Computational Physics GroupKarel Matous |
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Adaptive wavelet algorithm for solving nonlinear initial-boundary value problems with error control
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 AbstractWe 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. Conclusions
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.
AcknowledgmentThis 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. Download paper here
(c) 2018 University of
Notre Dame and Prof. Karel Matous
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