AMR for Hyperbolic Conservation Laws

Solution of systems of hyperbolic conservation laws was one of the first applications of adaptive mesh refinement. This is now a mature methodology which has been studied and used for almost two decades. However, the dynamic nature of adaptivity in time-dependent simulations makes it considerably more difficult to implement this type of methodology on modern parallel computers, particularly, distributed memory architectures. Over the past several years the Center for Computational Sciences and Engineering (CCSE) has developed and extended the AMR methodology for hyperbolic systems of conservation laws to support distributed-memory and shared-memory computing architectures. The work has resulted in several large-scale applications codes presently being used by CCSE researchers, others in the DOE community, and collaborators in DoD, academia and industrial research organizations.

In both examples shown here, the interaction of a shock with a gas bubble of a different density is calculated. In the first case, a Mach 1.25 shock in air hits an initially spherical bubble of helium. The density of the helium bubble is 0.139 times the density of the surrounding air, leading to acceleration of the shock as it enters the bubble and a subsequent generation of vorticity that dramatically deforms the bubble. In the two time snapshots shown here, the shock is incident from the left. The deformation of the bubble shows excellent agreement with laboratory experiments for the same configuration.

bubble 1 bubble 2

These numerical experiments were performed using the SGI/Cray T3E-900 at the NERSC facility at Lawrence Berkeley National Laboratory. The performance results showed nearly linear performance scaling up to 256 processors for both non-adaptive single-level computations and for dynamic multi-level adaptive simulations (see the reference below). The computing power that can be obtained using modern parallel architectures combined with the performance gains achievable by using AMR leads to a dramatic increase in computational capability.

Below is a visualization of the density field in a similar computational experiment, but one in which the bubble is composed of argon. By the time shown in this image, the bubble is greatly deformed through interaction with the baroclinic vorticity generated as the shock passes over the bubble. The adaptive grids, which define the region of highest resolution corresponding to regions in which the flow is the most complex, are shown in faint blue outlines. These grids illustrate the dynamic and irregular character of the load-balancing problem inherent in AMR applications. This numerical experiment was performed on an IBM Power3 architecture with 256 processing nodes, using the parallel application code, HyperCLaw. Despite the difficulty of load-balancing the calculation, HyperCLaw is able to efficiently utilize the parallel architecture.

density

For more about the details of parallel AMR, click here or see the reference below.

Reference

C.A. Rendleman, V.E. Beckner, M. Lijewski, W.Y. Crutchfield, J.B. Bell, Parallelization of Structured, Hierarchical Adaptive Mesh Refinement Algorithms, Computing and Visualization in Science, Volume 3, 2000. [pdf]

HyperCLaw is available as part of the CCSE Applications Suite. For more about the compressible adaptive methodology, or about these calculations, contact John Bell of CCSE.