The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using a $k$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results.
翻译:本文的目的是研究大约用一个微弱单一来源术语解决子扩散方程式的时间阶梯计划,在此情况下,许多流行的时间梯计划,包括纠正高阶BDF方法,可能会失去其高度的准确性。为了填补这一空白,我们在本文件中制定了一个新的时间阶计划,即源词通过使用千元倍的分解分解分解,而方程式则通过使用一个经过修改的BDF2演化二次曲线来分解。我们证明,拟议的时间阶梯计划是次等的,即使源词在时间上不时且与初始数据不相容。我们用数字结果来支持理论结果。