Efficient solution of one-dimensional time-dependent partial differential equations using a multi-step block method combined with compact finite difference schemes

Abstract In this paper, an $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> -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.