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518 chapter 1919.5.2 Algorithm for KdeV SolitonsThe KdeV equation is solved numerically using a finite-difference scheme with thetime and space derivatives given by central-difference approximations:∂u∂t ≃ u i,j+1 − u i,j−1,2∆t∂u∂x ≃ u i+1,j − u i−1,j. (19.23)2∆xTo approximate ∂ 3 u(x, t)/∂x 3 , we expand u(x, t) to O(∆t) 3 about the four pointsu(x ± 2∆x, t) and u(x ± ∆x, t),u(x ± ∆x, t) ≃ u(x, t) ± (∆x) ∂u∂x + (∆x)22!∂ 2 u∂ 2 x ± (∆x)33!∂ 3 u∂x 3 , (19.24)which we solve for ∂ 3 u(x, t)/∂x 3 . Finally, the factor u(x, t) in the second term of(19.19) is taken as the average of three x values all with the same t:u(x, t) ≃ u i+1,j + u i,j + u i−1,j. (19.25)3These substitutions yield the algorithm for the KdeV equation:u i,j+1 ≃ u i,j−1 − ɛ 3∆t∆x [u i+1,j + u i,j + u i−1,j ][u i+1,j − u i−1,j ]−µ ∆t(∆x) 3 [u i+2,j +2u i−1,j − 2u i+1,j − u i−2,j ] . (19.26)To apply this algorithm to predict future times, we need to know u(x, t) at presentand past times. The initial-time solution u i,1 is known for all positions i via theinitial condition. To find u i,2 , we use a forward-difference scheme in which weexpand u(x, t), keeping only two terms for the time derivative:u i,2 ≃ u i,1 − ɛ ∆t6∆x [u i+1,1 + u i,1 + u i−1,1 ][u i+1,1 − u i−1,1 ]− µ 2∆t(∆x) 3 [u i+2,1 +2u i−1,1 − 2u i+1,1 − u i−2,1 ] . (19.27)The keen observer will note that there are still some undefined columns of points,namely, u 1,j , u 2,j , u Nmax−1,j , and u Nmax,j , where N max is the total number of gridpoints. A simple technique for determining their values is to assume that u 1,2 =1 and u Nmax,2 =0. To obtain u 2,2 and u Nmax−1,2 , assume that u i+2,2 = u i+1,2 andu i−2,2 = u i−1,2 (avoid u i+2,2 for i = N max − 1, and u i−2,2 for i =2). To carry outthese steps, approximate (19.27) so thatu i+2,2 +2u i−1,2 − 2u i+1,2 − u i−2,2 → u i−1,2 − u i+1,2 .−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 518