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56
The Local Discontinuous Galerkin Method For TimeDependent ConvectionDiffusion Systems
"... In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, timedependent convectiondiffusion systems. These methods are an extension of the RungeKutta Discontinuous Galerkin methods for purely hyperbolic systems to convectiondiffusion systems and share with those methods the ..."
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Cited by 119 (18 self)
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In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, timedependent convectiondiffusion systems. These methods are an extension of the RungeKutta Discontinuous Galerkin methods for purely hyperbolic systems to convectiondiffusion systems and share with those methods their high parallelizability, their highorder formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L2stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.
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.
A Discontinuous Galerkin Finite Element Method For HamiltonJacobi Equations
 SIAM J. Sci. Comput
"... . In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear HamiltonJacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated g ..."
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Cited by 52 (10 self)
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. In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear HamiltonJacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. At least k th order of accuracy is observed for smooth problems when kth degree polynomials are used, and derivative singularities are resolved well without oscillations even without limiters. Key words. HamiltonJacobi Equations, discontinuous Galerkin, high order accuracy. AMS subject classifications. 65M60, 70H20 1. Introduction. In this paper we consider the numerical solutions of HamiltonJacobi (HJ) equations ' t + H(' x1 ; :::; ...
Adaptive Local Refinement with Octree LoadBalancing for the Parallel Solution of ThreeDimensional Conservation Laws
 J. Parallel Distrib. Comput
, 1997
"... Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive spacetime mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to method ..."
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Cited by 45 (16 self)
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Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive spacetime mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalaces, introduced at adaptive enrichment steps, are corrected by using traversals of an octtee representing a spatial decomposition of the domain. To accommodate the variable time steps, octtee partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octtee ad inertial partitioning procedures ae presented for the threedimensional Euler equations of compressible flow solved on an IBM SP2 computer.
Parallel Structures and Dynamic Load Balancing for Adaptive Finite Element Computation
 Applied Numerical Mathematics
, 1996
"... this paper, we have focused on describing and comparing several load balancing schemes. Comparisons by timing are difficult, since times vary between runs having the same parameters. The highspeed switch of the IBM SP2 computer is a shared resource that affects run times. More subtle effects can re ..."
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Cited by 39 (12 self)
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this paper, we have focused on describing and comparing several load balancing schemes. Comparisons by timing are difficult, since times vary between runs having the same parameters. The highspeed switch of the IBM SP2 computer is a shared resource that affects run times. More subtle effects can result from differences in the order in which messages used for migration are processed. Changes in the order in which those messages are received and integrated into the local MDB result in different traversal orders of the mesh entities. These differences cause small changes in load balancings and coarsenings. While such differences in meshes and partitionings do not affect the solution accuracy, they can cause sufficient changes in efficiency to make precise timings difficult. Qualitatively, PSIRB produced the best partitions (measured as a function of total analysis time). Octreegenerated partitions were comparable but resulted in slightly longer solution times. In both cases, one or two iterations of partition boundary smoothing led to a quality improvement. ITB by itself resulted in poorer partition quality, but is useful when mesh changes are small between computational stages. Predictive enrichment provided su21 perior performance to our current enrichment process with transient problems where there are frequent enrichment and balancing steps. Enhancements to the existing load balancing procedures and the implementation of new ones are under investigation. Improvements in the slicebyslice technique used by ITB for migration are necessary. Experiments with geometrical methods that use the spatial location of elements relative to the centroids of sending and receiving processors showed promise at reducing the number of processor interconnections. Vidwans et al. [39] pr...
QuadratureFree Implementation Of The Discontinuous Galerkin Method For Hyperbolic Equations
 AIAA Journal
, 1996
"... Introduction Computational methods for aeroacoustics must possess accuracy properties that exceed those of conventional secondorder computational fluid dynamics (CFD) methods. At the same time, many problems of interest involve complex geometries that are not easily treated by common highorder met ..."
