Adaptive Wavelet Algorithm for Solving Nonlinear Initial–boundary Value Problems With Error Control

Abstract

We present a numerical method which exploits the biorthogonal interpolating wavelet family, and second-generation wavelets, to solve initial-boundary value problems on finite domains. Our predictor-corrector algorithm constructs a dynamically adaptive computational grid with significant data compression, and provides explicit error control. Error estimates are provided for the wavelet representation of functions, their derivatives, and the nonlinear product of functions. The method is verified on traditional nonlinear problems such as Burgers' equation and the Sod shock tube. Numerical analysis shows polynomial convergence with negligible global energy dissipation.

A multiresolution adaptive wavelet method for nonlinear partial differential equations

“…Our algorithm uses the Deslauriers-Dubuc wavelet family, with second generation wavelets near spatial boundaries, as defined in [40]. Furthermore, a single parameter p defines the properties of this basis, such as the number of vanishing moments and the degree of continuity [36]. We discretize space by projecting each continuous field f ( x) onto the wavelet basis φ 0 k ( x) and λ ψ j k ( x), where • indicates a vector.…”

“…Leveraging properties of the Deslauriers-Dubuc wavelet family, the integrals in Eq. ( 3) can be solved exactly and are replaced with the matrix operator F , defined in terms of the filter coefficients, as shown in [36]. Repeated application of this operator yields all of the wavelet coefficients on each resolution level.…”

“…It has been shown that the Deslauriers-Dubuc wavelet family is continuous and differentiable, which allows spatial derivatives to operate directly on the basis functions [36,41],…”

“…To overcome some of the difficulties mentioned above, we have developed an algorithm which retains the advantages of other wavelet methods while attempting to overcome their limitations. Specifically, this work extends our one-dimensional solver described in [36] into multiple spatial dimensions. Since our proposed numerical method exploits the properties of wavelet basis functions, it is helpful to provide a brief outline of wavelet principles.…”

Wavelet based reduced order models for microstructural analyses

“…the Fast Fourier Transform [30] or wavelet bases to reduce the number of equations by projecting a fully discretised multi-scale problem onto a lower dimensional space, e.g. [3,9,17,19]. Some alternative wavelet-based reduction methods can be found in literature.…”