Solving Nonlinear PDEs with a priori accuracy using wavelets

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.

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

C. Harnis, K. Matous and D. Livescu.

International Journal for Multiscale Computational Engineering, 16(1):19–43 (2018)