A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation∗ Ren-Gui Yua)b)† , Ren-Hong Wanga) , and Chun-Gang Zhua) School of M athematical Sciences, Dalian U niversity of T echnology, Dalian 116024, Chinaa) Department of M athematics, Shangqiu N ormal College, Shangqiu 476000, Chinab)
In this paper, we use a multilevel quartic spline quasi-interpolation scheme to solve the onedimensional nonlinear KdV equation which exhibits a large number of physical phenomena. The presented scheme is obtained by using the second order central divided difference of the spatial derivative to approximate the third order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the proposed quasi-interpolation operator. Compared to other numerical methods, the main advantages of our scheme are the higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement. Numerical experiments in this paper also show that our scheme is feasible and valid.
Keywords: Korteweg-de Vries (KdV) equation, soliton, dispersion, multilevel Bspline quasi-interpolation
1. Introduction The Korteweg-de Vries (KdV) equation plays a significant role in the study of non-linear dispersive waves. It has been found to describe a large number of physical phenomena such as shallow water waves, ion acoustic plasma waves, bubble-liquid mixtures and wave phenomena in harmonic crystals. The KdV equation is a nonlinear partial differential equation of third order with the generalisation form[1] ut + εup ux + µuxxx = 0,
(1)
in which p, ε and µ are positive parameters. This equation exhibits both dispersion and nonlinearity. The nonlinear term in Eq.(1) causes the steeping of the wave form, whereas the dispersion term uxxx makes the wave form spread. In the case of p = 1, Eq.(1) becomes ut + εuux + µuxxx = 0,
(2)
which is known as one-dimensional nonlinear KdV equation and will be considered in this paper. This equation was originally first derived by Korteweg-de Vries[2] to describe the behavior of ∗
Project supported by National Natural Science Foundation of China (Grant Nos. U0935004, 11071031, 10801024), the Fundamental Research Funds for the Central Universities (DUT10ZD112, DUT11LK34), and National Engineering Research Center of Digital Life, Guangzhou 510006, China. † Corresponding author. E-mail:
[email protected]
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one-dimensional shallow water solitary waves to be called solitons. Solitons are localized waves with many excellent properties in physics, such as shape and velocity preserving in propagating, stability against mutual collisions. In recent times, there has been a considerable interest in the numerical solution of the KdV equation. It should be noted that obtaining the analytic solution of this equation is not a piece of cake except for appropriate initial conditions[3] . Many well-known numerical techniques, such as finite difference schemes, finite element schemes, Fourier spectral methods, heat balance integral method (HBIM), exponential finite difference method (EFDM), meshfree radial basis functions collation method[4,5,6] are employed to solve the KdV equation. Here EFDM has been shown to provide higher accuracy than classical explicit finite difference (CFDM) and HBIM in getting the numerical solution of the KdV equation at small time. Also a multiquadric (MQ) quasi-interpolation method[7] was proposed for the numerical solution of the KdV equation, in which LD operator[8] is employed. In this paper, we apply another technique, i.e., the multilevel B-spline quasi-interpolation, to establish a numerical scheme for the third order nonlinear KdV equation. As B-spline basis has compact support, our method is local and we do not need to solve any system of linear equations, hence we do not meet the question of the ill-condition of the matrix in [10]. All we need to do is simply to combine the nodal values according to certain rules. So the numerical solution is easily obtained and the computational time is also saved. In the proposed scheme, we use the derivative of the multilevel quartic spline quasi-interpolation to approximate the fist order spatial derivative and a first order forward divided difference to approximate the temporal derivative. In the meanwhile, we use the second order central divided difference of the first order spatial derivative to approximate the third order spatial derivative in order to get a better approximation order. The rest of this paper is organized as follows. In Section 2, the multilevel quartic spline quasi-interpolation is introduced, and the numerical technique for solving the KdV equation is deduced by using the multilevel quartic spline quasi-interpolation. To demonstrate the validness, efficiency and accuracy of this method, some numerical examples are given in Section 3. Also the numerical results are compared with the analytical solutions and results in [5, 7, 10]. Finally, a brief conclusion drawn from the present study is presented in Section 4.
