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

Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFGmethod 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.


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][2][3], finite element method (FEM) [4][5][6][7][8], Bspline methods [9][10][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][14][15][16][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 2 Advances in Mathematical Physics 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.

Numerical Algorithms
Let Ω ⊂ R nd be an open bounded region with piecewise smooth boundary Γ.The number of space dimensions, nd, is equal to 1 or 2. Burgers' equation is written as follows: where u 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 w represent the weight function for velocity u; then the standard weak form of (1) is given as follows: where u = u/, (⋅, ⋅) = ∫ Ω (⋅)Ω, that is, the  2 product of the indicated arguments over domain Ω.
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.

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 u and its weight function w can be decomposed into coarse scale and fine scale, respectively; namely, where u and û are the coarse scale and fine scale for velocity u and w and ŵ are the coarse scale and fine scale for weight function w, respectively.Similar to literatures [19,27], we further assume that û and ŵ, although nonzero within background integral cell Ω  , vanish identically over the Ω  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 formulations Next, employing the linearity between coarse scale w and fine scale ŵ, (7) can be split into coarse scale problem  and fine scale problem Ŵ as follows.
The Coarse Scale Problem .Consider The Fine Scale Problem Ŵ.Consider Obviously, 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 u + û in ( 8) and ( 9) by the last converged solution u  from the fixed point iteration.
The Linearized Coarse Scale Problem .Consider The Linearized Fine Scale Problem Ŵ.Consider Step 2 (solution of the fine scale problem).According to the literatures [19,27], we next deduce the fine scale solution û by using the bubble functions.Let us now consider the fine scale part of the weak form Ŵ. 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 get where r = − u − u  ⋅ ∇u + ∇ ⋅ ((1/Re)∇u).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 r 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 û 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 û and ŵ are represented via bubble functions over the Ω  of the EFG shape function when factor  approaches to 1: where   1 and   2 represent the bubble functions for the trial solution û and weight function ŵ over the Ω  , respectively, and  and  are their coefficients, respectively,  = 1, . . ., nd (nd 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   = −0.5.
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 (  ,   ) = (0.34, 0.34).Substituting ( 13) and ( 14) into (12), and taking the constant coefficient vector  and  out of the integral expression, we have Since  is arbitrary, consequently we get where S and R are defined as follows: where I is a nd × nd identity matrix and both ∇  1 and ∇  2 are nd×1 vectors of gradient of the bubble functions.Substituting ( 19) into ( 13), then we can obtain the fine scale û over the Ω  as follows: Step 3 (solution of the coarse scale problem ).Once we have obtained the fine scale solution û, now we turn our attention to the coarse scale problem .Exploiting the linearity of the solution slot between u and û, (10) can be written as Using the integration by parts for the terms containing û in (22) as well as the assumptions used in ( 5), (22) has the following forms: Substituting ( 18) into (23), subsequently assuming that the coarse scale residual r is a constant over the Ω  , after rearrangement we get the finally computed formulation: where Compared with the standard variational form (2), it is important to note that the third term on the left appears in (24) owing to the assumption that there exists fine scale in the problem, which in fact models the effect of the fine scale.

Low-Pass Filter.
Although the VMEFG method can play a role in stability for some moderate Re values [21], but for very large Re values, it will still produce spurious oscillations as we will see in the next section.Previous researches [22][23][24]28] indicate that a low-pass filter can suppress the spurious oscillations; thus in the paper we use the high order low-pass filter to smooth the solutions for very large Re values at each time loop.For one-dimensional problems, a high order tridiagonal filter can be formulated as [23,24] where   denotes the given value at point  and φ denotes the value after filtering.With proper choices of coefficients, ( 22) provides a 2th-order formulation on a 2+1 point stencil.
The  + 1 coefficients  0 ,  1 , . . .,   can be obtained in terms of   by Taylor expansion.In the paper, we use the following coefficients of a family of 10th-order filter derived in [24,28] For two-dimensional problems, the explicit high order filter can refer to [29].

