An integrated stairwise adaptive finite point scheme for the two-dimensional coupled Burgers' equation

This paper explores the potential of a streamlined adaptive finite point method (FPM) in tackling two-dimensional coupled Burgers' equations, employing them as a testbed for further advancements. Firstly the coupled system is transformed into a two-dimensional heat equation through Cole-Hopf transformation and then this transformed equation is split into one-dimensional heat equations at intermediate temporal levels along X and Y directions and these one-dimensional equations are finally to be treated with the adaptive FPM. The distinctive feature of the adaptive FPM used here lies in employing an implicit 4-point stencil within each local cell to compute the solution at a higher temporal level through a linear combination of solutions from the preceding temporal level. The coefficients involved in this linear combination are derived via the local fundamental solutions within that cell, thereby imbuing the formulations with the intrinsic essence of the exact solution. Moreover, the separation constant utilized is tailored to consistently integrate the influence of the initial solution, independent of the temporal level. The method's theoretical underpinnings ensure its conditionally stable, consistent, and convergent behavior. The accuracy of the scheme is substantiated by its proficient handling of diverse examples, attesting to its superior cost-effectiveness and time efficiency.