2. Multilevel B-spline quasi-interpolation method Multilevel quasi-interpolation operator LR was firstly proposed by Ling[12] . It was an improvement of quasi-interpolation operator LD proposed by Wu and Schaback[8] . It is practical as it does not require derivative values of the function interpolated and it has a higher degree of smoothness with a shape parameter c = O(h). For f ∈ C 2 [a, b], the quasi-interpolation LR f (x) converges to f (x) at a speed of O(h2.5 logh) under the L∞ -norm provided c = O(h). In [13], a two-level cubic spline quasi-interpolation operator (QI3 ) was proposed, which is based on the
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operator LR and cubic spline quasi-interpolation Q3 . The operator has a better performance than LD , LR and Q3 in curve fitting and numerical integration. Since the dispersion term of the KdV equation is uxxx , the spline quasi-interpolation needed for numerical solving should at least be of degree 4. In the following, we will give the quartic B-spline quasi-interpolation formula firstly, then present the two-level quartic spline quasi-interpolation which is employed to solving the KdV equation numerically. Given a bounded interval I = [a, b], denoted by Sd (Xn ) the space of splines of degree d and class C d−1 on the uniform partition Xn = {xi = a + ih, i = 0, · · · , n} with meshlength h = (b − a)/n. Let a basis of Sd (Xn ) be {Bj , j = 1, 2, . . . , n + d}. For application, we add multiple knots at the endpoints: a = x0 = x−1 = · · · = x−d and b = xn = xn+1 = · · · = xn+d . Then we can get the quartic quasi-interpolation as follows[14] : Q4 f =
n+4 ∑
µj (f )Bj ,
(3)
j=1
where the coefficient functionals are: µ1 (f ) = f1 , 17 35 35 21 5 µ2 (f ) = f1 + f2 − f3 + f4 − f5 , 105 32 96 160 224 19 377 61 59 7 µ3 (f ) = − f1 + f2 + f3 − f4 + f5 , 45 288 288 480 288 47 77 251 97 47 µ4 (f ) = f1 − f2 + f3 − f4 + f5 , 315 144 144 240 1008 47 107 319 µj (f ) = (fj−4 + fj+1 ) − (fj−3 + fj−1 ) + fj−2 , j = 5, . . . , n, 1152 288 192 47 77 251 97 47 µn+1 (f ) = fn+2 − fn+1 + fn − fn−1 + fn−2 , 315 144 144 240 1008 19 377 61 59 7 µn+2 (f ) = − fn+2 + fn+1 + fn − fn−1 + fn−2 , 45 288 288 480 288 17 35 35 21 5 µn+3 (f ) = fn+2 + fn+1 − fn + fn−1 − fn−2 , 105 32 96 160 224 µn+4 (f ) = fn+2 , and fi = f (ti ), ti = 21 (xi−2 +xi−1 ), i = 1, . . . , n+2. For f ∈ C 5 (I), we have the error estimate[14] ∥f − Q4 f ∥∞ = O(h5 ). Now for I, we give a finer partition X2n = {xi = a + ih, i = 0, · · · , 2n}, h = (b − a)/2n. ˆj , j = 1, . . . , 2n + 4} the quartic B-spline basis on X2n . Let the error function of Denote by {B the quasi-interpolation (3) be ε(x) = f (x) − Q4 f (x). Then for X2n , we will obtain another quartic spline quasi-interpolation as Q4X2n ε similarly as formula (3) with the following changes in the right of (3):
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(a) change n with 2n. ˆj . (b) change Bj with B (c) change µj (f ) with µ ˆj (ε), where µ ˆj (ε) is obtained by only substituting fi for εi in the coefficient functionals µj (f ), j = 1, · · · , 2n + 4. Here εi = ε(ti ), i = 1, · · · , 2n + 2. The two-level quartic spline quasi-interpolation operator is then composed by Q4 f and Q4X2n ε n+4 m+4 ∑ ∑ ˆj . QI4 f = Q4 f + Q4Xm ε = µj (f )Bj + µ ˆj (ε)B (4) j=1
j=1
Note that the error bound by the operator QI4 f is at least O(h5 ). Differentiating interpolation polynomials leads to classical finite differences for the approximate computation of derivatives. Therefore, it seems natural to approximate derivatives of f by derivatives of QI4 f up to the order h4 . Now, we present the numerical scheme for solving the KdV equation by using the multilevel quartic spline quasi-interpolation scheme. The KdV Eq.(2) is given by prescribed boundary conditions and a set of initial conditions[17,18] . Solution of Eq.(2) results in propogation of single solition, interaction, and evolution of two solitons over the specified time and space intervals. Here we consider the third-order nonlinear KdV equation[5,10] ut + εuux + µuxxx = 0, x ∈ Ω = [a, b] ⊂ R, t > 0
