NUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS

Transcription

1 NUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS Thiab R. Taha Computer Science Department University of Georgia Athens, GA 30602, USA USA Italy September 21,

2 Abstract In this talk a survey and a method of derivation of certain class of numerical schemes and an implementation of these schemes will be presented. These schemes are constructed by methods related to the Inverse Scattering Transform(IST) and can be used as numerical schemes for their associated nonlinear evolution equations. They maintain many of the important properties of their original partial differential equations such as infinite numbers of conservation laws and solvability by IST. Numerical experiments have shown that these schemes compare very favorably with other known numerical methods. 2

3 1. Introduction In 1975 Ablowitz and Ladik proposed a new discrete eigenvalue problem, an appropriate generalization of a discretized version of the eigenvalue problem of Zakharov and Shabat, as a basis for generating solvable discrete equations. They derived differential-difference versions (that is nonlinear evolution equations which are discrete in space and continuous in time) of the Nonlinear Schrödinger (NLS), Korteweg-de Vries (KdV), Modified KdV (MKdV), a nonlinear Self-dual network and Toda Lattice equations. In 1977 Ablowitz and Ladik extended their ideas to derive partialdifference equations (that is nonlinear evolution equations which are discrete in space and discrete in time). They found a class of such equations and further introduced an equation which has as a limiting form the NLS equation. 3

4 In 1984 Taha and Ablowitz derived differential-difference equations that have as limiting forms the KdV, and MKdV equations. In 1991 Herbst et al derived a differential-difference equation that has a limiting form the Complex MKdV equation (CMKdV). In 1992 and 1993 Taha derived differential-difference equations that have as limiting forms the NLS, Higher NLS (HNLS), and KdV-MKdV equations. All of the above difference equations are derived by methods related to the Inverse Scattering Transform (IST). We will refer to these difference equations as IST numerical schemes. These IST schemes maintain many of the important properties of the associated nonlinear equations (such as conserved quantities, solvability by IST, etc.). 4

5 In 1991 Herbst et al. implemented the IST numerical scheme for the CMKdV equation and compared it to a standard finite difference scheme. They showed that the standard scheme is subject to instability and numerical solution becomes unbounded in finite time. In contrast they showed that the IST scheme does not suffer from any instability. Also, in 1994 Taha implemented the IST scheme for the KdV-MKdV equation and compared it to a second scheme which is a combination of the IST schemes for the KdV and MKdV equations. 5

6 Numerical experiments have shown that the IST scheme is significantly more accurate than the second scheme. The only difference between the two later schemes, is in the discretization of the nonlinear terms, and the same is true in the case of the CMKdV equation. This demonstrates the importance of a proper discretization of the nonlinear terms when a numerical scheme is designed for simulating a nonlinear differential equation. In this talk a method of derivation of IST numerical schemes, numerical simulation of the CMKdV equation, numerical simulations of KdVlike equations by the Method of Lines (MOL), and parallel numerical methods for solving nonlinear evolution equations will be presented. 6

7 2. Deriving Differential-Difference Equations The key step in obtaining differential-difference equations which can be solved by the IST is to make an association between the nonlinear evolution equation and a linear eigenvalue (scattering) problem. In this discussion all of the differential-difference equations are related to the discrete eigenvalue problem. V 1,n+1 = zv 1,n + Q n V 2,n + S n V 2,n+1, V 2,n+1 = 1 z V 2,n + R n V 1,n + T n V 1,n+1, (1) 7

8 where the eigenvalue z and the potentials R n,q n,s n, and T n are defined on the interval n < and t>0, and the associated time dependences of the eigenfunctions V in (i =1, 2) are V 1,nt = A n V 1,n + B n V 2,n, V 2,nt = C n V 1,n + D n V 2,n, (2) where the functions A n,b n,c n, and D n depend in general on the potentials. The equations for determining the sets A n,b n,c n and D n and hence the evolution equation, are obtained by assuming z t =0 and letting t (EV i,n) =E(V i,nt ), i =1, 2 (3) where E is the shift operator defined by EV i,n = V i,n+1,i=1, 2. 8

