Correction of high-order BDF convolution quadrature for fractional evolution equations
This work addresses the convergence degradation issue in high-order BDF methods for fractional PDEs, providing a practical fix for computational scientists solving subdiffusion and diffusion-wave problems.
The authors developed correction formulas for the starting steps of BDF convolution quadrature to achieve k-th order convergence for fractional evolution equations, even with nonsmooth data and incompatible source terms. Numerical examples confirm the effectiveness.
We develop proper correction formulas at the starting $k-1$ steps to restore the desired $k^{\rm th}$-order convergence rate of the $k$-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired $k^{\rm th}$-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case $α\in (0,1)$, and sketch the proof for the diffusion-wave case $α\in(1,2)$. Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.