High order difference method for fractional convection equation

In this work, we propose a high order compact difference method for fractional convection equations (FCEs), where the Riesz derivative with order a is an element of (0, 1) is introduced in the spatial derivative. First, we prove that left and right Riemann-Liouville fractional operators are positive. Based on this, we provide an a priori estimate for the solution to FCEs, which implies the existence and uniqueness of the solution to FCEs. Then, we construct a 4th-order differential formula to approximate the Riesz derivative through a new generating function. Combining the formula with the Crank-Nicolson technique in time, we establish a high order compact difference scheme for the considered equation. A thorough analysis about the stability and convergence is conducted which shows that the proposed scheme is unconditionally stable and convergent with order O(z2 + h4). Finally, some numerical experiments are carried out to verify the theoretical analysis and to simulate the evolving process of anomalous process.