9 To derive differential-difference equations associated with the NLS, MKdV, CMKdV, KdV-MKdV equations we take T n = S n = 0. Performing the operations indicated in (3) results in four equations which are given by z(δ n A n ) C n Q n + R n B n+1 =0, 1 z B n+1 zb n + Q n (A n+1 D n )=Q nt, zc n+1 R n A n + R n D n+1 1 z C n = R nt, 1 z (Δ nd n )+C n+1 Q n R n B n =0, (4) where Δ n A n = A n+1 A n, etc. The coefficients for the time dependence of the eigenfunctions are expanded as follows: A n = 2 k= 2 z 2k A (2k) n, B n = 2 k= 1 z (2k 1) B (2k 1) n, 9

10 C n = 2 k= 1 z (2k 1) C (2k 1) n, D n = 2 k= 2 z 2k D (2k) n. (5) With the expanded form of A n,b n,c n,d n, (4) yields a system of twenty equations in eighteen unknowns corresponding to equating powers of z 5,z 5, z 4, z 4,...,z, z 1, all of which must be independently satisfied. Carrying out the algebra we find the values of A (4) n,...,d n ( 4) in terms of the potentials. The remaining two equations are the evolution equations. 10

11 Here are some nonlinear differential-difference equations associated with (4). 1. Discrete MKdV equation. Let R n = Q n, then the remaining two equations are consistent. With R n = hu n,γ = 1 (2h 3 ), δ = 1 h 3 we get the differential-difference equation u t = γ(u n+2 u n 2 )+δ(u n+1 u n 1 ) + γ [{ u 2 n+1(u n+2 + u n ) u 2 n 1(u n + u n 2 ) } (1 + u 2 n) + u 2 n(u n+2 u n 2 ) ] + δu 2 n(u n+1 u n 1 ), (6) that has as limiting form the MKdV equation u t +6u 2 u x + u xxx =0. (7) 11

12 2. Discrete NLS and HNLS equations. Let R n = Q n (where Q n is the complex conjugate of Q n ), and by a proper choice of the constants then the remaining two equations are consistent and give the nonlinear differential-difference equation Q nt = (1+ Q n 2 )[δ(q n+1 + Q n 1 ) + η{(q n+2 + Q n 2 ) + Q n+2 Q n Q n 2 Q n Q n(q 2 n+1 + Q 2 n 1) + Q n (Q n+1 Q n 1 + Q n 1 Q n+1)}] + γq n (8) Let Q n = hq n, and with η = i (12h 2 ), δ = 16i 30i, and γ = (12h 2 ) (12h 2 ), then (8) has as limiting form the NLS equation iq t = q xx +2 q 2 q, (9) 12

13 and with η = i h 4,δ = 4i h 4, and γ = 6i h 4, (8) has as limiting form the HNLS equation iq t = q xxxx +8q xx q 2 +4qq x q x +6q (q x ) 2 + 2q 2 q xx +6 q 4 q (10) 13

14 3. Discrete KdV-MKdV equation. Let Q n = u n,r n = αh βu n and by a proper choice of the constants then the remaining two equations are consistent and give the nonlinear differential-difference equation u nt = (1+αhu n )[γ(u n+2 u n 2 ) + δ(u n+1 u n 1 )] + γβ [{ u 2 n+1(u n+2 + u n ) u 2 n 1(u n + u n 2 ) } (1 + βu 2 n) + u 2 n(u n+2 u n 2 ) ] + δβu 2 n(u n+1 u n 1 ) + γαh(1 + αhu n ) { u 2 n+1 u 2 n 1 + u n+1 (u n + u n+2 ) u n 1 (u n + u n 2 )} + αβγh[u n u n+1 u n+2 (u n + u n+1 ) u n u n 1 u n 2 (u n + u n 1 ) 14

15 + 2u 2 n(u 2 n+1 u 2 n 1) + u 3 n(u n+1 u n 1 )] (11) Let u n = hu n, and with γ = 1 2h 3, and δ = 1 h 3, (11) has as its limiting form the KdV-MKdV equation u t +6αuu x +6βu 2 u x + u xxx =0,β >0 (12) 15

