Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional KdV and Burgers equations*

CHENG Si-rui （程思睿）, ZHAN Jie-min （詹杰民）

1. Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China, E- mail:csrpanda@163.com

2.Department of Applied Mechanics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China

Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional KdV and Burgers equations*

CHENG Si-rui （程思睿）1, ZHAN Jie-min （詹杰民）2

1. Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China, E- mail:csrpanda@163.com

2.Department of Applied Mechanics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China

(Received June 2, 2010, Revised April 26, 2015)

In this paper, the multi-scale Runge-Kutta_Galerkin method is developed for solving the evolution equations, with the spatial variables of the equations being discretized by the multi-scale Galerkin method based on the multi-scale orthogonal bases in(a, b)and then the classical fourth order explicit Runge-Kutta method being applied to solve the resulting initial problem of the ordinary differential equations for the coefficients of the approximate solution. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection-diffusion problem), the KdV equation (single solitary and 2-solitary wave problems) and the KdV-Burgers equation, where analytical solutions are available for estimating the errors. Numerical results show that using the algorithm we can solve these equations stably without the need for extra stabilization processes and obtain accurate solutions that agree very well with the corresponding exact solutions in all cases.

multi-scale Galerkin method, fourth order Runge Kutta method, Burgers equation KdV equation, KdV-Burgers equation

Introduction

In this paper, we numerically solve one-dimensional Burgers equations, the KdV equations and the KdV-Burgers equations by applying the multi-scale Galerkin method combined with the classical fourth order Runge-Kutta method. Both the Burgers equation and the KdV equation are important model equations in hydrodynamics. The Burgers equation was first introduced by Bateman and later treated by Burgers as a mathematical model for turbulence (Burgers 1948). The KdV equation was derived by Korteweg and de Vries as a model for long waves propagating in a channel (Korteweg and De Vries 1895). The Korteweg-de Vries type equations can be used to describe a variety of phenomena in physical sciences, and they are well studied. Many methods were proposed to study the nonlinear water wave equations, such as the spectral method[1,2], the group method[3], and the expansion method[4].

Multi-scale methods enjoy many advantages and become standard approaches in solving integral equations[5-15]. They lead to matrix compression schemes which would generate the coefficient matrix efficiently and allow us to design fast solvers for solving the resulting discrete systems of the integral equations.

Chen et al.[16]constructed multi-scale orthonormal bases in(0， 1)spaces and developed the multilevel augmentation methods for solving linear differential equations with zero boundary conditions. In a recent paper[17], Chen et al. developed the multilevel augmentation method for solving nonlinear operator equations of the second kind and applied it to solve the one dimensional sine-Gordon equation.

A great advantage of the multi-scale methods to solve differential equations is the stability of the multi-scale bases. In this paper, we use the multi-scaleGalerkin method based on the bases constructed in paper[16]to solve the one dimensional Burgers equations and the KdV equations. After discretizing the spatial variables by the Galerkin method, we apply the fourth order Runge-Kutta method to solve the resulting ordinary differential equations. The numerical results show that the proposed method has two main competitive advantages. Firstly, it is stable enough to solve the KdV equations without the need for extra stabilization processing. Secondary, it can be used to obtain numerical solutions with high accuracy without the use of a large number of bases.

1. The multi-scale Runge-Kutta-Galerkin scheme for solving KdV-Burgers equations

We consider the initial-boundary value problem of the general Korteweg-de Vries Burgers equation

where ε,νandµare parameters.

First we introduce three functions l( x),r( x)and q( x)satisfying

and let u( x, t)=u˜( x, t)+l( x) ua( t)+r( x) ub( t)+ q( x) u1b(t). Thus Eqs.(1) can be reformulated as an initial-boundary value problem with respect to the functionu˜( x, t)

Equation (5) can be reformulated into a variational problem as[18]: find u˜ = u˜(·,t)∈ X , such that

when µ=0. And(·,·)denotes the L2inner product.