(5)
with the initial condition u(x, t) = u0 (x), t = 0,
(6)
and boundary conditions u(x, t) = f (t), x ∈ ∂Ω, t > 0,
(7)
ux (b, t) = g(t), t > 0. where ε and µ are positive parameters, and u0 (x), f (t), and g(t) are known functions. Discretizing the KdV equation (5) in the time domain with step τ yields un+1 = unj − τ (εunj (ux )nj + µ(uxxx )nj ), j where unj is the approximation of the value of u(x, t) at point (xj , tn ), xj = jh, tn = nτ . Then, we use the derivative of QI4 f (x) defined by (4) to approximate ux ′
(ux )nj = (QI4 f ) (xj ) =
n+4 ∑
′
µi (f )Bi (xj ) +
i=1
m+4 ∑
′
ˆi (xj ). µ ˆi (ε)B
(8)
i=1
In order to get a better approximation of uxxx , we employ the following technique instead of the third derivative of QI4 f (x) directly (uxxx )nj =
(ux )nj+1 − 2(ux )nj + (ux )nj−1 . h2 4
(9)
Thus, the resulting numerical scheme (MBQI) is un+1 j
=
unj
−
n+4 ∑
τ (εunj (
′
µi (f )Bi (xj ) +
i=1
m+4 ∑
ˆi′ (xj )) + µ ˆi (ε)B
i=1
(ux )nj+1 − 2(ux )nj + (ux )nj−1 ). h2
(10)
By iterating this scheme, we obtain the numerical solution for the KdV equation.
3. Numerical examples In this section, some examples are employed to test the accuracy and efficiency of MBQI (10). These examples include the propagations of single solitary wave and two solitary waves. Three norms are used in these numerical results which are num L∞ = max uexact − u , i i 1≤i≤N
v u N u∑ − unum )2 , L2 = t (uexact i j i=1
v( ) u N u ∑ RM S = t (uexact − unum )2 /N , i j i=1 exact
num
where u is the exact solution, u is the numerical solution of the KdV equation by the proposed scheme. Example 1 Propagation of two solitary waves[15] . In this example, we study the interaction of two solitary waves resulting from the thirdorder nonlinear KdV equation (5) with ε = 6 and µ = 1. The initial condition at t = 0 is given by 3 + 4 cosh(2x) + cosh(4x) u0 (x) = 12 · . (3 cosh(x) + cosh(3x))2 and the exact solution is u(x, t) = 12 ·
3 + 4 cosh(2x − 8t) + cosh(4x − 64t) . (3 cosh(x − 28t) + cosh(3x − 36t))2
The boundary functions also can be obtained from the exact solution above. In Table 1, we compute the norms L∞ , L2 and RM S of errors at t = 0.01, 0.05, 0.1, respectively. In the meanwhile, the comparison of the L∞ -error with the results in [7,10] is given in Table 2. The profiles of the exact and numerical solutions at t = 0.01, 0.05, 0.1 are illustrated in Figure 1. These plots show evolution and propagation of two soliton after interaction at t=0. Example 2 A special model problem of KdV equation[4,5,16] is considered for illustrating the performance and efficiency of multilevel quartic spline quasi-interpolation scheme. Consider KdV Eq.(5) with initial condition u0 (x) = u(x, 0) = 3C1 sech2 (A1 x + D1 ), 0 ≤ x ≤ 2,
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Table 1: The norm t 0.01 0.05 0.10
errors of Example 2 at t = 0.01, 0.05, 0.1 L∞ -error L2 -error −3 3.84 × 10 1.37 × 10−2 4.13 × 10−3 1.45 × 10−1 9.29 × 10−2 3.40 × 10−1
with τ = 0.00001, h = 0.1. RMS 1.92 × 10−4 2.06 × 10−3 4.64 × 10−3
