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370
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 160 (19 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...
Nonoscillatory central differencing for hyperbolic conservation laws
 J. Comput. Phys
, 1990
"... Many of the recently developed highresolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the fieldbyfield decomposition which is required in orde ..."
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Cited by 151 (30 self)
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Many of the recently developed highresolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the fieldbyfield decomposition which is required in order to identify the “direction of the wind. ” Instead, we propose to use as a building block the more robust LaxFriedrichs (LxF) solver. The main advantage is simplicity: no Riemann problems are solved and hence fieldbyfield decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. We compensate for it by using highresolution MUSCLtype interpolants. Numerical experiments show that the quality of the results obtained by such convenient central differencing is comparable with those of the upwind schemes. c○Academic Press, Inc.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 125 (17 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
New HighResolution Central Schemes for Nonlinear Conservation Laws and ConvectionDiffusion Equations
 J. Comput. Phys
, 2000
"... this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution ..."
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Cited by 100 (14 self)
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this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semidiscrete formulation
Nonoscillatory Central Schemes For Multidimensional Hyperbolic Conservation Laws
 SIAM J. Sci. Comput
, 1998
"... We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory ..."
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Cited by 80 (14 self)
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We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewiselinear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a secondorder, nonoscillatory central scheme, a natural extension of the onedimensional secondorder central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408448]. As in the onedimensional case, the main feature of our twodimensional scheme is simplicity. In particular, this central scheme does not require the intricate and timeconsuming (approximate) Riemann solvers which are essential for the highresolution upwind schemes; in fact, even the com...
The development of discontinuous Galerkin methods
, 1999
"... In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational ..."
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Cited by 79 (13 self)
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In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible NavierStokes equations, and HamiltonJacobilike equations.
Analysis and Design of Numerical Schemes for Gas Dynamics 1 Artificial Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence
 INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS
, 1995
"... The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, w ..."
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Cited by 78 (40 self)
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The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for onedimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multidimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with a modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.
A ‘‘vertically Lagrangian’’ finitevolume dynamical core for global models
 Weather Rev
, 2004
"... A finitevolume dynamical core with a terrainfollowing Lagrangian controlvolume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow wate ..."
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Cited by 74 (9 self)
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A finitevolume dynamical core with a terrainfollowing Lagrangian controlvolume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow water system. The 2D horizontaltoLagrangiansurface transport and dynamical processes are then discretized using the genuinely conservative fluxform semiLagrangian algorithm. Time marching is splitexplicit, with large time steps for scalar transport, and small fractional steps for the Lagrangian dynamics, which permits the accurate propagation of fast waves. A mass, momentum, and total energy conserving algorithm is developed for remapping the state variables periodically from the floating Lagrangian controlvolume to an Eulerian terrainfollowing coordinate for dealing with ‘‘physical parameterizations’ ’ and to prevent severe distortion of the Lagrangian surfaces. Deterministic baroclinic wavegrowth tests and longterm integrations using the Held–Suarez forcing are presented. Impact of the monotonicity constraint is discussed. 1.
A Massively Parallel Adaptive Finite Element Method with Dynamic Load Balancing
 Appl. Numer. Math
, 1993
"... We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta meth ..."
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Cited by 70 (13 self)
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We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We demonstrate parallel efficiency through computations on a 1024processor nCUBE/2 hypercube. We present results using adaptiverefinement to reduce the computational cost of the method, and tiling, a dynamic, elementbased data migration system that maintains global load balance of the adaptive method by overlapping neighborhoods of processors that each perform local balancing. 1. Introduction We are studying massively parallel adaptive finite element methods for solving systems ofdimensional hyper...
Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
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