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689
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...
Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
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Aspects of unstructured grids and finitevolume solvers for the Euler and NavierStokes equations
, 1992
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Third Order Nonoscillatory Central Scheme For Hyperbolic Conservation Laws
"... . A thirdorder accurate Godunovtype scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: #1. A nonoscillatory piecewisequadratic reconstruction of pointvalues from their given cell averages; and #2. A central differencing ..."
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Cited by 66 (15 self)
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. A thirdorder accurate Godunovtype scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: #1. A nonoscillatory piecewisequadratic reconstruction of pointvalues from their given cell averages; and #2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a thirdorder central scheme, an extension along the lines of the secondorder central scheme of Nessyahu and Tadmor [NT]. The scalar scheme is nonoscillatory (and hence  convergent), in the sense that it does not increase the number of initial extrema (as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, fieldbyfield characteristic decompositions, etc., are required. Numerical experiments confirm the highresolution content of...
Numerical Methods For Hyperbolic Conservation Laws With Stiff Relaxation I. Spurious Solutions
 SIAM J. Sci. Comput
, 1992
"... . We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the ..."
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Cited by 61 (2 self)
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. We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system. A second criterion is that a certain "subcharacteristic" condition be satisfied by the hyperbolic system. We support our conjecture with analytical and numerical results for a specific example, the solution of generalized Riemann problems of a model system of equations with a fractional step scheme in which Godunov's method is coupled with the backward Euler method. We also apply our ideas to the numerical solution of stiff detonation problems. 1. Introduction. Hyperbolic systems of conservation laws with relaxation ...
An Adaptive Cartesian Grid Method For Unsteady Compressible Flow In Irregular Regions
 J. Comput. Phys
, 1993
"... In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit secondorder Godunov algorithm fo ..."
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Cited by 61 (15 self)
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In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit secondorder Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the fluidbody interface is based on a volumeoffluid approach with a redistribution procedure to maintain conservation while avoiding time step restrictions arising from small cells where the boundary intersects the mesh. The single grid Cartesian mesh integration scheme is coupled to a conservative adaptive mesh refinement algorithm that selectively refines regions of the computational grid to achieve a desired level of accuracy. Examples showing the results of the combined Cartesian grid integration/adaptive mesh refinement algorithm for both two and threedimensional flows are presented. (This page intent...
Relaxation of energy and approximate Riemann solvers for general pressure laws in dynamics
 SIAM J. Num. Anal
, 1998
"... Abstract. We consider the Euler equations for a compressible inviscid fluid with a general pressure law p(ρ, ε), where ρ represents the density of the fluid and ε its specific internal energy. We show that it is possible to introduce a relaxation of the nonlinear pressure law introducing an energy d ..."
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Cited by 53 (6 self)
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Abstract. We consider the Euler equations for a compressible inviscid fluid with a general pressure law p(ρ, ε), where ρ represents the density of the fluid and ε its specific internal energy. We show that it is possible to introduce a relaxation of the nonlinear pressure law introducing an energy decomposition under the form ε = ε1 + ε2. The internal energy ε1 is associated with a (simpler) pressure law p1(ρ, ε1); the energy ε2 is advected by the flow. These two energies are also subject to a relaxation process and in the limit of an infinite relaxation rate, we recover the initial pressure law p. We show that, under some conditions of subcharacteristic type, for any convex entropy associated with the pressure p, we can find a global convex and uniform entropy for the relaxation system. From our construction, we also deduce the extension to general pressure laws of classical approximate Riemann solvers for polytropic gases, which only use a single call to the pressure law (per mesh point and time step). For the Godunov scheme, we show that this extension satisfies stability, entropy, and accuracy conditions.
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 50 (7 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.
RungeKutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms
 J. Comput. Phys
, 1995
"... Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally wellresolved. We design a second order RungeKutta type splitting method that posse ..."
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Cited by 50 (14 self)
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Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally wellresolved. We design a second order RungeKutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit, thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. Key words. Hyperbolic conservation laws with stiff relaxation, shock capturing difference method, RungeKutta methods, asymptotic limit AMS(MOS) subject classifications. 35L65, 35B40, 65M60 Typeset by A M ST E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [42,36], in relaxing gas flow with thermal and chemical nonequilibrium [41,9], in kinetic theory of ra...
HighResolution Nonoscillatory Central Schemes With Nonstaggered Grids For Hyperbolic Conservation Laws
 SIAM J. Numer. Anal
, 1998
"... We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recent ..."
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Cited by 49 (14 self)
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We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 18921917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemannsolverfree recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.