In order to solve Eqs.(6) numerically, we choose Xn，the finite dimensional subspace ofX，as a piecewise polynomial subspace with knots a+(b-a) j kn,j∈N, wherek is an integer larger than1，kn-1 and the notation Nn:={1,2,…,n}forn∈N. Then the semi-discrete scheme is: findu˜n:=u˜n(·,t)∈Xnsuch that

where u˜0(x):=u0(x)-l( x) ua(0)-r( x) ub(0)-q( x)· u1b(0), and u˜0n(x)is a certain approximation of u˜0(x)usually taken as its interpolation projection or L2-projection or elliptic projection on Xn.

Since the subspaces Xnare nested, i.e.,Xn⊂Xn+1,n∈N0:={0,1,2,…},Xn+1, can be expressed as an orthogonal direct sum of Xnand Wn+1. It follows that for n∈N0,

where W:=X, and the notation A⊕⊥Bstands for

00 the orthogonal direct sum of the spacesAandB .

Let w( i) denote the dimension of the i-th level subspace W( i )of Xn,Zn:={0,1,2,…,n-1}, Un:={(i, j):i∈Zn+1,j∈Zw(i)}. Then the multi-scale orthonormal bases forXnconstructed in Ref.[7] can be represented as {wij:(i, j)∈Un}. By utilizing the bases, we rewrite the problem (7) as: find coefficients uij(t),(i, j)∈Un, such that the solution u˜n:=, satisfies for all (i′,j′)∈Un,t∈ (0,]T,

Equations (8) express an initial value problem of ordinary differential equations with coefficients {uij(t): (i, j)∈Un}.

Denote matrices

and vectors

then Eqs.(8) can be written in the matrix-vector form as

Chooseτas the time step and apply the classical fourth order explicit Runge-Kutta method to solve Eq.(9), i.e., for p=1,2,…,T/τ, compute

Table1 Numerical results for Example 1

Thus for p∈{1,2,…,T/τ}, we obtain the coefficientsand the approximate solution of Eqs.(1) can be expressed as

2. Numerical examples

2.1 Numerical results for Burgers equations

When µ=0and ε=1, Eqs.(1) is reduced to the following Burgers equation

Example 1: Burgers equation with zero boundary values

Let u0(x)=4x(1-x),ua( t)=ub( t )=0,a =0, b =1,T=1in Eqs.(1). Then the corresponding exact solution is

where

The solution space for the variational problem is(0,1)We chooseXnas the piecewise linear polynomial subspace with knots j/2n,j-1∈Z2n-1. Then W0=∅,dimWi=2i-1,i>0, and the base of W1is[16]

The bases for Wi,i>1, can be obtained by the translation and the dilation ofw10. Details can be found in Ref.[16].

Since ua( t)=ub( t )=0, we let l( x)=r( x)=0. We takeu0nas the elliptic projection on Xnof u0，i.e.,uij(0)=′). Several cases of vare considered. The numerical results are presented in Table 1, whereτis the time step,nand D( n)stand for the level and the dimension of the subspaceXn, respectively, and Er stands for the root mean square error of 1001 equal distant points on[a, b], including the endpointsaandb.The same notations will be used in the following all tables in this paper.

Table2 Numerical results for Example 2 with α=0. 2,β=0. 3,λ=0

Table3 Numerical results for Example 2 with α=0. 4,β=0. 6,λ=0. 125

Example 2: Burgers equation with nonzero boundary values

Let

and the parametersα,β,λandνare arbitrary constants. The exact solution for this problem is

where ζ=α/ ν(x-βt-λ).

The solution space for this variation problem is(a, b). We choose Xas the piecewise linear po

n lynomial subspace with knotsa+j( b-a)/2n,j-1∈Z2n-1. As in Example 1,W0=∅,dimWi=2i-1, i>0. By an affine transformation of Eqs.(12), we obtain the following base of W1for(a, b)

1

Table4 Numerical results for Example 3

The bases for Wi， i>1, can also be obtained by the translation and the dilation of Eqs.(13) as before.

We choosel( x)and r( x)as the Lagrange interpolation functions onx=aand x=b , i.e.,

We also take u˜0nas the elliptic projection on Xnof u˜0. Let a=-10,b =10,T=10We compute two cases ofα,βandλ. The numerical results for α=0. 2,β=0. 3,λ=0are presented in Table 2. The numerical results forα=0. 4,β=0. 6,λ= 0.125 are presented in Table 3.

