Simulations of the EW undular bore

Abstract

SUMMARYThe equal width equation is solved by a Petrov±Galerkin method using quadratic B-spline spatial ®nite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary dierential equations is obtained via a Crank±Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an EW undular bore is investigated and compared with that of the RLW bore.

Spectral method for solving the equal width equation based on Chebyshev polynomials

“…In a corresponding simulation using a collocation method with quartic spline elements [9], the L 2 -error norm at t = 20 is less than 0.6463 × 10 −3 and the quantity I 1 does not change from the analytic, I 2 varies by less than 0.043%, and I 3 varies by less than 0.005%. Second, we model the motion of a single solitary wave with the three different amplitudes 0.3, 0.09, and 0.03 and compare with results given in [6,7,9] at times t = 40, 80; see Tables 3-7, using the region 0 ≤ x ≤ [9] x = 0.03, t = 0.2 40 1.199992 0.292159 0.057599 0.0795 Least-square [7] x = 0.03, t = 0.03 40 1.1967 0.2860 0.0570 3.475 Least-square [7] x = 0.03, t = 0.03 80 1.1964 0.2858 0.0569 7.444 Petrov-Galerkin [6] x = 0.03, t = 0.05 80 1.1910 0.2855 0.0558 3.849 Table 4 Error norms for a single solitary wave: Table 3 where the L 2 -error norm is less than 0.0038 × 10 −3 and the constants of motion vary little from the analytic value: I 1 varies by less than 0.002%, I 2 and I 3 are conserved during the experiment. In a corresponding simulation using a collocation method with quartic spline elements [9], the L 2 -error norm is less than 0.0796 × 10 −3 and the quantity I 1 changes by less than 0.0007%, I 2 by less than 1.5%, and I 3 varies by less than 0.002%, and using a least-squares method with linear spline elements [7] the L 2 -error norm is less than 3.476 × 10 −3 and the quantity I 1 changes by less than 0.28%, I 2 by less than 0.7%, and I 3 varies by less than 1.1%.…”

“…Approximate solutions for solving EWE using Galerkin's method with both cubic B-spline finite elements [4,5], a Petrov-Galerkin method using quadratic B-spline finite elements [6], Zaki [7,8] has solved EW equation by a leastsquare technique using linear space-time finite elements and Petrov-Galerkin finite element scheme with shape functions taken as quadratic B-spline functions, respectively. Recently, Raslan [9] has solved the EW equation using collocation method with quartic Bspline finite elements and solved the resulting system of first-order ordinary differential equations using the fourth-order Runge-Kutta method.…”

“…In a corresponding simulation using a collocation method with quartic spline elements [9], the L 2 -error norm is less than 0.0796 × 10 −3 and the quantity I 1 changes by less than 0.0007%, I 2 by less than 1.5%, and I 3 varies by less than 0.002%, and using a least-squares method with linear spline elements [7] the L 2 -error norm is less than 3.476 × 10 −3 and the quantity I 1 changes by less than 0.28%, I 2 by less than 0.7%, and I 3 varies by less than 1.1%. At time t = 80, the L 2 -error norm is less than 0.2053 × 10 −3 , which is smaller than the previous results [6,7] 3.849 × 10 −3 and 7.444 × 10 −3 , respectively. The invariant value I 1 changes by less than 0.094%, I 2 by less than 0.0004%, and I 3 does not change and in the corresponding simulation using Petrov-Galerkin method with cubic spline finite elements [6] the invariant values I 1 , I 2 , and I 3 change by less than 0.75, 0.87, and 3.13%, respectively, and using a least-squares method with linear spline elements [7] the quantity I 1 changes by less than 0.3%, I 2 by less than 0.74%, and I 3 varies by less than 1.22%.…”

“…At time t = 80, the L 2 -error norm is less than 0.2053 × 10 −3 , which is smaller than the previous results [6,7] 3.849 × 10 −3 and 7.444 × 10 −3 , respectively. The invariant value I 1 changes by less than 0.094%, I 2 by less than 0.0004%, and I 3 does not change and in the corresponding simulation using Petrov-Galerkin method with cubic spline finite elements [6] the invariant values I 1 , I 2 , and I 3 change by less than 0.75, 0.87, and 3.13%, respectively, and using a least-squares method with linear spline elements [7] the quantity I 1 changes by less than 0.3%, I 2 by less than 0.74%, and I 3 varies by less than 1.22%. Tables 6 and 7, respectively, with the corresponding previous results.…”

Evaluation of two-dimensional ZK-MEW equation using the Exp-function method

“…The goal of the present paper is to apply the Exp-function method to the two-dimensional ZK-MEW equation [22][23][24][25][26][27][28][29][30][31][32][33] to finding some new solitonary solutions, and shows the great effectiveness of the introduced method.…”