Advances in Mathematical Physics

Volume 2016 (2016), Article ID 1836978, 12 pages

http://dx.doi.org/10.1155/2016/1836978

## Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion

College of Science, China Three Gorges University, Yichang 443002, China

Received 22 October 2015; Revised 27 January 2016; Accepted 2 February 2016

Academic Editor: Stephen C. Anco

Copyright © 2016 Ping Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients.

#### 1. Introduction

Burgers’ equation, which is a nonlinear partial differential equation, can model several physical phenomena such as traffic, shock waves, gas dynamics, longitudinal elastic waves in an isotropic solid, and turbulence problems. In general, the exact solution of Burgers’ equation can be only obtained in special case. Moreover, in many cases these solutions involve infinite series which may converge very slowly for small diffusivity coefficients. Therefore, finding accurate and efficient numerical methods for solving Burgers’ equation has been an attractive research undertaking. The main challenge of numerical solution of Burgers’ equation lies in nonphysical oscillations for convection-dominated problem. Thus, many numerical schemes are constructed or developed in the last few years to solve Burgers’ equation, such as finite difference method (FDM) [1–3], finite element method (FEM) [4–8], B-spline methods [9–11], and lattice Boltzmann method (LBM) [12]. A comprehensive review of the different methods for solving Burgers’ equation can refer to [10].

In recent years, meshless methods emerge as a new numerical method for solving various mathematical-physical problems. Compared with the traditional computational methods based on mesh, meshless methods have many outstanding advantages. In these methods, the approximation is built only based on nodes and no predefined nodal connectivity is required; meanwhile, the removal and the addition of nodes in the domain are easily performed.

Previous studies [13, 14] show that the numerical solution may be corrupted by nonphysical oscillations when meshless methods are directly used to solving Burgers’ equation with small diffusion. Usually, spatial stabilization technique is required for the meshless methods and many meshless stabilized methods are developed in recent years. Most of these stabilized methods are a generalization of the stabilized FEM [13–17], such as streamline-upwind Petrov-Galerkin (SUPG), Galerkin least-square (GLS), and subgrid scales (SGS). As Fries and Matthies [16] had pointed out, standard stabilization approaches in FEM can be directly applied to the meshless methods; however, the choice of the stabilization parameter required special attention. But in practice, the choice of the stabilization parameter is more or less dependent on the researches’ experience. In order to avoid the selection of the stabilization parameter, Zhang and his coworkers [18] presented variational multiscale element free Galerkin method (VMEFG) which is free of user-defined stability parameters and a consistent stabilized method. Subsequently, they applied the method to solve Stokes problems [18], water wave problems [19], MHD flows problems [20], and convection-diffusion-reaction problem [21] successfully. But, as far as the solution of Burgers’ equation is concerned, for the very small diffusivity coefficients, the VMEFG method also produces the numerical pseudo oscillations as we will see in the paper; therefore, we must find another technique to deal with it.

In recent years, high order low-pass filters have been proposed to treat hyperbolic conservation laws [22, 23] and some experiences show that a low-pass filter is ideal for suppressing the amplitude of undesirable high frequency components and does not affect the remaining components of the solution [24]. To the best of our knowledge, most numerical applications are restricted to problem involving moderate and larger diffusion. Moreover, no published results with meshless methods in conjunction with low-pass filter are used to solve Burgers’ equations with very small diffusivity coefficients. Therefore, the main goal of the paper is to evaluate VMEFG coupled with high order low-pass filter for Burgers’ equation with very small diffusion.

This paper is organized as follows. In Section 2, we list some elementary knowledge about the VMEFG method and low-pass filter. In Section 3, numerical experiments are performed using our proposed method. And the conclusions are presented in the Section 4.

#### 2. Numerical Algorithms

Let be an open bounded region with piecewise smooth boundary . The number of space dimensions, , is equal to 1 or 2. Burgers’ equation is written as follows:where is the dependent variable resembling the flow velocity and Re is the Reynolds number characterizing the size of viscosity. From (1), it can be obviously seen that Burgers’ equation is a simplified form of the Navier-Stokes equation having viscosity and nonlinear convection term. When the diffusivity coefficient is very small, Burgers’ equation models convection-dominated problem.

