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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. [ABSTRACT FROM AUTHOR]

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios rk := τk/τk-1 ≤ (3 + √17)/2 ≈ 3.561, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the L2norm. The second-order temporal convergence can be recovered if almost all of time-step ratios rk ≤ 1 + √2 or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the H1 seminorm) and the L2 norm monotonicity at the discrete levels. An example is included to support our analysis. [ABSTRACT FROM AUTHOR]