Publications

Journal of Computational Physics

Banks, Jeffrey W.; Aslam, T.; Rider, William J.

A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t1/2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of Δt1/(p+1) which yields an estimate for the convergence rate as p/(p + 1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations. © 2008 Elsevier Inc.

Rider, William J.; Love, Edward L.; Voth, Thomas E.; Mosso, Stewart J.; Niederhaus, John H.; Strack, Otto E.; Robinson, Allen C.

Abstract not provided.

46th AIAA Aerospace Sciences Meeting and Exhibit

Robinson, Allen C.; Brunner, Thomas A.; Carroll, Susan; Richarddrake; Garasi, Christopher J.; Gardiner, Thomas; Haill, Thomas; Hanshaw, Heath; Hensinger, David; Labreche, Duane; Lemke, Raymond; Love, Edward; Luchini, Christopher; Mosso, Stewart; Niederhaus, John; Ober, Curtis C.; Petney, Sharon; Rider, William J.; Scovazzi, Guglielmo; Strack, O.E.; Summers, Randall; Trucano, Timothy; Weirs, V.G.; Wong, Michael; Voth, Thomas

ALEGRA is an arbitrary Lagrangian-Eulerian (multiphysics) computer code developed at Sandia National Laboratories since 1990. The code contains a variety of physics options including magnetics, radiation, and multimaterial flow. The code has been developed for nearly two decades, but recent work has dramatically improved the code's accuracy and robustness. These improvements include techniques applied to the basic Lagrangian differencing, artificial viscosity and the remap step of the method including an important improvement in the basic conservation of energy in the scheme. We will discuss the various algorithmic improvements and their impact on the results for important applications. Included in these applications are magnetic implosions, ceramic fracture modeling, and electromagnetic launch. Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc.