Quasi-compact fourth-order approximations for fractional derivatives and applications

Two fourth-order approximations for fractional derivatives are presented. These approximations are constructed from an approximation of order 2 through convex combinations of two different shifts eliminating the third-order term and then approximating the remaining second-order term with a central difference approximation. The constructed approximations are applied in Crank- Nicolson schemes to the one-dimensional space fractional diffusion equation. Numerical tests confirm that the schemes are unconditionally stable for any choice of space discretization and converge with order 4.