Adaptive blocking schemes for two-dimensional diffusion problems

Keywords

Numerical analysis

Abstract

In this work, the two-dimensional diffusion equation is solved using a multi-rate numerical algorithm. The algorithm divides the system into different spatial parts or blocks, allowing each block to take different time steps. Three blocking formats, or the so called 'Square', 'Tight', and 'Elliptical' schemes, are used to keep the working grid and Jacobian as compact as possible. Technical complications also pertain to the implementation of the Jacobian computations. To the extent possible, an attempt is made to automate the procedure for arbitrary initial concentration profiles and diffusion coefficients such that each block takes the largest timestep consistent with the specified accuracy tolerance. The various blocking schemes and the adaptive updating of the block boundaries are carefully considered. For most of the various concentration profiles studied in tandem with 'slow', 'mixed', 'fast', and 'dual' diffusion coefficient dependences, there is a very significant speedup over a regular trapezoidal rule/ second-order backward difference (TRBDF) time-stepping scheme without blocking. The greatest speedup (by a factor of 4 in certain cases) is obtained for the ' low' diffusion together with the 'tight' blocking. Typically there is somewhat less speedup with the other two spatial blocking schemes. Diffusion speed and block size are found to major factors affecting the performance of this algorithm. In particular the greatest speedups occur where the diffusion coefficients increase relatively slowly with concentration, or when the initial concentration profile occupies a small fraction of the spatial domain. In both of these cases, the fast block, where concentration is already relatively high and timestepping must be rapid, expands relatively slowly. This results in greater speedup from the significantly slower timestepping in the slow block(s) which .. 11 occupy appreciable fractions of the spatial domain. The converse occurs for either faster diffusion and/ or widely dispersed initial profiles, for both of which the slow block(s) occupy smaller portions of the overall domain, thus allowing for less speedup. With some combinations, notably the 'fast' diffusion and deeply embedded initial concentration, the various blocking schemes lose much of their speedup and some even take slightly longer than without blocking.

Notes

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Graduation Date

2004

Degree

Master of Science (M.S.)

College

College of Arts and Sciences

Format

PDF

Pages

72 p.

Language

English

Length of Campus-only Access

None

Access Status

Masters Thesis (Open Access)

Identifier

DP0029480

Subjects

Arts and Sciences -- Dissertations, Academic; Dissertations, Academic -- Arts and Sciences

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