16 4. Discrete CMKdV equation. Let R n = Q n, and by a proper choice of the constants, the remaining two equations are consistent and give the nonlinear differentialdifference equation Q nt = (1+ Q n 2 )[δ(q n+1 Q n 1 ) + γ{q n+2 (1 + Q n+1 2 ) + Q n(q 2 n+1 Q 2 n 1)} α{q n 2 (1 + Q n 1 2 ) + Q n (Q n 1 Q n+1 Q n+1 Q n 1)}] (13) Let Q n = hu n, and with γ = 1 2h 3 and δ = 1 h 3, (13) has as its limiting form the CMKdV equation u t +6 u 2 u x + u xxx =0. (14) 16

17 To derive a differential-difference equation associated with the KdV equation u t +6uu x + u xxx = 0 (15) we take Q n = R n =0,T n = 1 in (1). The coefficients for the time dependence are expanded as A n = 2 k= 2 z 2k A (2k) n,b n = 1 k= 2 z 2k B (2k) n, C n = 2 k= 1 z 2k C (2k) n,d n = 2 k= 2 z 2k D (2k) n. Following the same procedure discussed above to derive other differentialdifference equations, and by a proper choice of the constants then the remaining equations give the nonlinear differential-difference equation S nt = (1 S n )[δ(s n+1 S n 1 )+γ{s n+2 S 2 n+1 S n+1 S n+2 S n S n+1 + S 2 n 1 + S n 1 S n 2 17

18 S n 2 + S n S n 1 }] (16) Let S n =1 e h2 u n, and with γ = 1 2h 3, and δ = 1 h 3, (16) has as its limiting form the KdV equation given in (15). The differential-difference equations for the MKdV, and KdV equations can be derived from (11) and by letting α = 0 and β 0 respectively. 18

19 III. Deriving Nonlinear partial difference equations The key step in obtaining partial difference equations which can be solved by inverse scattering transform is to make an association between the nonlinear evolution equation and a linear eigenvalue (scattering) problem. To find a nonlinear partial difference equation associated with the CMKdV equation, it is essential to use (a) a suitable eigenvalue problem e.g., V m 1n+1 = zv m 1n + Q m n V m 2n, V m 2n+1 = 1 z V m 2n + R m n V m 1n (17) where z is the eigenvalue and the potentials R m n,q m n are defined on the spacelike interval n < and the timelike interval m>0, and (b) the associated time (m) dependance of the eigenfunctions 19

20 Δ m V m 1n Δ m V m 2n = A m n (z)v m 1n + B m n (z)v m 2n, = C m n (z)v m 1n + D m n (z)v m 2n, (18) where Δ m Vin m = Vin m+1 V m in, (i = 1, 2), and the functions A m n,b m n,c m n,d m n depending in general on the potentials. The equations for determining the sets A m n,...,d m n, and hence the evolution equation are obtained by requiring the eigenvalue z to be invariant with respect to m and by forcing the consistency Δ m (E n V m in )=E n (Δ m V m in ),i=1, 2, (19) where E n is the shift operator in the spatial coordinate defined by E n V m in = V m in+1,i=1, 2. 20

21 Performing the operations indicated in (19) results in four equations which are given by zδ n A m n = Q m+1 n Cn m Rn m Rn m Bn+1, m 1 z Bm n+1 zb m n + A m n+1q m n D m n Q m+1 n =Δ m Q m n, zc m n+1 1 z Cm n + D m n+1r m n A m n R m+1 n =Δ m R m n, (20) 1 z Δ ndn m = Rn m+1 Bn m Q m n Cn+1, m where Δ n A m n = A m n+1 A m n, etc. 21

22 Using the ideas in [1,3,6], the coefficients in the equations for the time dependence of the eigenfunctions are expanded as follows: A m n = 2 k= 2 z 2k A (2k) n,b m n = 2 k= 1 z (2k 1) B (2k 1) n, C m n = 2 k= 1 z (2k 1) C (2k 1) n,d m n = 2 k= 2 z 2k D (2k) n. (21) With the expanded form of A m n,b m n,c m n,d m n, Eq. (20) yield a sequence of twenty equations in eighteen unknowns corresponding to equating powers of z 5,z 5,z 4,z 4,...,z,z 1, all of which must be independently satisfied. To solve these equations it is most convenient to solve the resulting equations corresponding to z 5 and z 5 first, then the equations corresponding to z 4,z 4, etc. Carrying out the algebra we find the values of A (4) n,...,d n ( 4) potentials [3]. in terms of the 22

