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In this article, an unconditionally stable compact high-order iterative finite difference scheme is developed on solving the two-dimensional fractional Rayleigh–Stokes equation. A relationship between the Riemann–Liouville (R–L) and Grunwald–Letnikov (G–L) fractional derivatives is used for the time-fractional derivative, and a fourth-order compact Crank–Nicolson approximation is applied for the space derivative to produce a high-order compact scheme. The stability and convergence for the proposed method will be proven; the proposed method will be shown to have the order of convergence $O(\tau + h^{4})$. Finally, numerical examples are provided to show the high accuracy solutions of the proposed scheme.

High-order compact scheme for the two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid