Computational Physics Group

Karel Matous










Adaptive wavelet algorithm for solving nonlinear initial-boundary value problems with error control

C.Harnish1,2, K. Matous1,2 and D. Livescu3

1Department of Aerospace and Mechanical Engineering,
2Center for Shock Wave-processing of Advanced Reactive Materials,
University of Notre Dame, Notre Dame, IN, 46556, USA.
3Computer and Computational Sciences Division, Los Alamos National Laboratory,
Los Alamos, NM 87545, USA


    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.


    In this work, we have developed an adaptive algorithm for solving nonlinear PDEs. We have incorporated a matrix notation to simplify the fundamental wavelet operations, and utilized a matrix- free computational implementation. We have shown that our numerical method is capable of solving initial-boundary value problems on finite domains with an explicit error control and negligible global energy growth. The algorithm takes advantage of the regularity of the biorthogonal interpolating wavelet family and evaluates spatial derivatives directly on the wavelet basis functions. We have advanced the state of wavelet based algorithms by deriving bounds on the spatial error of PDE solutions and developing a predictor-corrector strategy to ensure that the spatial error stays bounded at each time step. We have verified these error estimates through numerical analysis of nonlinear shock problems with analytical solutions. Furthermore, we have defined each field in the governing equations on its own dynamically adaptive computational grid and fine scale features, such as shock waves, are well resolved with no spurious numerical oscillations.


    This work was supported by the Department of Energy, National Nuclear Security Administration, under the award No. DE-NA0002377 as part of the Predictive Science Academic Alliance Program II. We would also like to acknowledge support from Los Alamos National Laboratory under the award No. 369229. Cale Harnish and Karel Matous would like to thank Dr. S. Paolucci for fruitful discussions regarding the wavelet solutions of PDEs.

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