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145
The Relaxation Schemes for Systems of Conservation Laws in Arbitrary Space Dimensions
 Comm. Pure Appl. Math
, 1995
"... We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with ..."
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Cited by 250 (21 self)
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We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solver temporally. Numerical results for 1D and 2D problems are presented. The second order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. 1. Introduction In this paper we present a class of nonoscillatory numerical schemes for systems of conservation laws in several space dimensions. The basic idea is to use a local relaxation approximation. For any given system of conservation laws, we will construct a corresponding linear hyp...
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 79 (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...
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 73 (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 ...
QuasiMonotone Advection Schemes Based on Explicit Locally Adaptive Dissipation
, 1998
"... The authors develop and test computational methods for advection of a scalar field that also include a minimal dissipation of its variance in order to preclude the formation of false extrema. Both of these properties are desirable for advectively dominated geophysical flows, where the relevant scala ..."
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Cited by 67 (4 self)
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The authors develop and test computational methods for advection of a scalar field that also include a minimal dissipation of its variance in order to preclude the formation of false extrema. Both of these properties are desirable for advectively dominated geophysical flows, where the relevant scalars are both potential vorticity and material concentrations. These methods are based upon the sequential application of two types of operators: 1) a conservative and nondissipative (i.e., preserving first and second spatial moments of the scalar field), directionally symmetric advection operator with a relatively high order of spatial accuracy; and 2) a locally adaptive correction operator of lower spatial accuracy that eliminates false extrema and causes dissipation. During this correction phase the provisional distribution of the advected quantity is checked against the previous distribution, in order to detect places where the previous values were overshot, and thus to compute the excess. Then an iterative diffusion procedure is applied to the excess field in order to achieve approximate monotone behavior of the solution. In addition to the traditional simple flow tests, we have made longterm simulations of freely evolving twodimensional turbulent flow in order to compare the performance of the proposed technique with that of previously known algorithms, such as UTOPIA and FCT. This is done for both advection of vorticity and passive scalar. Unlike the simple test flows, the turbulent flow provides nonlinear cascades of quadratic moments of the advected
A HighOrder Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows
, 1997
"... We present a numerical method for computing solutions of the incompressible Euler or NavierStokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties. The method is based on a secondorder projection method for variable d ..."
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Cited by 54 (8 self)
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We present a numerical method for computing solutions of the incompressible Euler or NavierStokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties. The method is based on a secondorder projection method for variable density flows using an "approximate projection" formulation. The boundary between the fluids is tracked with a secondorder, volumeoffluid interface tracking algorithm. We present results for viscous RayleighTaylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm. We also present computational results for the RayleighTaylor instability in airhelium and for bubbles and drops in an airwater system without interfacial tension to demonstrate the behavior of the algorithm on problems with larger density and viscosity contrasts. 1. Introduction Fluid flows with free surfaces or material interfaces occur in a large number of natural and ...
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 52 (10 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 cellcentered adaptive projection method for the incompressible NavierStokes equations
"... We present an algorithm to compute adaptive solutions for incompressible flows using blockstructured local refinement in both space and time. This method uses a projection formulation based on a cellcentered approximate projection, which allows the use of a single set of cellcentered solvers. Bec ..."
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Cited by 51 (14 self)
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We present an algorithm to compute adaptive solutions for incompressible flows using blockstructured local refinement in both space and time. This method uses a projection formulation based on a cellcentered approximate projection, which allows the use of a single set of cellcentered solvers. Because of refinement in time, additional steps are taken to accurately discretize the advection and projection operators at grid refinement boundaries using composite operators which span the coarse and refined grids. This ensures that the method is approximately freestream preserving and satisfies an appropriate form of the divergence constraint. c ○ 2000 Academic Press Key Words: adaptive mesh refinement; incompressible flow; projection methods. 1.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
"... ..."
A Cartesian grid projection method for the incompressible Euler equations in complex geometries
 SIAM J. Sci. Comput
, 1998
"... Abstract. Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating timedependent incompressible inviscid flow which combines a projection method with a “Cartesian grid ” approach for representing ..."
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Cited by 44 (6 self)
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Abstract. Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating timedependent incompressible inviscid flow which combines a projection method with a “Cartesian grid ” approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volumeoffluid representation. A redistribution procedure is used to eliminate timestep restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the timedependent algorithm in two dimensions. The method is also demonstrated on flow past a halfcylinder with vortex shedding. Key words. Cartesian grid, projection method, incompressible Euler equations
A Projection Method for Locally Refined Grids
, 1996
"... A numerical method for the solution of the twodimensional Euler equations for incompressible flow on locally refined grids is presented. The method is a second order Godunovprojection method adapted from Bell, Colella, and Glaz. Second order accuracy of the numerical method in time and space is ..."
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Cited by 41 (2 self)
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A numerical method for the solution of the twodimensional Euler equations for incompressible flow on locally refined grids is presented. The method is a second order Godunovprojection method adapted from Bell, Colella, and Glaz. Second order accuracy of the numerical method in time and space is established through numerical experiments. The main contributions of this work concern the formulation and implementation of a projection for refined grids. A discussion of the adjointness relation between gradient and divergence operators for a refined grid MAC projection is presented, and a refined grid approximate projection is developed. An efficient multigrid method which exactly solves the projection is developed, and a method for casting certain approximate projections as MAC projections on refined grids is presented. Subject Classification Index Numbers: 65M06, 65M12, 76D05 Keywords: Incompressible Flow, Mesh Refinement, Projection Methods, Godunov Methods As a draft, thi...