Let represent the weight function for velocity ; then the standard weak form of (1) is given as follows:where , , that is, the product of the indicated arguments over domain .

In the paper, the moving least-square (MLS) method is used to construct the shape functions of meshless method and the detailed MLS approximation can refer to [25]. In general, owing to that the MLS approximation does not possess the Kronecker Delta condition property, this makes the imposition of essential boundary conditions more complicated and time-consuming than that of FEM. Here we used a simple technique proposed by Zhang et al. [19] to enforce the essential boundary conditions; that is, the nodal influence domain of meshless method is extended to have arbitrary polygon shape. Their numerical results indicated that the shape functions almost possess the Kronecker delta function property when the dimensionless size of the nodal influence domain approaches to 1.

##### 2.1. Variational Multiscale Element Free Galerkin Method

The VMEFG method is inheritance of the variational multiscale FEM which was offered by Hughes et al. [26] and the main idea of the method is to divide the unknown physical quantity into fine scale and coarse scale. Then it can use bubble functions to determine fine scale solution analytically and gets stabilization parameter naturally. Last, it substitutes the fine scale solution into coarse scale problem and then obtains the coarse scale solution numerically. In the following, the brief introduction of VMEFG is presented.

*Step 1 (multiscale decomposition). *Assume that the velocity and its weight function can be decomposed into coarse scale and fine scale, respectively; namely,where and are the coarse scale and fine scale for velocity and and are the coarse scale and fine scale for weight function , respectively.

Similar to literatures [19, 27], we further assume that and , although nonzero within background integral cell , vanish identically over the boundary when the factor approaches to 1:We now substitute the trial solutions equation (3) and the weighting functions equation (4) in the standard variational equation (2), and this becomes the point of departure from the conventional Galerkin formulationsNext, employing the linearity between coarse scale and fine scale , (7) can be split into coarse scale problem and fine scale problem as follows.*The Coarse Scale Problem *. Consider*The Fine Scale Problem *. ConsiderObviously, the coarse and fine scale equations are in fact nonlinear equations because of the convection term. In general, to solve nonlinear equations we need to linearize them. Here, we substitute the nonlinear convective coefficient in (8) and (9) by the last converged solution from the fixed point iteration.*The Linearized Coarse Scale Problem *. Consider*The Linearized Fine Scale Problem *. Consider

*Step 2 (solution of the fine scale problem). *According to the literatures [19, 27], we next deduce the fine scale solution by using the bubble functions. Let us now consider the fine scale part of the weak form . Exploiting linearity of the solution slot in the first term and employing integration by parts to the second term on the left-hand side in (11), we getwhere . In this process, we have applied the integration by parts as well as the relevant assumptions such as (6). In addition, it is important to note that is the residual of Burgers’ equation for the coarse scale. This is a crucial ingredient of the VMEFG method and ensures that the resultant formulation yields a consistent stabilized method.

In order to obtain fine scale solution analytically from (12) and deduce the structure of the stability parameter , we use bubble functions similar to FEM context [27]. Without loss of generality, we assume that the fine scale and are represented via bubble functions over the of the EFG shape function when factor approaches to 1:where and represent the bubble functions for the trial solution and weight function over the , respectively, and and are their coefficients, respectively, ( indicates the dimension of the problem).

In the paper, for the 1D problem, the bubble functions fine scale trial solution and weighting function are given as follows [27]:where is the location of the internal virtual node for the piecewise linear bubble and in the paper .

For the 2D problem, when the triangle background cells are used and in a reference background cell, the bubble for the fine scale trial solution is a quadratic bubble defined as [21]For the fine scale weighting function, the reference background cell is divided into three regions (see Figure 1) and the bubble functions on these regions are given as follows:where and represent the location of the internal virtual node in the background cell and in the paper .

Substituting (13) and (14) into (12), and taking the constant coefficient vector and out of the integral expression, we haveSince is arbitrary, consequently we getwhere and are defined as follows:where is a identity matrix and both and are vectors of gradient of the bubble functions. Substituting (19) into (13), then we can obtain the fine scale over the as follows: