Global Dissipativity for A-Stable Methods

Abstract

This paper concerns the discretization of the initial value problem $u_{t}=f(u)$ under the three structural conditions: (i) $f:{\Bbb C}^{N}\longrightarrow {\Bbb C}^{N},\;\;\;$ $\Re e\langle f(u),\, u\rangle\leq a-b|u|^{2},\;\;\;\;a\geq 0,\;b>0$ for all $u\in {\Bbb C}^{N}$; (ii) $f:{\Bbb C}^{N}\longrightarrow {\Bbb C}^{N},\;\;\;$ $\Re e\langle f(u),\,u\rangle 0$; (iii) $f:W\longrightarrow H,\;\;\;$ $\Re e\langle f(w),\,w\rangle_{H} \leq a -b|w|_{H}^{2},\;\;\;\;a\geq 0,\,b>0$ for all $w\in W$ for complex Hilbert spaces $W\subseteq H$. Dahlquist's G-stability theory is used to show that linear multistep and one-leg methods yield dissipative discretizations for all $f$ satisfying (i) if and only if the method $(\rho, \sigma)$ is A-stable. Extensions of G-stability theory are made to find necessary and sufficient conditions on $(\rho,\,\sigma)$ for similar properties to hold in cases (ii) and (iii). In every case, conditions are found for the strict contractivity of solutions for large initial data, and bounds for the asymptotic rate of decay are calculated in cases (i) and (iii).