A higher order implicit adaptive finite point method for the Burgers' equation

ABSTRACT

The method proposed in this manuscript is constructed for the classical non-linear Burgers' equation on the basis of the tailored finite point method. Attempts have been made in the direction to enhance the tailored finite point method with appealing features like cost-effectiveness and rapidity through the usage of a minimum machinery algorithm, ability to handle shocks and boundary layers without mesh refinement, and continuous dependence on the initial conditions throughout the algorithm. Also, the foundation of the construction of the algorithm is completely based on the local properties of the solution which thereby brings in the influence of localized behaviours even when the kinematic viscosity is infinitesimal. Moreover, the usage of a 4-point stair stencil makes it an implicit scheme, and the adaptation of adequate linearization techniques reduces the usage of higher-order machinery to handle non-linearity. The conditional stability, consistency, and convergence of the method have been deliberated with appropriate theories. The method is second-order convergent in space and first-order convergent in time. The credibility of the method is tested through diverse test examples by comparing it with the exact solutions and other methods in the literature and the results are found to be at par excellence with the classical methods.

KEYWORDS:

• Burgers' equation
• TFPM
• 4-point stair stencil
• localized basis function
• global separation constant

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M50

Disclosure statement

No potential conflict of interest was reported by the author(s).