23 The remaining two equations are consistent under the conditions A (i) = D ( i),i=4, 2, 0, 2, 4 (22) and R m n = (Q ) m n (where (Q ) m n is the complex conjugate of Q m n ), and the nonlinear partial difference equation is Δ m Q m n = Q m n+2a (4) Q m+1 n+2 γ n+1 D (4) + Q m n+1s n+1 Q m+1 n+1 P n Q m+1 n 2 A (4) + Q m n 2γ n 2 D (4) Q m+1 n 1 S n 2 + Q m n 1P n 1 + Q m n (A (0) + n l= T l ) Q m+1 n (A (0) + n 1 l= T l ), (23) where Δ m Q m n = Q m+1 n Q m n,γ n = n i= β n = γ n δn+1 m, S n = A (2) + A (4) F n + D (4) F n = ( Q ) m n Q m n+1 + n j= δi m+1 δi m,δi m =1+ Q m i 2, n j= H j, Δ m (( Q ) m j 1Q m j ), H n = {( Q ) m n Q m+1 n+1 δ m+1 n ( Q ) m n 1Q m+1 n δ m n }β n 1, 23

24 P n = (D (2) + n [A (4) j= E j + D (4) G j ]η j )γ n, η j = γ 1 j,e δn m n = Q m n+1( Q ) m+1 n δn m Q m n ( Q ) m+1 n 1 δn m+1, G j = (Q m j ( Q ) m j 1 Q m+1 j+1 ( Q ) m+1 j )δ m+1 j γ j 1, T l = Q m+1 l [( Q ) m+1 l 2 A (4) ( Q ) m l 2γ l 2 D (4) + ( Q ) m+1 l 1 S l 2 ( Q ) m l 1P l 1 ]+( Q ) m l [ Q m l+2a (4) + Q m+1 l+2 γ l+1 D (4) Q m l+1s l+1 + Q m+1 l+1 P l ] (24) 24

25 In the limit as Δt 0, Eq. (23) becomes Q nt = (1+ Q n 2 )[β(q n+1 Q n 1 )+α{q n+2 (1 + Q n+1 2 ) + Q n(q 2 n+1 Q n 1 ) 2 } α{q n 2 (1 + Q n 1 2 ) + Q n (Q n 1 Q n+1 Q n+1 Q n 1)}] (25) where α = A (4) D (4),β = A (2) D (2). With Q n =Δxq n, and a proper choice (see below) of the constants, taking limit as Δx 0 in Eq. (25) yields the CMKdV equation q t +6 q 2 q x + q xxx = 0 (26) 25

26 From Eq. (23) let us consider the linear part which can be written as Q m+1 n Q m n = (Q m n+2 Q m+1 n 2 )A (4) +(Q m n 2 Q m+1 n+2 )D (4) + (Q m n+1 Q m+1 n 1 )A (2) +(Q m n 1 Q m+1 n+1 )D (2) + (Q m n Q m+1 n )A (0) (27) 26

27 To choose the constants, one requires a scheme of order ((Δt) 2, (Δx) 2 ), (i.e., by expanding Q m+1 n+2,,q m n 2 in a Taylor series). With this requirement one finds A (2) = 2 3 A(0) σ, D (2) = 2 3 A(0) 1 2 σ, A (4) = 1 6 A(0) 1 4 σ, D (4) = 1 6 A(0) + 1 σ, (28) 4 where σ = Δt (Δx),andA (0) 3 = arbitrary constant. In order to get a local scheme of order 0((Δt) 2, (Δx) 2 ) for the CMKdV equation from (23), let Q m n =Δxq m n, keep the terms through order 0((Δx) 3 ) and then drop the sum terms of the form n j= Δ m (q m j q m j 1), n j= Δ m ( q m j 2 ) and replace γ n by 1. 27