We conclude from Table 1, Table 2 and Table 3 that the algorithm is stable and accurate results can be obtained in relatively low dimensional subspaces. Furthermore, the root mean square errors decrease regularly with the increase of the leveln . Whennis changed ton+1, the root mean square error is reduced by a factor of 4.

Fig.1 Comparison of exact solution with approximate solution for Example 3 with n =5,dimXn=62,τ=0. 005, t=1.0

2.2 Numerical results for KdV equations

When ν=0and µ=1, Eqs.(1) is reduced to the following KdV equation

The solution space for the variation problem of Eqs.(15) is(a, b). We choose Xnas the piecewise cubic polynomial subspace with knotsa+j( b-a)/ 2n,j-1∈Z2n-1. Then X0=∅,dimWi=2i,i>0 By an affine transformation of the bases of(0,1) constructed as in Ref.[7], we obtain the following bases of Wfor(a, b)

1

Table5 Numerical results for Example 4

where

E=b-a,F=a+b

The bases of Wifor i>1can be obtained by the translation and the dilation ofw10and w11.

Fig.2 Comparison of exact solution with approximate solution for Example 4 with n =6,dimXn=126,τ=0.0005, t=0.5

We choose l( x)and r( x)as the Hermitien interpolation functions onx=aand x=b , i.e.,

which satisfy

And we choose

which satisfies q( a)=q( b)=0,q′( a)=0,q′( b)=1.

The vectorV0of the initial approximate function

n u˜0nis defined by

where F(0)=[(u˜0,wij):(i, j)∈Un].

Example 3: KdV equation with solitary wave solution

Let

Then the corresponding exact solution of Eqs.(15) is u( x, t)=-2sech2(x-4t)

Let a=-10,b =10,T=1. The numerical results are presented in Table 4. Figure 1 shows the comparison of the analytic solution with the approximate solution withn =5,dimXn=62,τ=0. 005, t=1.0. We use 1001 point values to plot all figures in this section.

Example 4: KdV equation with 2-soliton solution

Table6 Numerical results for Example 5 with ε=1.0,ν=0.1,µ=0.01

where

G=2b-8t,H=4b-64t,I=b-28t,D=3b-36t

Then the corresponding analytical solution of Eqs.(15) is

Let a=-10,b =10,T=0. 5. The numerical results are presented in Table 5. And Fig.2 shows the comparison of the analytic solution with the approximate solution withn =6,dimXn=126,τ=0.0005, t=0.5.

The numerical results of Example 3 and Example 4 show that with the algorithm developed in Section 1, the KdV equations are solved stably and accurately without the need for extra stabilization processes and in relatively low dimensional subspaces. When nis changed ton+1, the root mean square error is almost reduced by a factor of 10.

2.3 Numerical results for KdVB equation

Example 5: KdV-Burgers equation

Consider the initial-boundary value problem

where A=ν2/25εµ,B=ν/10µ,C=6ν2/25µ.

The corresponding analytical solution is

We use the same approximate subspace and bases as in Example 3, choose l( x),r( x)as Eqs.(18), chooseq( x)as Eq.(21), and computeby Eq.(22). Leta=-10,b =10,T =20,ε=1.0,ν=0.1, µ=0.01. The numerical results are presented in Table 6, which shows the high stability and accuracy of the method. Figure 3 shows the comparison of the analytic solution with the approximate solution with n =4,dimXn=30,τ=0.2,t =20.

Fig.3 Comparison of exact solution with approximate solution for Example 5 with n =4,dimXn=30,τ=0.2, t=20

3. Conclusion

In this paper we combine the multi-scale Galerkin method based on the multi-scale bases constructed as in Ref.[16] and the classical fourth order Runge-Kutta method to solve the KdV equation, the Burgers equation and the KdV-Burgers equation. The algorithm presented in Section 1, is then applied to solve several equations and the numerical results are presented in Section 2. It is shown that with the algorithm, the KdV equations, the Burgers equations andthe KdV-Burgers equations can be solved stably without any stabilization technique and accurately with a limited number of bases. Moreover, the results of Example 2, Example 3 and Example 4 show the same accuracy as the generalized finite spectral method[1]. Figure 1, Fig.2 and Fig.3 show that the approximate solutions agree very well with the corresponding exact solutions by using a limited number of bases.

