Numerical solutions of the generalized equal width wave equation using Petrov Galerkin method
For researchers in numerical analysis and applied sciences needing accurate solutions to nonlinear wave equations, this work provides a validated numerical scheme with proven stability and superior performance.
The paper develops a Petrov-Galerkin method for the generalized equal width wave equation, proving stability and error bounds, and demonstrates through numerical experiments that the scheme outperforms existing methods in terms of accuracy and conservation of invariants.
In this article we consider a generalized equal width wave (GEW) equation which is a significant nonlinear wave equation as it can be used to model many problems occurring in applied sciences. As the analytic solution of the (GEW) equation of this kind can be obtained hardly, developing numerical solutions for this type of equations is of enormous importance and interest. Here we are interested in a Petrov-Galerkin method, in which element shape functions are quadratic and weight functions are linear B-splines. We firstly investigate the existence and uniqueness of solutions of the weak form of the equation. Then we establish the theoretical bound of the error in the semi-discrete spatial scheme as well as of a full discrete scheme at $t=t^{n}$. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of single and double solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed numerical scheme by calculating the error norms (in $L_{2}(Ω)$ and $L_{\infty}(Ω)$). The three invariants ($% I_{1},I_{2}$ and $I_{3})$ of motion have been commented to verify the conservation features of the proposed algorithms. Our proposed numerical scheme has been compared with other published schemes and demonstrated to be valid, effective and it outperforms the others.