A Comperative Numerical Study Based on Cubic Polynomial and Trigonometric B-splines for the Gardner Equation
An incremental comparison of two numerical methods for solving a specific nonlinear partial differential equation (Gardner equation) in computational physics.
This study compares cubic polynomial and trigonometric B-spline methods for solving the Gardner equation, demonstrating that both methods produce accurate and stable numerical solutions for various wave propagation models, with error measured by discrete maximum norm and conservation laws.
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.