Numerical method for fractional sub-diffusion equation with space–time varying diffusivity and smooth solution

Using a new generalized $L2$ formula and a time varying compact finite difference operator, we construct a high order numerical scheme for a class of generalized fractional diffusion equation with space–time varying diffusivity that admits a smooth solution. The convergence order is shown to be $O({\tau }_z^{3-\alpha }+h^4)$ via the energy method and demonstrated by numerical experiments. Our contributions, which improve some previous work, focus primarily on two aspects: (i) we develop a novel generalized $L2$ formula achieving $O\left({\tau }_z^{3-\alpha }\right)$ accuracy; (ii) we derive an essential a priori estimate for a time-varying compact finite difference operator, ensuring the new numerical scheme is stable and convergent.