if you become a c ..."
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Cited by 7 (0 self)
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o TA a class while simultaneously taking it for credit. More importantly, for being an extremely valuable mentor, taking special care to introduce me to his colleagues. Further, it was he who provided the initial impetus for the work in Chapter V. 1 Whatever you do, do well. Even if you become a crook, just make sure you're a good one. 2 When one of my projects is going nowhere, I leave it (in the magic drawer) and work on a totally different project. When I return and start over, the answers "magically" jump out. iii Thanks to Professors Sichel, Van Leer and Powell for inviting me to Michigan. I have never regretted my decision  hopefully, they have never done so either. My sincere gratitude to Professors Roe, Van Leer, Sichel, Powell and Harabetian, for serving on my committee, for reading through my dissertation at very short notice, and for their valuable insights and comments. Special thanks go, first, to Rosemary, who quickl
Nonlinear Conservation Laws and Finite Volume Methods for Astrophysical Fluid Flow
 Computational Methods for Astrophysical Fluid Flow, 27th SaasFee Advanced Course Lecture Notes
, 1998
"... Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Cited by 7 (0 self)
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Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.3 Classification of differential equations : : : : : : : : : : : : : : : : : : : : : : : 7 2. Derivation of conservation laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 The Euler equations of gas dynamics : : : : : : : : : : : : : : : : : : : : : : : 13 2.2 Dissipative fluxes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Source terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.4 Radiative trans
The Reconstruction of Upwind Fluxes for Conservation Laws: Its Behavior in Dynamic and Steady State Calculations
, 1998
"... this paper, we consider a simple and efficient finite volume method for the Euler equation for compressible flows. The 1D system of the Euler equation is given by # t U + # x F(U ) = 0, U = 2 4 # m E 3 5 , F = 2 4 m #u 2 + p u(E + p) 3 5 , (1.1) where #,u,m=#u, and E are density, ..."
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Cited by 5 (3 self)
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this paper, we consider a simple and efficient finite volume method for the Euler equation for compressible flows. The 1D system of the Euler equation is given by # t U + # x F(U ) = 0, U = 2 4 # m E 3 5 , F = 2 4 m #u 2 + p u(E + p) 3 5 , (1.1) where #,u,m=#u, and E are density, velocity, momentum, and total energy, respectively, and the pressure is obtained from the equation of state, p = (# 1)(E#u 2 /2). The equation 1 Current address: Courant Institute, New York, NY 10012. 2 Current address: Institute for Physical Sciences and Technology, and Department of Mathematics, University of Maryland, College Park, MD 20742. Research was supported in part by NSF Grant DMS9505275. 237 00219991/98 $25.00 Copyright c # 1998 by Academic Press All rights of reproduction in any form reserved. 238 CHOI AND LIU in multidimensional space is given similarly. Most modern shock capturing schemes for the solution of Eq. (1.1) are of Godunovtype, which reconstructs the solution after fieldby field decomposition and solves a Riemann problem for time evolution. The wellknown Godunovtype schemes are MUSCL [30], PPM [32], and ENO schemes [7, 26]. We explore an efficient implementation by using a flux splitting for the time evolution for dynamic and steady state computations. In the flux splitting, F = F + + F  , the Jacobians of the split fluxes F have only positive or negative eigenvalues. That is, each split flux always keeps only one wind direction. For this reason, they are sometimes called upwind fluxes. The high order accuracy of the scheme is achieved by directly reconstructing each component of upwind fluxes by a piecewise polynomial. Then the numerical flux in the finite volume method is determined by evaluating the reconstructed upwind ...
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 3 (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.
A Robust and EntropySatisfying Numerical Scheme for Fluid Flows in Discontinuous Nozzles
, 2013
"... We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the crosssection and on a approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the us ..."
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Cited by 1 (1 self)
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We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the crosssection and on a approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous with arbitrary large jumps. To do so, we introduce in the first step of the relaxation solver a singular dissipation measure superposed on the standing wave which enables us to control the approximate speeds of sound and thus, the time step, even for extreme initial data. Keywords: Discontinuous nozzle flows, relaxation techniques, Riemann problem. AMS subject classifications: 76S05, 35L60, 35F55. 1
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...