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International Journal for Multiscale Computational Engineering

Impact factor: 1.103

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2018024915
pages 19-43

ADAPTIVE WAVELET ALGORITHM FOR SOLVING NONLINEAR INITIAL–BOUNDARY VALUE PROBLEMS WITH ERROR CONTROL

C. Harnish
Department of Aerospace and Mechanical Engineering, Center for Shock Wave-processing of Advanced Reactive Materials, University of Notre Dame, Notre Dame, Indiana 46556, USA
Karel Matous
Department of Aerospace and Mechanical Engineering, Center for Shock Wave-processing of Advanced Reactive Materials, University of Notre Dame, Notre Dame, Indiana 46556, USA
D. Livescu
Computer and Computational Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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.