In this paper, an $$\mathcal {A}$$ A -stable four-step block method of order seven has been developed to numerically approximate the solution of first-order ordinary differential equations obtained through the semi-discretization of first-order time-dependent physically relevant partial differential equations (PDEs). The spatial-derivatives involved in the PDE are discretized utilizing conventional compact finite difference schemes (CFDS) of order four, and the developed block method is then been implemented to advance the solution in the time direction. The implementation in variable step-size mode of the proposed strategy is also discussed. The generalized form of the Burgers-Huxley equation, along with some of its special cases, is examined as benchmark problems to demonstrate the efficacy of the proposed strategy. A special case of the generalized Burgers-Huxley equation, known as the Chafee-Infante problem, involving Neumann boundary conditions, is solved accurately using this approach. Additionally, it has efficiently solved a challenging case of the Burgers’ equation for small viscosity coefficients, as well as a complex case involving the FitzHugh-Nagumo equation with stiff parameter values. The variable step-size implementation on the considered test problems has turned out to be more computationally efficient than the fixed step-size approach. It can be observed that this approach provides a very efficient algorithm compared to some existing techniques in the literature.

In this article, we have investigated the performance of a well-known predictor-corrector MacCormack method coupled with compact finite difference scheme for solving third order nonlinear Korteweg-de Vries equation.The numerical scheme has second order accuracy in time and the order of accuracy in space is the same as that of the order of a compact finite difference scheme used for approximating the spatial derivatives present in the equation. This compact version of the classical MacCormack method has a very good performance in comparison with its classical counterpart and some other existing approaches for solving the nonlinear KdV equation.