Numerical Examples
To show the effectiveness of the combination VMEFG and low-pass filter, numerical solutions of Burgers' equation with small diffusivity coefficients for three problems will be considered.
Example 1.Consider the following 1D Burgers' problem:  The exact solution of the problem is given by [10,11] where (31) For this problem, extensive numerical results indicated that the solutions are smooth when Re ≤ 200 [11]; as the Re number increases, the solutions will appear discontinuous.
Figure 2 shows the VMEFG solutions without filter and with filter at different times and Re = 2000.It can be seen that VMEFG solutions have some spurious oscillation under this larger Re number, while the solutions obtained by VMEFG method with filter are still smooth.This example also illustrates that VMEFG method produces the numerical pseudo oscillations for very small diffusivity coefficient.Thus, other techniques need to overcome the difficulty.
Figure 3 presents the comparison of solutions obtained by VMEFG method with filter and exact solutions at different times.It can be seen that the VMEFG method with filter solutions are in well agreement with the exact solutions.From Figures 2 and 3, it can be seen that the VMEFG method coupled high order low-pass filter is one of the techniques which can work out the difficulty well.
In order to investigate the accuracy of the VMEFG method with filter, the following  2 norm error is used: where ℎ is the space step,  is the number of node,  num is the numerical solution, and  exact is the exact solution.Table 1 shows the  2 norm errors by taking different numbers of node at different times for the simulation and observes that the numerical results are satisfactory.Figure 4 illustrates these errors plotted against ℎ on a log-log scale.It can be noted that the order of convergence of the VMEFG method with filter is linear.
Example 2. In this example, we consider the 1D Burgers' equation with the initial condition The exact solution of this problem is [10,30]  (, ) where the Fourier coefficients are Although the example has the exact solution, the convergence is very slow for larger value of Re owing to that the exact solution of the equation involves Fourier series solutions.Therefore, in order to show the computational accuracy of the proposed method for small diffusivity coefficients, that is, large Re values, the numerical solutions are compared with the results available in the literatures [3,30].
In this example, 200 nodes are distributed uniformly in the domain and Δ = 0.001.Figure 5 presents the numerical solutions obtained by EFG method and VMEFG method for Re = 10 3 .It can be seen that there exists some oscillation in the solution of EFG, but, in contrast, the solutions of VMEFG are always smooth, which indicates that VMEFG method has the ability to accommodate the exact solution to the problem.Meanwhile, this demonstrates that the VMEFG has better stability and accuracy than EFG method for moderate large Re number indeed.  1 plotted against ℎ on a log-log scale.Figure 6 illustrates the numerical results obtained by VMEFG without and with low-pass filter at different time for Re = 10 4 .It can be observed that the profiles of  develop extremely sharp gradients near the right-hand boundary for  ≥ 0.4, and the numerical results of VMEFG method without filter show the nonphysical oscillations near the righthand boundary for  ≥ 0.6, which indicates that the VMEFG method still has spurious oscillations for very large Re numbers, whereas the numerical results obtained by VMEFG method with low-pass filter are still smooth and meanwhile the sharp gradients of the profiles near  = 1, especially for  = 0.6, are well distinguished.
Table 2 shows the results obtained by VMEFG method with low-pass filter at  = 1.0 for Re = 10 4 and those available in the literatures.It can be found that our results are in agreement with the results presented in [30] for Re = 10 4 .From Figure 6 and Table 2, we can conclude that the VMEFG method with low-pass filter gives remarkable accuracy for large values of Re.
The numerical solution of Burgers' equation at a high Reynolds number (Re ≥ 10 5 ) is very difficult due to the presence of shock [21].A direct application of the VMEFG method for Re = 10 5 leads to highly oscillatory results as shown in Figure 7.It can be seen that the oscillation starts at  = 0.4 and is accumulated and amplified in the following times.Figure 7 also shows the results obtained by using the low-pass filter under the same condition.It can be found that the oscillations are eliminated and sharp shock profiles are resolved.Meanwhile, we consider Re = 10 9 further.Figure 8 shows the results obtained without and with low-pass filter for Re = 10 9 .Figure 8(a) is obtained without filter; we see some numerical oscillations obviously.But the low-pass filter eliminates those high frequency oscillations very well.

Figure 1 :
Figure 1: Schematic diagram showing the domains for the bubble functions for a reference triangle cell [21].

Figure 2 :
Figure 2: The numerical solutions of Re = 2000 obtained by VMEFG method with and without filter, respectively.(a) is obtained without filter and (b) is with low-pass filter.

ErrorsFigure 4 :
Figure4: The errors from Table1plotted against ℎ on a log-log scale.