We introduce a multilevel PDE solver for equations whose solutions exhibit large gradients. Expanding on Ami Harten's ideas, we construct an alternative to wavelet-based grid refinement, a multiresolution coarsening method capable of capturing sharp gradients across different scales and thus improving PDE-based simulations by concentrating computational resources in places where the solution varies sharply. Our scheme is akin to Finite Differences in that it computes derivatives explicitly and then uses the derivative information to march the solution in time. However, we utilize meshfree methods to compute derivatives and integrals in space-time to increase the robustness of our solver and tailor the basis functions to the Kd-tree structure provided by the multiresolution analysis.

Date Published: 17 September 2007
PDF: 11 pages
Proc. SPIE 6700, Mathematics of Data/Image Pattern Recognition, Compression, Coding, and Encryption X, with Applications, 670007 (17 September 2007); doi: 10.1117/12.735051

Published in SPIE Proceedings Vol. 6700:
Mathematics of Data/Image Pattern Recognition, Compression, Coding, and Encryption X, with Applications
Gerhard X. Ritter; Mark S. Schmalz; Junior Barrera; Jaakko T. Astola, Editor(s)