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Cited by 28 (11 self)
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Introduction Computational methods for aeroacoustics must possess accuracy properties that exceed those of conventional secondorder computational fluid dynamics (CFD) methods. At the same time, many problems of interest involve complex geometries that are not easily treated by common highorder methods that usually require a smooth, structured grid. In addition to the geometrically complex problem, we are particularly interested in strongly nonlinear flows that contain shock waves as a major source of sound generation, such as in the case of jet noise. In an effort to satisfy these requirements, the relatively untried discontinuous Galerkin (DG) method is being tested for hyperbolic problems. Some advantages of this approach include the ease with which the method can be applied to both structured and unstructured grids and its suitability for parallel computer architectures. The approach also has several useful mathematical properties. Johnson and Pitkarata 1 proved
A Hierarchical Partition Model for Adaptive Finite Element Computation
 Comput. Methods Appl. Mech. Engrg
, 1998
"... Introduction The finite element method (FEM) has become a standard analysis tool for solving partial differential equations (PDEs). Computationally demanding threedimensional problems make adaptive methods and parallel computation essential. Adaptive FEMs provide reliability, robustness, and time an ..."
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Cited by 23 (5 self)
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Introduction The finite element method (FEM) has become a standard analysis tool for solving partial differential equations (PDEs). Computationally demanding threedimensional problems make adaptive methods and parallel computation essential. Adaptive FEMs provide reliability, robustness, and time and space efficiency. In such a method, the computational domain is discretized into a mesh. During the adaptive solution process, portions of the mesh may be refined or coarsened (hrefinement) or moved to follow evolving phenomena (rrefinement). The method order may also be varied (prefinement). Each adaptive process concentrates the computational effort in areas where the solution resolution would otherwise be inadequate [7]. Conventional arraybased data representations, which work well for fixedmesh solutions, are not wellsuited to solutions involving mesh adaptivity [1]. Traversal of the data must be efficient in all cases, but w
Parallel Adaptive hpRefinement Techniques for Conservation Laws
, 1996
"... We describe an adaptive hprefinement local finite element procedure for the parallel solution of hyperbolic systems of conservation laws on rectangular domains. The local finite element procedure utilizes spaces of piecewisecontinuous polynomials of arbitrary degree and coordinated explicit Run ..."
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Cited by 21 (10 self)
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We describe an adaptive hprefinement local finite element procedure for the parallel solution of hyperbolic systems of conservation laws on rectangular domains. The local finite element procedure utilizes spaces of piecewisecontinuous polynomials of arbitrary degree and coordinated explicit RungeKutta temporal integration. A solution limiting procedure produces monotonic solutions near discontinuities while maintaining highorder accuracy near smooth extrema. A modified tiling procedure maintains processor load balance on parallel, distributedmemory MIMD computers by migrating finite elements between processors in overlapping neighborhoods to produce locally balanced computations. Grids are stored in tree data structures, with finer grids being offspring of coarser ones. Within each grid, AVL trees simplify the transfer of information between neighboring processors and the insertion and deletion of elements as they migrate between processors. Computations involving Burger...
An adaptive discontinuous galerkin technique with an orthogonal basis applied to compressible flow problems
 SIAM Review
"... Abstract. We present a highorder formulation for solving hyperbolic conservation laws using the Discontinuous Galerkin Method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit RungeKutta time discretization. Some results of higherorder adaptive refinement cal ..."
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Cited by 21 (2 self)
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Abstract. We present a highorder formulation for solving hyperbolic conservation laws using the Discontinuous Galerkin Method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit RungeKutta time discretization. Some results of higherorder adaptive refinement calculations are presented for inviscid Rayleigh Taylor flow instability and shock reflexion problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities. Key words. Discontinuous Galerkin, adaptive meshing, Orthogonal Basis.
A Parallel hpAdaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws
 Appl. Numer. Math
, 1994
"... This paper describes a parallel algorithm based on discontinuous hpfinite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques ..."
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Cited by 17 (0 self)
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This paper describes a parallel algorithm based on discontinuous hpfinite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent, thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems. 1