*Result*: Compact ADI methods for the generalized two-dimensional Oldroyd-B type multi-term fractional mixed diffusion and diffusion-wave model.

*This article provides uniform and efficient numerical approaches for the generalized two-dimensional Oldroyd-B type multi-term fractional mixed diffusion and diffusion-wave model. Firstly, a compact alternating direction implicit (ADI) method is proposed with convergence accuracy O τ min 3 − γ l , 2 − β s , 2 − α , 1 + γ l − 2 α + h x 4 + h y 4 , where γ l , β s and α are orders of the fractional derivatives, τ , h x and h y are the time and space step sizes, respectively. The convergence analysis of the proposed compact ADI method is investigated strictly utilizing the energy estimation technique. Secondly, in order to improve the accuracy in the temporal direction, an improved compact ADI method with convergence accuracy O τ min 3 − γ l , 2 − β s , 2 − α + h x 4 + h y 4 is constructed, the convergence analysis is also clarified using the energy estimation method. Lastly, the proposed compact ADI methods are implemented on MATLAB platform. The numerical simulation results are listed in tabular forms, which manifest the validity of the derived numerical methods. • The research object of this paper consists of several complex fractional mechanical models. • Two compact ADI methods with higher spatial and temporal accuracy are derived uniformly. • The convergence analyses of the two compact ADI methods are investigated rigorously using a novel analytical technique. [ABSTRACT FROM AUTHOR]*