Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines

Abstract

In this work, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by the method based on collocation of quintic B-splines over the finite elements. A linear stability analysis shows that the numerical scheme based on Von Neumann approximation theory is unconditionally stable. Test problems including the solitary wave motion, the interaction of two and three solitary waves and the Maxwellian initial condition are solved to validate the proposed method by calculating error norms L 2 and L ∞ that are found to be marginally accurate and efficient. The three invariants of the motion have been calculated to determine the conservation properties of the scheme. The obtained results are compared with other earlier results. MSC: 97N40; 65N30; 65D07; 76B25; 74S05

An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method

“…Simulations are run up to time = 20. Error norms 2 and ∞ and conserved quantities are tabulated in Table 3, together with the results obtained in [6,30,32,[34][35][36]. These results show high degree of accuracy and efficiency of the method.…”

“…For the computational work, two sets of parameters have been chosen and discussed. First of all, we have taken the parameters = 1, = 1, ℎ = 0.2, 0 = 40, = 0.025 over the interval [0, 100] to compare our results with [6,29,30,35,36]. Thus, the solitary wave has an amplitude 1.0 and the computations are done up to time = 10 to obtain the invariants and error norms 2 and ∞ .…”

“…Secondly, we have taken the parameters = 1, = 0.3, ℎ = 0.1, = 0.01 and 0 = 40 with range [0, 100] to enable comparison with [6,30,32,[34][35][36]. So the solitary wave has amplitude 0.547723.…”

“…Ali [35] has formulated a classical radial basis functions (RBFs) collocation method for solving the equation. Karakoc et al [36] have obtained a type of the quintic B-spline collocation procedure in which nonlinear term in the equation is linearized by using the form introduced by the Rubin and Graves [37] to solve the equation. A Petrov-Galerkin method using cubic B-spline function as trial function and a quadratic B-spline function as the test function is set up to solve the equation by Karakoc and Geyikli [38].…”

Numerical solutions of the MRLW equation using moving least square collocation method

“…These include …nite element, …nite di¤erence, Fourier method and meshless methods. The MRLW equation was solved by various types of B-spline functions by using …nite element method such as the collocation method with quintic B-splines …nite element method in [3], the collocation method using cubic B-splines …nite element in [7], a numerical scheme based on quartic B-spline method in [17], based on collocation of quintic B-splines …nite elements in [18] was presented. Also the collocation method with quadratic, cubic, quartic and quintic B-splines [10], cubic B-spline lumped Galerkin …nite element method [11], a Petrov-Galerkin method [13] were used.…”