Global Dissipativity for A-Stable Methods
Global Dissipativity for A-Stable Methods
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DOI：
10.1137/s0036142994270971
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发表时间：
1997-02
影响因子：
2.9
通讯作者：
A. Hill
中科院分区：
文献类型：
--
作者：
A. Hill
本文讨论了初值问题$u_{t}=f(U)$在三种结构条件下的离散化：$f：{\bbb C}^{N}\long right tarrow{\bbb C}^{N}，\；\；$$\re e\langf(U)，\，u-leq a-b|u|^{2}，\；；(Ii)$f：{\bbb C}^{N}\long right tarrow{\bbb C}^{N}，\；$$\re e\langf(U)，\，u\Range0$；\\(Iii)$f：w\Longright tarrow H，\；\；$$\re e\langf(W)，\，w\Rangle_{H}\leq a-b|w|_{H}^{2}，\；\；对于复Hilbert空间$W\subseteq H$，对W$中的所有$w，都有0$。利用Dahlquist的G-稳定性理论证明了线性多步法和单支法对满足(I)条件的所有$f$产生耗散离散化当且仅当方法$(Rho，Sigma)$是A-稳定的.对G-稳定性理论进行了推广，得到了类似性质在情形(II)和(III)中成立的充要条件。在每种情况下，对于大的初始数据，找到了解的严格可缩性的条件，并在情况(I)和(III)中计算了渐近衰减率的界。
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).