A note on the stability of two families of two-step schemes

Abstract

We investigate the stability of two families of three-level two-step schemes that extend the classical second order BDF (BDF2) and second order Adams-Moulton (AM2) schemes. For a free parameter restricted to an appropriate range that covers the classical case, we show that both the generalized BDF2 and the generalized AM2 schemes are A-stable. We also introduce the concept of uniform-in-time stability which characterizes a scheme's ability to inherit the uniform boundedness over all time of the solution of damped and forced equation with the force uniformly bounded in time. We then demonstrate that A-stability and uniform-in-time stability are equivalent for three-level two-step schemes. Next, these two families of schemes are utilized to construct efficient and unconditionally stable IMEX schemes for systems that involve a damping term, a skew symmetric term, and a forcing term. These novel IMEX schemes are shown to be uniform-in-time energy stable in the sense that the norm of any numerical solution is bounded uniformly over all time, provided that the forcing term is uniformly bounded time, the skew symmetric term is dominated by the dissipative term, together with a mild time-step restriction. Numerical experiments verify our theoretical results. They also indicate that the generalized schemes could be more accurate and/or more stable than the classical ones for suitable choice of the parameter.

Figures (6)

• Figure 1: Loglog plot for the relative error of the generalized BDF2 method
• Figure 2: Loglog plot for the relative error of the generalized AM2 method
• Figure 3: Loglog plot for the relative error of the generalized AM2 method with $T_{end}=10^4$
• Figure 4: Approximation solution of the generalized IMEX BDF2 method with different parameter $\alpha$. Left panel $h=0.1, T=100$. Right panel $h=10, T=10000$.
• Figure 5: Solution of the generalized BDF2 scheme with different parameter $\alpha$. Left panel $h=100, T=2\times 10^4$. Right panel $h=200, T=1\times 10^5$.

Theorems & Definitions (15)

• proposition thmcounterproposition
• proposition thmcounterproposition
• theorem 1
• proof
• theorem 2
• proof
• definition thmcounterdefinition: Uniform-in-time (energy) stability
• theorem 3: Equivalence between A-stability and Uniform-in-time (energy) stability
• proof
• theorem 4: Unconditional and uniform-in-time energy bound for the generalized BDF2-IMEX method