Using wavelets to solve the Burgers equation: A comparative study
R. L. Schult, H. W. Wyld · Physical Review A · 1992
The Burgers equation is solved for Reynolds numbers \ensuremath{\lesssim}8000 in a representation using coarse-scale scaling functions and a subset of the wavelets at finer scales of resolution. Situations are studied in which the solution develops a shocklike discontinuity. Extra wavelets are kept for several levels of higher resolution in the neighborhood of this discontinuity. Algorithms are presented for the calculation of matrix elements of first- and second-derivative operators and a useful product operation in this truncated wavelet basis. The time evolution of the system is followed using an implicit time-stepping computer code. An adaptive algorithm is presented which allows the code to follow a moving shock front in a system with periodic boundary conditions.