A Two-Stage Coupled Exponential-Runge-Kutta Time Integration Scheme with Compact Spatial Discretization for Mixed Convective Thermal Flow Problems

Abstract

Applications where unsteady mixed convective flow across a stationary surface are significant, including metal processing, polymer extrusion, cooling of electronic devices, and thermal insulation systems. The coexistence of natural convection, driven by buoyancy, and forced convection, driven by external forces, results in mixed convection. Accurately capturing the dynamics of such flows, particularly when coupled with heat and mass transfer, remains a challenging task due to the nonlinear, time-dependent nature of the governing Partial Differential Equations (PDEs). A computational two-stage scheme that is a combination of the modified exponential time integrator and the Runge-Kutta method for solving time-dependent partial differential equations. A high-order compact scheme is used for spatial discretization. The stability analysis of the proposed scheme is demonstrated for scalar partial differential equations, and convergence analysis is deliberated for systems of convection-diffusion problems. In addition, a mathematical model for mixed convective flow over a stationary sheet is presented, and the proposed scheme solves its dimensionless form. The scheme is also compared with the existing Runge-Kutta scheme, and it is shown that the proposed scheme performs better than the existing Runge-Kutta scheme on the chosen Step sizes. Comparative results show that at a time step of  the proposed scheme yields an L₂-error of 2.43 × 10⁻⁴ with a runtime of 0.077 seconds, while the classical Runge-Kutta method results in a higher error of 8.47 × 10⁻⁴ and a slower runtime of 0.084 seconds, highlighting both the efficiency and stability of the proposed scheme. This work presents a robust and accurate numerical framework for simulating mixed convective transport phenomena, offering a versatile strategy for solving a broader class of time-dependent PDEs in fluid mechanics and heat transfer.