Stability and convergence of the L1 formula on nonuniform time grids are studied for solving linear reaction-subdiffusion equations with the Caputo derivative. A discrete fractional Grönwall inequality is developed for the nonuniform L1 formula by introducing a discrete convolution kernel of Riemann--Liouville fractional integral. To simplify the consistency analysis of the nonuniform L1 formula, we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete Riemann--Liouville integral kernel. With the help of discrete fractional Grönwall inequality and global consistency error analysis, a sharp error estimate reflecting the regularity of solution is obtained for a simple L1 scheme. Numerical examples are provided to verify the sharpness of the error analysis.

We construct and analyze a class of extrapolated and linearized Runge--Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen--Cahn and Cahn--Hilliard ...

Spatial discretizations of time-dependent partial differential equations usually result in a large system of semi-linear and stiff ordinary differential equations. Taking the structures into account, we develop a family of linearly implicit and high order accurate schemes for the time discretization, using the idea of implicit-explicit Runge-Kutta methods and the relaxation techniques. The proposed schemes are monotonicity-preserving/conservative for the original problems, while the previous linearized methods are usually not. We also discuss the linear stability and strong stability preserving (SSP) property of the new relaxation methods. Numerical experiments on several typical models are presented to confirm the effectiveness of the proposed methods.