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12.8.4 Conclusion

Grid-based particle simulation algorithms continue to provide an effective technique for studying systems of pointlike particles in addition to continuum systems. These methods are a useful alternative to grid-less simulations which cannot incorporate fluid interactions or complicated boundary conditions as easily or effectively. While the approach is quite different, the tree-structure and enhanced accuracy criterion which are the bases of multipole methods are equally applicable as the fundamental structure of an adaptive refinement mesh algorithm. The two techniques complement each other well and can provide a useful environment both for studying mixed particle-continuum systems and for comparing results even when a mesh is not necessitated by the physically interesting aspects of the modelled system. The hierarchical structure naturally occurs in problems which demonstrate locality such as systems governed by Poisson's Equation.

Implementations for parallel, distributed-memory computers gain direct benefit from the locality. Because both the grid-based and particle-based methods form the same hierarchical structure, common data partitioning can be employed. A hybrid simulation using both techniques implicitly has the information for both components-particle and fluid-at hand on the local processor node, simplifying the software development and increasing the efficiency of computing such systems.

Considerations such as the efficiency of a deep, grid-based hierarchy with few or even one particle per grid cell need to be explored. Current particle-based algorithm research comparing computational accuracy against grid resolution (i.e., one can utilize lower computational accuracy with a finer grid or less refinement with higher computational accuracy), will strongly influence this result. Also, the error created by interpolating the particles onto a grid and then solving the discrete equation must be addressed when comparing gridless and grid-based methods.

Guy Robinson
Wed Mar 1 10:19:35 EST 1995