HAM-Based Adaptive Multiscale Meshless Method for Burgers Equation

Shu-Li Mei

doi:10.1155/2013/248246

Abstract

Based on the multilevel interpolation theory, we constructed a meshless adaptive multiscale interpolation operator (MAMIO) with the radial basis function. Using this operator, any nonlinear partial differential equations such as Burgers equation can be discretized adaptively in physical spaces as a nonlinear matrix ordinary differential equation. In order to obtain the analytical solution of the system of ODEs, the homotopy analysis method (HAM) proposed by Shijun Liao was developed to solve the system of ODEs by combining the precise integration method (PIM) which can be employed to get the analytical solution of linear system of ODEs. The numerical experiences show that HAM is not sensitive to the time step, and so the arithmetic error is mainly derived from the discrete in physical space.

Highlights

Burgers equation is a typical nonlinear partial differential equation, which was constructed to describe a kind of hydromechanical phenomenon
There are many analytical methods for nonlinear problems that have been proposed in recent years
In order to further improve the properties of homotopy analysis method (HAM), Liao [14] and some other researchers [15, 16] studied the choice rules about the auxiliary parameter h and the auxiliary function H(t) on different nonlinear problems, which requires the users to have high-level skill about it

Summary

Introduction

Burgers equation is a typical nonlinear partial differential equation, which was constructed to describe a kind of hydromechanical phenomenon. We try to develop HAM to solve the matrix ODEs without decoupling by combining the precise integration method (PIM) and meshless method, which can be used to solve the nonlinear PDEs. Meshless methods eliminate some or all of the traditional mesh-based view of the computational domain and rely on a particle (either Lagrangian or Eulerian) view of the field problem. Collocation method and Galerkin method are the two discretization methods which have been dominant in existing meshless methods, in which the radial basis function is often employed to construct the interpolation operator. The purpose of this study is to construct a HAM-based multi-scale meshless method for Burgers equation. We construct a multi-scale interpolation operator with RBF, which can discretize PDEs adaptively into a system of nonlinear ODEs. we develop HAM to solve the system of nonlinear ODEs obtained in the first step, in which the adaptability was introduced to the HAM. The numerical experiments show that this is an efficient second-order time-marching solver for timedependent problems as long as a factorization of the differential operator is available

Construction of Meshless Adaptive Multiscale Interpolation Operator

Numerical Result and Discussion

Conclusions