In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobilike equations.

Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

Runge-Kutta discontinuous Galerkin methods for convection-dominated problems

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 Navier-Stokes equations, and Hamilton-Jacobi-like equations.

/pdf/the-development-of-discontinuous-galerkin-methods-17hxfhour7.pdf

The Development of Discontinuous Galerkin Methods

This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.

/pdf/discontinuous-galerkin-methods-2ilov2v146.pdf

Discontinuous Galerkin methods

High resolution schemes for hyperbolic conservation laws

Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws

Parallel structures and dynamic load balancing for adaptive finite element computation

Sparse matrix-vector multiplication forms the heart of iterative linear solvers used widely in scientific computations (e.g., finite element methods). In such solvers, the matrix-vector product is computed repeatedly, often thousands of times, with updated values of the vector until convergence is achieved. In an SIMD architecture, each processor has to fetch the updated off-processor vector elements while computing its share of the product. In this paper, we report on run-time optimization of array distribution and off-processor data fetching to reduce both the communication and computation time. The optimization is applied to a sparse matrix stored in a compressed sparse row-wise format. Actual runs on test matrices produced up to a 35 percent relative improvement over a block distribution with a naive multiplication algorithm while simulations over a wider range of processors indicate that up to a 60 percent improvement may be possible in some cases.

/pdf/run-time-optimization-of-sparse-matrix-vector-multiplication-2zij9hbbzb.pdf

Run-Time Optimization of Sparse Matrix-Vector Multiplication on SIMD Machines

Conservation laws are solved by a local Galerkin finite element procedure with adaptive space-time mesh refinement and explicit time integration. A distributed octree structure representing a spatial decomposition of the domain is used for mesh generation, and later may be used to correct for processor load imbalances introduced at adaptive enrichment steps. A Courant 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. To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size. Computational results are presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer. The problem examined is the flow inside a perforated shock tube.

Distributed Octree Data Structures and Local Refinement Method for the Parallel Solution of Three-Dimensional Conservation Laws

We examine the quality of partitions produced by an iterative load balancer in parallel adaptive finite element calculations. We present several metrics which we use to evaluate the quality of a mesh partitioning, and report statistics generated from our analysis of adaptively refined meshes produced during the solution of computational fluid dynamics problems. Timings from the finite element solution phase for runs involving these meshes on 16 and 32 processors of an IBM SP2 are also presented.

Louis H. Ziantz

Papers

Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws

Parallel structures and dynamic load balancing for adaptive finite element computation

Run-Time Optimization of Sparse Matrix-Vector Multiplication on SIMD Machines

Distributed Octree Data Structures and Local Refinement Method for the Parallel Solution of Three-Dimensional Conservation Laws

The Quality of Partitions Produced by an Iterative Load Balancer