Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws

We present a novel implementation of the modal DG method for hyperbolic conservation laws in two dimensions on graphics processing units (GPUs) using NVIDIA's Compute Unified Device Architecture. Both flexible and highly accurate, DG methods accommodate parallel architectures well as their discontinuous nature produces element‐local approximations. High‐performance scientific computing suits GPUs well, as these powerful, massively parallel, cost‐effective devices have recently included support for double‐precision floating‐point numbers. Computed examples for Euler equations over unstructured triangle meshes demonstrate the effectiveness of our implementation on an NVIDIA GTX 580 device. Profiling of our method reveals performance comparable with an existing nodal DG‐GPU implementation for linear problems. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient local time-stepping scheme for solution of nonlinear conservation laws

We develop an efficient local time-stepping algorithm for the method of lines approach to numerical solution of transient partial differential equations. The need for local time-stepping arises when adaptive mesh refinement results in a mesh containing cells of greatly different sizes. The global CFL number and, hence, the global time step, are defined by the smallest cell size. This can be inefficient as a few small cells may impose a restrictive time step on the whole mesh. A local time-stepping scheme allows us to use the local CFL number which reduces the total number of function evaluations. The algorithm is based on a second order Runge–Kutta time integration. Its important features are a small stencil and the second order accuracy in the L² and L^∞ norms.     

Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems

We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h ^2p+1) compared to a global rate of O(h ^p+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.