A robust numerical scheme based on uniform hyperbolic polynomial B-spline and compact finite difference method for the numerical treatment of the Burgers–Huxley equation

The primary aim of this work is to determine the numerical solution of the nonlinear Burgers–Huxley equation, which has several applications in the field of fluid mechanics, plasma physics, and mathematical biology. As this equation has a high degree of nonlinearity, finding precise numerical solutions is crucial in understanding its physical process. So, in this work, we apply a higher-order hybrid approach that integrates a compact finite difference scheme with a uniform hyperbolic polynomial (UHP) B-spline collocation method. Here, the Crank–Nicolson method is used for temporal discretization, and the resulting semi-discrete equation is further discretized using a higher-order compact finite difference approach. The quasi-linearization formula is applied to linearize the nonlinear Burgers-Huxley equation. This methodology generates a pentadiagonal system of linear equations, which is solved effectively via a pentadiagonal solver. The stability of the suggested scheme is analyzed using von Neumann analysis, and it is identified as conditionally stable. Various problems, along with comparisons to existing methods, are displayed in the tables and graphical illustrations to validate the scheme’s performance.