An Adaptive Multi-resolution Method for Solving PDE's

Abstract

In this paper, we present a multi-resolution adaptive algorithm for solving problems described by partial differential equations. The technique is based on the collocation method using Fup basis functions, which belong to a class of Rvachev's infinitely differentiable finite functions. As it is possible to calculate derivation values of Fup basis functions of high degree in a precise yet simple way, so it is possible to efficiently apply strong formulation procedures. The mesh free method developed in this work is named Adaptive Fup Collocation Method (AFCM). The distribution of collocation points within the observed area is changed adaptively, depending on the character of the solution function and the accuracy criteria. The numerical procedure is designed through a method of lines (MOL). The basic characteristic of the method is an adaptive multi-resolution approach in solving problems with different spatial and temporal scales and with a desired level of accuracy using the entire family of Fup basis functions. Good performance of the proposed method is shown through the numerical examples by using a few general advection dominated problems. The results demonstrate that the method is very convenient for solving engineering problems which require extensive computational resources, especially in describing sharp fronts or high gradients and changes of narrow transition zones in space and time.