28 Eq. (23) gives the local scheme qn m+1 qn m = {(qn+2 m qn 2 m+1 )A (4) +(qn 2 m qn+2 m+1 )D (4) + (qn+1 m qn 1 m+1 )A (2) + (q m n 1 q m+1 n+1 )D (2) +(q m n q m+1 n )A (0) } (Δx) 2 [A (4) {qn+1( q m ) m n+1qn+2 m+1 qn 2( q m ) m+1 n 1 qn 1 m+1 qn 1( q m ) m n qn 1 m+1 + qn+1 m+1 qn+1( q m ) m+1 n qm n 2 ((q ) m+1 n qn+2 m+1 +(q ) m+1 n 1 qn+1 m+1 + (q ) m n q m n+2 +(q ) m n 1q m n+1) + qm+1 n2 (qm+1 n 1 (q ) m+1 n+1 + qn 2 m+1 (q ) m+1 n + q m n 1(q ) m n+1 + q m n 2(q ) m n )} + D (4) {q m+1 n 1 ( q ) m+1 n 1 q m n 2 q m n+1q m+1 n+2 ( q ) m n+1 + q m+1 n (q m n 1( q ) m+1 n+1 + q m n 2( q ) m+1 n ) q m n (( q ) m n q m+1 n+2 +( q ) m n 1q m+1 n+1 ) + qm n 1 2 [( q ) m+1 n q m+1 n 1 q m n 1(q ) m n ] qm+1 n+1 2 [( q ) m+1 n q m+1 n+1 q m n+1(q ) m n ]} 28

29 + A (2) { qm n 2 (qm n+1(q ) m n + q m+1 n+1 (q ) m+1 n ) qm+1 n2 (qm+1 n 1 (q ) m+1 n + qn 1(q m ) m n )} D (2) {q m+1 n q m n 1(q ) m+1 n q m n q m+1 n+1 (q ) m n }] (29) 29

30 where A (4),D (4),A (2),D (2),A (0),D (0) are given in Eqs. (22) and (28). Eq. (29) is consistent with the CMKdV equation (26), with the truncation error of order 0((Δt) 2, (Δx) 2 ). This truncation error holds also for the full scheme given in Eq. (23). 30

31 3. MOL solution of the IST numerical schemes. The esssential features of a MOL solution are 1. The discretization of the spatial derivatives. 2. The integration of the temporal derivative which requires the integration of a system of ordinary differential equations (ODEs) in t as a result of the spatial discretization of 1. above [9]. The IST schemes are very well suited for such method. The left hand side of a differential-difference equation is the temporal derivative while the right hand side is the spatial discretization of the other terms of the evolution equation. As an example, the IST scheme for the KdV-MKdV equation with α = β =1 has been implemented and compared to a combination IST scheme which is [ 1 u nt = (1+h 2 u n ) 2h 3(u n 2 2u n 1 + 2u n+1 u n+2 )] 1 [ (1 + h 2 u 2 2h n)(u 2 n+1(u n+2 + u n ) u 2 n 1(u n + u n 2 )) + u 2 n(u n+2 u n 2 2u n+1 +2u n 1 ) + (1+h 2 u n )(u 2 n+1 u 2 n 1 + u n+1 (u n + u n+2 ) 31

32 u n 1 (u n + u n 2 ))] (30) In this combination scheme the discretization of the nonlinear part of (12), namely: 6uu x +6u 2 u x is the sum of the discretization of 6uu x from the differentialdifference equation for the KdV equation and 6u 2 u x from the differential-difference equation for the MKdV equation [8]. It is to be noted that the only difference between the IST scheme and the combination IST scheme is the discretization of the nonlinear part in (12). The truncation error of the two schemes given in (11) and (30) are of order O(h 2 ). The two schemes are applied to the KdV-MKdV equation (12) subject to the following conditions: (a) The initial condition. Eq. (12) has the exact soliton solution u(x, t) = λ/{c cosh 2 ( 1 2 λ(x λt ξ0 )) + D sinh 2 ( 1 2 λ (x λt ξ 0 ))}, (31) where ξ 0 is the integration constant and C = α 2 + βλ + α, 32

