NUMERICAL SOLUTIONS OF EQUAL WIDTH EQUATION USING HYBRID CUBIC B-SPLINE METHOD

Mehreen Zahid; Tayyba Arif; Fakhar Waqqas; Ayesha Tanveer

doi:10.71146/kjmr668

Authors

Mehreen Zahid The Superior University Lahore, Pakistan. Author
Tayyba Arif The Superior University Lahore, Pakistan. Author
Fakhar Waqqas The Superior University Lahore, Pakistan. Author
Ayesha Tanveer The Superior University Lahore, Pakistan. Author

DOI:

https://doi.org/10.71146/kjmr668

Keywords:

Equal Width equation, Hybrid Cubic B-spline, Finite difference scheme, Solitary waves, Von Neumann stability, Conserved invariants

Abstract

This study presents a strong and very accurate numerical solution of the non-linear Equal Width (EW) equation. The numerical method proposed here is based on a finite difference method for the time discretization and a Hybrid Cubic B-spline (HCBS) spatial approximation method. The Rubin and Graves quasilinearization method is used to handle the non-linear term that is part of the EW equation. Proposed finite difference scheme is rigorously proven to be unconditionally stable by applying Von Neumann (Fourier) stability analysis method. Moreover, a formal convergence analysis is presented that shows that the second-order accuracy is reached both in time and space . For the validation of the theoretical framework three classic numerical examples are studied: propagation of a single solitary wave, interaction of two solitary waves and the evolution of wave trains from a Maxwellian initial condition. Three conserved motion invariants ( ) are used to evaluate the numerical performance along with the error norms ( ) and ( ). Numerical results show the superiority of the Hybrid Cubic B-spline scheme in terms of spatial resolution and superior conservation properties, over the standard B-spline schemes reported in the literature.

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