An implicit tailored finite point method for the Burgers’ equation: leveraging the Cole-Hopf transformation

Abstract

Any expedition in designing numerical methods besides aiming at accuracy, also equally steers for simplicity and ease in implementation. This paper brings in one such algorithm the tailored finite point method (TFPM) in tandem with the Cole Hopf transformation. At the initiation, the non-linear Burgers’ equation is transformed into a linear heat equation to which TFPM is applied. The proffered TFPM functions on an explicit stencil on the left boundary of the domain and on an implicit stair stencil throughout the rest of the domain. On these stencils, the nodal solutions at the advanced temporal level are written as a linear combination of the solutions at the remaining nodes within the stencil. The scalars involved in the linear combination are identified by the application of fundamental solutions into the stencil resultantly infusing the essential nature of the local exact solutions into the approximations. The foundation of such a linear combination avoids the need for complex computations involving matrix multiplication and inversion. The numerical accuracy of the method is established through comparisons of TFPM solutions of classical examples with the exact solutions and solutions from other contemporary methodologies. The theoretical correctness of the method is established through analyses of consistency, stability, and convergence. Furthermore, the method exhibits the potential for extension to higher dimensions and other complex modalities.