Adaptive multiresolution mesh refinement for the solution of evolution PDEs

Abstract

In this paper we propose a novel multiresolution-based adaptive mesh refinement method for solving Initial-Boundary Value Problems (IBVP) for evolution partial differential equations. The proposed algorithm dynamically adapts the grid to any existing or emerging irregularities in the solution, thus refining the grid only at places where the solution exhibits sharp features. The main advantage of the proposed grid adaptation method is that it results in a grid with a fewer number of nodes when compared to adaptive grids generated by other waveletor multiresolution-based mesh refinement techniques. Several examples show the robustness, stability and savings, in terms of the CPU time, of our algorithm.

Trajectory Optimization Using Multiresolution Techniques

“…Motivated by the previous observations in [29,30], we have introduced a novel multi-resolution-based mesh refinement technique for the solution of initial boundary-value problems for evolution equations. The algorithm results in a fewer number of nodes than with similar grid adaptation schemes, while maintaining the same overall accuracy of the final solution.…”

“…Several challenging examples (namely, Burgers's equation and Euler's equations of gas dynamics) have demonstrated the stability and robustness of the mesh refinement algorithm for the solution of these types of problems in 1-D [30]. In the current paper, we use the ideas introduced in [29,30] to design a novel, fully automated, adaptive multiresolution trajectory optimization technique to solve optimal control problems quickly and accurately. The criterion for deciding the region to refine the mesh is based on simple interpolations, which tends to speed up the whole process.…”