[1] ZHAN Jie-min, LI Yok-sheung. Generalized finite spectral method for 1D Burgers and KdV equations[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(12): 1635-1643.

[2] ZHAO Xi-zeng, SUN Zhao-chen and LIANG Shu-xiu et al. A numerical method for nonlinear water waves[J]. Journal of Hydrodynamics, 2009, 21(3): 401-407.

[3] MINA B. A., MEDHAT M. H. Group method solutions of the generalized forms of Burgers, Burgers-KdV and KdV equations with time-dependent variable coefficients[J]. Acta Mechanica, 2011, 221 (3-4): 281-296.

[4] HASIBUN N., FARAH A. A. and AHMET B. Abundant traveling wave solutions of the compound KdVBurgers equation via the improved (G’/G)-expansion method[J]. AIP Advances, 2012, 2(4): 042163.

[5] CHEN Z., MICCHELLI C. A. and XU Y. A multilevel method for solving operator equations[J]. Journal of Mathematical Analysis and Applications, 2001, 262(2): 688-699.

[6] CHEN Z., MICCHELLI C. A. and XU Y. Fast collocation methods for second kind integral equations[J]. SIAM Journal on Numerical Analysis, 2002, 40(1): 344-375.

[7] CHEN Zhong-ying, WU Bin and XU Yue-sheng. Multilevel augmentation methods for solving operator equations[J]. Numerical Mathematics: A Journal of Chinese Universities (English Series), 2005, 14(1): 31-55.

[8] CHEN Z., WU B. and XU Y. Fast multilevel augmentation methods for solving Hammerstein equations[J]. SIAM Journal on Numerical Analysis, 2009, 47(3): 2321-2346.

[9] CHEN Z., XU Y. and YANG H. A multilevel augmentaion method for solving ill-posed operator equations[J]. Inverse Problems, 2006, 22(1): 155-174.

[10] FANG W., MA F. and XU Y. Multilevel iteration methods for solving integral equations of the second kind[J]. Journal of Integral Equations and Applications, 2002, 14(4): 355-376.

[11] CHEN X., CHEN Z. and WU B. Multilevel augmentation methods for nonlinear boundary integral equations II: Accelerated quadratures and Newton iterations[J]. Journal of Integral Equations and Applications, 2012, 24(4): 545-574.

[12] CHEN X., CHEN Z. and WU B. Multilevel augmentation methods with matrix compression for solving reformulated Hammerstein equations[J]. Journal of Integral Equations and Applications, 2012, 24(4): 513-544.

[13] DING S., YANG H. Fast multiscale Galerkin methods for solving ill-posed integral equations via a coupled system under general source conditions[J]. Journal of Mathematical Analysis and Applications, 2013, 408(1): 213-224.

[14] CHEN J. Fast multilevel augmentation method for nonlinear integral equations[J]. International Journal of Computer Mathematics, 2012, 89(1): 80-89.

[15] LUO X., FAN L. and WU Y. et al. Fast multi-level iteration methods with compression Technique for solving ill-posed integral equations[J]. Journal of Computational and Applied Mathematics, 2014, 256: 131-151.

[16] CHEN Z., WU B. and XU Y. Multilevel augmentation methods for differential equations[J]. Advances in Computational Mathematics, 2006, 24(1-4): 213-238.

[17] CHEN J., CHEN Z. and CHENG S. Fast multilevel augmentation methods for solving the sine-Gordon equation[J]. Journal of Mathematical Analysis and Applications, 2011, 375(2): 706-724.

[18] STRUWE M. Variational methods[M]. New York, USA: Springer-Verlag, 2000.

* Project supported by the National Marine Public Welfare Research Projects of China (Grant No. 201005002), the National Natural Science Foundation of China (Grant No. 11071264) and the Fundamental Research Funds for the Central Universities.

Biography: CHENG Si-rui (1979-), Female, Ph. D., Lecturer

ZHAN Jie-min,

E-mail: stszjm@mail.sysu.edu.cn