An adaptive finite point scheme for the two-dimensional coupled burgers’ equation

The dawn of this new era is marked by a profound shift in how challenging destinations are approached, as it is now centered around the art of unlocking simpler routes to explore seemingly insurmountable places. Within this transformative age, the emphasis lies on devising innovative strategies to navigate through the complexities and this paper is one such attempt to tackle the two-dimensional coupled Burgers’ equation with an emerging adaptive FPM (Finite Point Method), distinguished by its humble formulations, synergizing along with the classical Hopf-Cole transformation and operator splitting. Succinctly, the idea is to transform the coupled two-dimensional Burgers’ system into a linear two-dimensional heat equation post which this equation is split via operator splitting into one-dimensional heat equations along X and Y directions into different temporal levels, and the solutions of these one-dimensional equations are navigated through a TFPM (Shyaman et al.: J. Comput. Sci. 62, 101744 (2022)) finally with the solution of the original coupled system extracted through inverse transforms. The finest aspect of the algorithm is that the adaptive FPM involved, on an explicit 4-point stencil in every local cell, approximates the solution at the higher temporal level as a linear combination of the solutions at the preceding temporal level, and the scalars involved in the linear combination are determined using the local fundamental solutions in that cell obtained through the method of separation of variables which thereby brings in the essence of the exact solution in the formulations. The method’s theoretical foundations guarantee its conditional stability, consistency, and convergence. Analysis reveals that it exhibits first-order convergence in time and second-order convergence in space. The accuracy of the scheme is demonstrated through the successful resolution of various examples, validating its superior cost-cutting and time efficiency. Remarkably, this straightforward algorithm, devoid of complex techniques, rivals several advanced methods in the existing literature. All aspects considered the adaptive FPM offers notable benefits in solving partial differential equations, providing precise numerical solutions for complex geometries, efficiently managing high Reynolds numbers with coarse mesh resolutions, optimizing computational resources through localized stencils, and being relatively easy to implement and parallelize, making it a cost-effective and adaptable approach compared to more resource-intensive methods.