Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations

摘要

We propose and study a class of numerical schemes to approximate timefractional differential equations. The methods are based on the approximationof the Caputo fractional derivative by continuous piecewise polynomials, whichis strongly related to the backward differentiation formulae for theinteger-order case. We investigate their theoretical properties, such as thelocal truncation error and global error analyses with respect to a sufficientlysmooth solution, and the numerical stability in terms of the stability regionand $A(\frac{\pi}{2})$-stability by refining the technique proposed in\cite{LubichC:1986b}. Numerical experiments are given to verify the theoreticalinvestigations.