33 D = α 2 + βλ α. For initial condition, (31) is used at t =0. Wetakeα =1,β =1,λ =1 and ξ 0 =0. (b) The boundary conditions. Periodic boundary conditions on the interval [-30 70] are imposed. The numerical solution is compared to the exact solution. In addition three of the conserved quantities are computed, namely; 1 2 (αu + βu2 )dx, β 8 (u2 ) x dx, and 1 8 ((αu + βu2 ) 2 + βuu xx )dx. The implementation is carried out by using the MOL code provided by Dr. Schiesser for the solution of the KdV equation [10]. He uses the RKF45 as an ODE integrator [11]. The time integration tolerance is taken to be , and RKF45 is an adaptive method. The code is modified to take into account the spatial discretization of (12) given by the IST and the Combination IST schemes. Our numerical results show that the IST scheme is significantly more accurate 33

34 than the combination IST scheme (Table 1-2). This shows that the direct discretization of a nonlinear term in a nonlinear differential equation is crucial. Also, it shows that the numerical schemes that have been derived by methods related to the IST can play a major role in the proper discretization of nonlinear terms in nonlinear differential equations. 34

35 4. Summary In this paper a systematic method has been established for deriving differentialdifference equations, based on IST, that have as limiting forms the MKdV, NLS, HNLS, CMKdV, KdV-MKdV, and KdV nonlinear evolution equations. These differential-difference equations can be used as numerical schemes, called IST numerical schemes, for their associated nonlinear evolution equations. Moreover, these IST numerical schemes are continuous in time and discrete in space which make them very well suited to be implemented by the MOL. As an example, the IST numerical scheme for the KdV-MKdV equation was implemented by the MOL and compared with a combination IST scheme. Numerical experiments have shown that the IST numerical scheme is more accurate than the combination IST scheme. Table1showsthevaluesofL and L 2 norms at t =5andt =35fortheIST and the Combination IST schemes utilized in solving the KdV-MKdV equation with N = 401. Where L =maxσ ũ n u n, ũ n is the numerical solution and u n is the exact solution at the point (nδx, t) for all n, Δx is the increment in x. L 2 =Σ(ũ n u n ) 2 for all n, andn is the number of grid points on the interval [-30 70]. 35

36 Table 1 IST scheme Combination IST scheme t =5 L L t =35 L L

37 Table2showsthevaluesofL and L 2 norms at t =5andt =35fortheIST and the Combination IST schemes utilized in solving the KdV-MKdV equation with N = 201. L and L 2 aredefinedasintable1above. 37

38 Table 2 IST scheme Combination IST scheme t =5 L L t =35 L L Acknowledgements The author would like to thank W. E. Schiesser for providing his MOL code for the KdV equation and for helpful discussions. References [1] M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, Jour. Math. Phys. 16, #3 (1975), [2] M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, [3] M. J. Ablowitz and J. F. Ladik, On the solution of a class of nonlinear partial difference equations, Stud. Appl. Math., 57 (1977),

39 [4] T. R. Taha and M. J. Ablowitz, Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations. I. Analytical, J. Comput. Phys. 55, #2 (1984) [5] M. Herbst, M. J. Ablowitz and E. Ryan, Numerical homoclinic instabilities and the complex modified Korteweg-de Vries equation, Comput. Phys. Commun. 65 (1991) [6] T. R. Taha, A Differential-difference equation for higher order nonlinear Schrödinger equation, Computational and Applied Mathematics II. Differential Equations, (W. F. Ames and P. J. Van der Houwen, eds.), North- Holland, Amsterdam (1992) [7] T. R. Taha, A differential-difference equation for a KdV-MKdV equation, Jour. Mathematics and Computers in Simulation 35 (1993) [8] T. R. Taha, Numerical simulation of the KdV-MKdV equation, International Journal of Modern Physics C, 5, No. 2 (1994) [9] W. E. Schiesser, The Numerical Methods of Lines Integration of Partial Differential Equations, (Academic Press, San Diego 1991). 39

40 [10] W. E. Schiesser, Method of Lines Solution of the Korteweg-de Vries Equation, Computers and Math. Applic., 28, no , (1994) [11] G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematics Computations (Prentice-Hall, Englewood Cliffs, NJ 1977). [12] M. Wadati, Wave Propagation in Nonlinear Lattice I, J.Phys.Soc.Japan, 38 (1975)