On the Adaptive Numerical Solution of Nonlinear Partial Differential Equations in Wavelet Bases

✍ Scribed by Gregory Beylkin; James M. Keiser

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 765 KB
• Volume: 132
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.5562

✦ Synopsis

This work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the form u t ϭ Any wavelet-expansion approach to solving differential L u ϩ N f (u), where L and N are linear differential operators and equations is essentially a projection method. In a projection f (u) is a nonlinear function. These equations are adaptively solved method the goal is to use the fewest number of expansion by projecting the solution u and the operators L and N into a coefficients to represent the solution since this leads to wavelet basis. Vanishing moments of the basis functions permit a sparse representation of the solution and operators. Using these efficient numerical computations. The number of coeffisparse representations fast and adaptive algorithms that apply opercients required to represent a function expanded in a Fouators to functions and evaluate nonlinear functions, are developed rier series (or similar expansions based on the eigenfuncfor solving evolution equations. For a wavelet representation of the tions of a differential operator) depends on the most solution u that contains N s significant coefficients, the algorithms singular behavior of the function. We are interested in update the solution using O(N s ) operations. The approach is applied to a number of examples and numerical results are given. ᮊ 1997 solutions of partial differential equations that have regions Academic Press of smooth, nonoscillatory behavior interrupted by a number of well-defined localized shocks or shock-like structures. Therefore, expansions of these solutions, based upon