A B‐spline finite element method for the regularized long wave equation

Abstract

SUMMARYA B-spline finite element method is used to solve the regularized long wave equation numerically. This approach involves a Galerkin method with quadratic B-spline finite elements so that there is continuity of the dependent variable and its first derivative throughout the solution range. Time integration of the resulting system of ordinary differential equations is effected using a Crank-Nicolson approximation. Standard problems are used to validate the algorithm, which is then used to model the smooth development of an undular bore.

Numerical integration of the RLW equation using cubic splines

“…Finite difference methods were proposed based on both cubic and quintic splines in the papers [3,11]. Some variants of finite element and collocation methods were constructed for the RLW equation by using B-splines as weight and trial functions [6][7][8][9]. The use of cubic splines in numerical methods for finding solutions of PDEs leads to a matrix system which is tridiagonal, thus permitting the use of the Thomas algorithm.…”

Numerical study of the Benjamin-Bona-Mahony-Burgers equation

“…In addition, we can see that the graph shows the same behavior as in [7]. T ime Table 7: Numerical results for Example 4.1 with h = 0.1, N = 1000 and c = 0.03 with the algorithim of [4,5]. …”

“…The initial condition is taken from the solution. Also the solution satisfy three conservation laws: Table 3 and Table 6 give C 1 , C 2 , C 3 , L ∞ and L 2 found by our method in different times for c = 0.1, 0.03 and Table 4 and Table 7 give numerical results from method in [4,5]. Figure.…”

“…Also from Figure 3 we can see that numerical solutions show the same behavior as solutions. T ime Table 4: Numerical results for Example 4.1 with h = 0.1, N = 1000 and c = 0.1 with the algorithim of [4,5]. Table 9 and Table 10 give numerical results found by our method in different times.…”

“…In recent years, many different methods have been used to estimate the solution of the Benjamin-Bona-Mahony-Burgers equation and the BBM equation, for example, see [3,4,5,7,11]. The paper is organized as follows.…”