A Novel Scheme to Capture the Initial Dramatic Evolutions of Nonlinear Subdiffusion Equations

Abstract

The solution of the nonlinear subdiffusion equation has the initial layer and its initial energy may decay very fast. Therefore, it is important to investigate the evolution of the solution at the beginning. In the paper, a transformed L1 scheme is proposed to capture the initial dramatic evolution. It is proved that the temporal error of the new method is ${\mathcal {O}}(\tau ^{2-\alpha })$, where $\tau$ is the temporal stepsize and $0<\alpha <1$. The error estimate holds even when $t\rightarrow 0$. In contrast, the maximum errors of the uniform L1 scheme, the convolution quadrature (CQ) Euler method, CQ BDF method, and their corrected forms are usually ${\mathcal {O}}(\tau ^{\alpha })$ at the beginning. Meanwhile, the proposed time discretization is particularly effective for models with the small $\alpha$ and their interesting numerical phenomena are reported for the first time. Finally, numerical comparison are provided with widely used L1-type methods, the CQ methods, and their corrected forms.