A Comperative Numerical Study Based on Cubic Polynomial and Trigonometric B-splines for the Gardner Equation

TL;DR

This paper compares cubic polynomial and trigonometric B-splines in collocation methods for numerically solving the Gardner equation, analyzing various wave models and validating results through error measurement and conservation laws.

Contribution

It introduces a comparative study of two B-spline based collocation methods for the Gardner equation, including stability and conservation law analysis.

Findings

Both methods accurately model wave propagation scenarios.

The error norms confirm the effectiveness of the proposed methods.

Conservation laws remain nearly constant, indicating stability.

Abstract

Two cubic B-spline functions from different families are placed to the collocation method for the numerical solutions to the Gardner equation.Four models describing propagation of bell shaped single solitary, travel of a kink type wave, wave generation and interaction of two positive bell shaped solitaries propagating in the opposite directions are studied by both methods. The error between the numerical and the analytical solutions ismeasured by using the discrete maximum norm when the analytical solutions exist. The absolute changes of the lowest three conservation laws are also good indicators of valid results even when the analytical solutions do not exist. The stability of the proposed method is investigated by the Von Neumann analysis.