Seminars & Events

Argonne Leadership Computing Facility
"An Iterated, Multipoint Differential Transform Method for Numerically Evolving PDE IVPs"

DATE: to April 29, 2011
TIME: 10:00 AM - 11:00 AM
SPEAKER: Hal Finkel
LOCATION: Building 240 - Confernence Room 4301, Argonne National Laboratory
HOST: Timothy J. Williams, Ph.D.

Description:
Traditional numerical techniques for solving time-dependent partial-differential-equation (PDE) initial-value problems (IVPs) store a truncated representation of the function values and some number of their time derivatives at each time step. What if spatial derivatives were also stored? Although redundant in the dx --> 0 limit, this paper demonstrates that stored spatial derivatives can be propagated in an efficient and self-consistent manner. They can effectively contribute to the evolution procedure in a way which can confer several advantages, including aiding solution verification. Specifically, three novel contributions will be presented: First, a concept for constructing verifiably-self-consistent numerical evolution schemes for PDE IVPs; second, an iterated, multipoint differential transform method (IMDTM) for numerically evolving PDE IVPs (the IMDTM can be used to efficiently implement verifiably-self-consistent PDE evolution); and finally, in order to efficiently implement the IMDTM scheme, a generalized finite-difference stencil formula is derived which can take advantage of multiple higher-order spatial derivatives when computing even-higher-order derivatives. As will be demonstrated, the performance of these techniques compares favorably to other explicit evolution schemes in terms of speed, memory footprint and accuracy.

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