A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity

Abstract Accurately modelling non-Newtonian fluid flow through porous media is vital in several industrial and engineering processes, particularly where inertial effects and temperature-dependent properties are critical. This study aims to develop a high-order numerical scheme for simulating Darcy–Forchheimer flow of Williamson fluids with temperature-dependent thermal conductivity. The proposed methodology combines a second-order exponential time integrator with a sixth-order compact finite difference scheme for spatial discretization. A modified predictor–corrector structure is formulated, and its conditional stability is verified through Fourier (von Neumann) analysis. The method avoids linearization and iterative solvers, enhancing computational efficiency. Quantitative validation demonstrates that the scheme achieves second-order accuracy in time and sixth-order convergence in space, with a reduction in error norm of up to 18% compared to the classical second-order Runge–Kutta method. Parametric studies confirm the strong influence of Weissenberg number, Forchheimer number, and non-linear conductivity on velocity, temperature, and concentration fields, validating the robustness and applicability of the scheme in complex transport phenomena. All computations were carried out using MATLAB R2023a, where the proposed scheme was implemented and validated against benchmark solutions to confirm its numerical accuracy and computational efficiency.