Dynamically knots setting in meshless method for solving time dependent propagations equation

Abstract

In order to display a function with some finite discrete sampling data efficiently, we require more sampling data where the function is more oscillatory, and less sampling data where the function is more flat. If the function is happen to be a solution of partial differential equation and is solved by finite elements method, then we should construct finer element near the singularity. However, we do not know where the oscillation or even shocks will happen to a function, which is a solution of non-linear partial differential equation. Therefore we cannot preset the finer elements near such points. A trivial method is taking very dense knots or very fine elements everywhere to keep the accuracy. This strategy cost the computation time. Another idea is to move the position of the sampling points according to the varying of the function with the time. We observed that, this approach could be achieved by Lagrangian formulation for finite difference method. However the Lagrangian formulation does not possess shape preserving and variation diminishing properties. It is difficult to achieve the approach for finite elements methods too, because the moving knots will destroy the topology of the mesh. Recently the meshless method becomes to topic to solve partial differential equation numerically. The meshless method does not require any mesh or any structure of the knots (sampling points); therefore we can move the knots freely to simulate the problem. The only restriction is to keep the knots no overlapping. This paper is a test of our approach for the Burger's equation with radial basis quasi-interpolation.