Efficient numerical simulation and physical insights of tempered fractional magnetohydrodynamic flow and heat transfer

Purpose The purpose of this study is to develop an efficient and reliable numerical framework for simulating coupled magnetohydrodynamic flow and heat transfer processes described by tempered time-fractional derivatives, which are capable of modeling memory effects in both momentum and energy transport. Design/methodology/approach A fully discrete numerical scheme is constructed by combining the L1 approximation for the temporal discretization of the tempered fractional derivatives with a spectral method for spatial discretization. To reduce the substantial computational cost induced by the nonlocal history terms, a fast algorithm based on the sum-of-exponentials approximation is incorporated. Rigorous stability and convergence analyses are carried out to assess the theoretical performance of the proposed scheme. Findings The theoretical analysis demonstrates that the numerical method is unconditionally stable and achieves convergence rates for both velocity and temperature variables. Numerical experiments confirm the accuracy and computational efficiency of the fast method and illustrate the effects of key physical parameters on the velocity profiles and temperature distributions in the coupled magnetohydrodynamic system. Originality/value This work provides a unified numerical and analytical framework for tempered fractional magnetohydrodynamic flow and heat transfer problems. By combining high-order spatial accuracy, fast temporal convolution techniques and rigorous error analysis, the proposed approach offers an effective and practical computational tool for studying complex heat and fluid flow phenomena with memory effects.