Convergence of consistent and inconsistent schemes for fractional diffusion problems with boundaries

An implicit numerical method for a fractional diffusion problem in the presence of an absorbing boundary is analyzed. The discretization chosen for the spatial fractional differential operator is known to be second-order accurate, when the problem is defined in the real line. The main purpose of this work is to show how the presence of the boundary can change the properties of the scheme, namely its consistency and convergence. We establish that the order of accuracy of the spatial truncation error can be lower than two in the presence of the boundary and in some cases we have inconsistency, depending not necessarily on the regularity of the solution but on the values of its derivatives at the boundary. Furthermore, we prove the rate of convergence will be higher than the order of accuracy given by the consistency analysis and sometimes we can recover the order two. In particular, the convergence is achieved for some of the inconsistent cases.

Automated summary

This study analyzes an implicit numerical method for a fractional diffusion problem with an absorbing boundary. The presence of the boundary is shown to change the properties of the scheme, affecting its consistency and convergence. The authors establish that the accuracy of the method can be lower than second-order in the presence of the boundary, and the convergence rate can be higher than the order of accuracy. In some cases, the second-order accuracy can still be achieved.