Table 2: Comparison of L∞ -error with the results in [7,10] t MBQI MQQI[7] RBF(MQ)[10] RMF(IMQ)[10] 0.01 3.84 × 10−3 7.74 × 10−3 9.21 × 10−4 2.21 × 10−2 0.05 4.13 × 10−3 6.38 × 10−2 2.96 × 10−2 7.23 × 10−2 0.10 9.29 × 10−2 1.62 × 10−1 1.28 × 10−2 1.01 × 10−1 and boundary conditions u(0, t) = u(2, t) = ux (2, t) = 0, t > 0. The exact solution of this problem is taken from [17] and is given by u(x, t) = 3C1 sech2 (A1 x − B1 t + D1 ), 0 ≤ x ≤ 2, √ where C1 , D1 are real constants, A1 = 12 εC1 /µ and B1 = εA1 C1 . The new scheme MBQI (10) is applied to Example 2 and the results of percent error are compared with those given in [5,7,10]. These techniques include RBF(MQ) scheme, HBIM scheme and MQQI scheme with τ = 0.001, h = 0.0125, C1 = 0.3, D1 = −6.0, ε = 1.0 and µ = 4.84 × 10−4 . The percent errors are given in Table 3, which shows the better performance of the proposed scheme. In Fig.2, we have plotted the profiles of the exact and numerical solution at t=0.005 and t=0.01. From the tables and figures above, we can say that the results of our scheme are acceptable, and conclude that the proposed scheme is feasible and valid. 7
7 Exact Solution Numerical Solution
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8 Exact Solution Numerical Solution
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5
5
4
4
Exact Solution Numerical Solution
7 6 5
t=0.1
t=0.05
t=0.01 3
4
3 3
2
2
1
1
0
0
0
−1 −5
−1 −5
−1 −5
2
0
5 x−axis
(a)
10
15
1
0
5 x−axis
(b)
10
15
0
5 x−axis
10
15
(c)
Figure 1: Analytical and estimated function of Example 1 at t = 0.01, 0.05, 0.1 with τ = 0.00001, h = 0.1.
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Table 3: Percentage error using different schemes of Example 2 at time t=0.005 and t=0.01. t=0.005 t=0.01 x MBQI MQQI[7] RBF(MQ)[10] HBIM[5] MBQI MQQI[7] HBIM[5] 1 0.0007 0.1082 0.0 3.8033 0.0011 1.7176 7.7548 2 0.0008 0.1019 0.0003 3.7984 0.0020 0.2049 7.7418 3 0.0001 0.0597 0.0003 3.7243 0.0007 0.1310 7.5905 4 0.0009 0.1095 0.0103 2.9326 0.0023 0.2491 5.9806 5 0.0014 0.0691 0.0060 0.7865 0.0032 0.1299 1.5010 0.0003 0.0821 0.0101 3.2960 0.0009 0.1626 6.4706 6 7 0.0000 0.0534 0.0015 3.6331 0.0005 0.1004 7.1332 8 0.0003 0.0748 0.0007 3.6626 0.0000 0.1488 7.1911 0.0002 0.0767 0.0088 3.6656 0.0002 0.1533 7.1904 9 10 0.0002 0.0769 0.0 3.7353 0.0004 0.1538 7.2016 1 0.9
Exact Solution Numerical Solution
0.9 Exact Solution Numerical Solution
0.8
0.8
0.7
0.7
0.6
0.6
0.5
t=0.01
0.5
t=0.005 0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0 0
0.5
1 x−axis
1.5
2
(a)
0
0.5
1 x−axis
1.5
2
(b)
Figure 2: Analytical and estimated function of Example 2 at t = 0.005, 0.01 with τ = 0.00001, h = 0.1.
4. Conclusions In this paper, the multilevel quartic spline quasi-interpolation method is used for solving one-dimensional nonlinear KdV equation. The numerical results given in the previous section demonstrate the good performance of the scheme. Although the accuracy of our scheme is not higher than RBF(MQ) method at some nodes, we do not need to deal with the shape parameter c, which is unresolved until now. At the same time, the proposed scheme is simple and easy to ′ implement. In application, we can calculate each Bi (xj ), i = 1, . . . , n + 4, j = 1, 2, . . . , N and ˆi′ (xj ), i = 1, . . . , m + 4, j = 1, 2, . . . , N in advance for the given data set {xi , i = 1, . . . , N } in B order to obtain a higher efficiency for computation. The scheme can also be used for non-equidistant grids, although we have used equidistant grids in our numerical experiments. Moreover, we can improve the accuracy by selecting three or even higher levels of the quartic spline quasi-interpolation, while the computation may be
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complicated in these cases. Another technique would be also employed by using quintic spline quasi-interpolation for the improvement of the accuracy of our method. For high-dimensional nonlinear KdV equations, we believe our scheme can also be applicable. In this case, we would use multivariate spline quasi-interpolation instead of univariate spline quasi-interpolation. We will consider these problems in our future work.
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