A fourth-order numerical scheme for the spatial Riesz tempered fractional diffusion equation

This paper proposes a high-order numerical scheme for the spatial Riesz tempered fractional diffusion equation, which characterizes transient anomalous diffusion phenomena with exponentially tempered power-law jumps. By introducing a tempered compact differential operator, a fourth-order compact difference approximation is constructed for the Riesz tempered fractional derivative. The temporal discretization employs the (2,2) Padé approximation, yielding a fully discrete scheme that achieves fourth-order accuracy in both space and time. The stability and convergence of the scheme are established by the energy method. Numerical experiments confirm the effectiveness and advancement of the proposed scheme.