ABSTRACT

Fractional PDEs and related nonlocal models provide an adequate and accurate description of transport processes that exhibit anomalous diffusion and/or long-range space-time interactions. Computationally, because of the nonlocal property of these models, the numerical methods often generate dense stiffness matrices. Traditionally, direct methods were used to solve these problems, which require O(N3 ) computations (per time step) and O(N2 ) mememy, where N is the number of unknowns.

We go over the development of accurate and efficient numerical methods for these nonlocal models, which has an optimal order storage and almost linear computational complexity. These methods were developed by utilizing the structure of the stiffness matrices. No lossy compression or approximation was used. Hence, these methods retaining the same accuracy and approximation/conservation property of the underlying numerical methods.

We will also discuss the open problems in the development and our future direction of research.