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198
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.
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...
A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows
 SIAM J. Sci. Comput
"... Abstract. We consider the SaintVenant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when com ..."
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Cited by 42 (4 self)
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Abstract. We consider the SaintVenant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the socalled wellbalanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a wellbalanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast wellbalanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.
SpaceTime Discontinuous Galerkin Finite Element Methods
"... In these notes an introduction is given to spacetime discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The spacetime DG discretization is explained in detail, including the definition of the numerical fluxes and stabilizati ..."
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Cited by 28 (4 self)
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In these notes an introduction is given to spacetime discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The spacetime DG discretization is explained in detail, including the definition of the numerical fluxes and stabilization operators necessary to maintain stable and nonoscillatory solutions. In addition, a pseudotime integration method for the solution of the algebraic equations resulting from the DG discretization and the relation between the spacetime DG method and an arbitrary Lagrangian Eulerian approach are discussed. Finally, a brief overview of some applications to aerodynamics is given.
Numerical Solution Of Reservoir Flow Models Based On Large Time Step Operator Splitting Algorithms
 FILTRATION IN POROUS MEDIA AND INDUSTRIAL APPLICATIONS, LECTURE NOTES IN MATHEMATICS
, 1999
"... During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multiphase flow in heterogeneous porous media. The purpose of these lecture notes is to revie ..."
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Cited by 25 (14 self)
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During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multiphase flow in heterogeneous porous media. The purpose of these lecture notes is to review some of this activity. We illustrate the main ideas behind these novel operator splitting algorithms for a basic twophase flow model. Special focus is posed on the numerical solution algorithms for the saturation equation, which is a convection dominated, degenerate convectiondiffusion equation. Both theory and applications are discussed. The general background for the reservoir flow model is reviewed, and the main features of the numerical algorithms are presented. The basic mathematical results supporting the numerical algorithms are also given. In addition, we present some results from the BV solution theory for quasilinear degenerate parabolic equations, which provides the correct ...
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.
Contact discontinuity capturing schemes for linear advection and compressible gas dynamics
 J. Sci. Comput
, 2002
"... Abstract We present a nondiffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the UltraBee limiter of [20], [24]. We prove for the UltraBee scheme a property of exact advection for a large s ..."
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Cited by 22 (6 self)
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Abstract We present a nondiffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the UltraBee limiter of [20], [24]. We prove for the UltraBee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e. piecewise constant) initial data, and state a conjecture of nondiffusion at infinite time based on some local overcompressivity of the scheme for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.
Entropy satisfying flux vector splittings and kinetic BGK models
, 2000
"... We establish forward and backward relations between entropy satisfying BGK models such as those introduced previously by the author and the first order flux vector splitting numerical methods for systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of ..."
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Cited by 15 (1 self)
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We establish forward and backward relations between entropy satisfying BGK models such as those introduced previously by the author and the first order flux vector splitting numerical methods for systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of entropies we can associate an entropy flux vector splitting. We prove that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes. We deduce a new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case. In particular, this gives a stability condition for numerical kinetic methods with noncompact velocity support. A new interpretation of general kinetic schemes is also provided via approximate Riemann solvers. We deduce the construction of finite velocity relaxation systems for gas dyn...
ADER schemes for threedimensional nonlinear hyperbolic systems
 J. Comput. Phys
, 2005
"... In this paper we carry out the extension of the ADER approach to multidimensional nonlinear systems of conservation laws. We implement nonlinear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order o ..."
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Cited by 15 (4 self)
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In this paper we carry out the extension of the ADER approach to multidimensional nonlinear systems of conservation laws. We implement nonlinear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order of accuracy and nonoscillatory properties of the new schemes. Compared to the stateofart finitevolume WENO schemes the ADER schemes are faster, more accurate and need less computer memory. Key words: highorder schemes, weighted essentially nonoscillatory, ADER, generalized Riemann problem, three space dimensions. 1 1