Abstract:
Computational kinetic theory has long been plagued by the curse of dimensionality, leading to the development of particle-based methods like particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC). In each case, the curse persists because the statistical figure of merit is the number of particles per cell, and the number of cells is typically enormous in 2- and 3-D.
Sparse grids represent an alternative approach to circumventing the curse of dimensionality. We propose a new scheme that merges the "combintation technique" from the sparse grid literature with PIC by using a sequence of computational grids, each of which is finely resolved in at most one direction. By combining these approximations intelligently, one can achieve accuracy near that of a regular grid at dramatically reduced cost. The end result is a PIC scheme whose computational complexity depends only logarithmically on dimension.
Results from test cases in 2- and 3-D will be presented, demonstrated the substantial speedups achieved relative to standard schemes. |