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Wavelet Methods for Nonlinear Partial Differential Equations

Bradley K. Alpert, ACMD
Gregory Beylkin, University of Colorado, Boulder

Fluid flow, multiphase systems, and chemically reacting flows, to name a few examples, lead to field quantities that develop sharp gradients and highly localized phenomena. It is now well established that the accurate numerical modeling of these systems requires adaptive methods for integrating the associated partial differential equations (PDE). Wavelets and other hierarchical bases can be used to construct adaptive schemes that represent fields with sharp wavefronts and complex local structure without excessive redundancy, and offer the promise of very accurate and reasonably efficient solution of nonlinear problems.

This project has concentrated on the development of numerical methods using Alpert's multiwavelet bases. They avoid many of the difficulties in representing boundary conditions faced by other wavelet constructions, can be used for a variety of geometries, and are relatively simple to implement. The project's success on a model wave problem, Burgers' equation, leads to confidence that the method can be extended to systems of PDE and to higher dimensional problems. Phase-field models of solidification, extensively studied at NIST by the Applied and Computational Mathematics and Metallurgy Divisions, represent a future testbed (and an important motivation) for this work. It is anticipated that the methods will also be applicable to problems in the Thermophysics Division and elsewhere at NIST.

A significant recent advance is expected to permit explicit, rather than implicit, time stepping, leading to improved efficiency and simplicity. The project implementations are currently being extended to systems of equations in two spatial dimensions.

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