Dissipativity of linear θ-methods for integro-differential equations
Computers & Mathematics with Applications
Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations
Journal of Computational and Applied Mathematics
Dissipativity of θ-methods for nonlinear delay differential equations of neutral type
Applied Numerical Mathematics
Dissipativity of one-leg methods for neutral delay integro-differential equations
Journal of Computational and Applied Mathematics
Dissipativity of Runge-Kutta methods for Volterra functional differential equations
Applied Numerical Mathematics
Dissipativity of multistep runge-kutta methods for dynamical systems with delays
Mathematical and Computer Modelling: An International Journal
Mathematics and Computers in Simulation
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This paper concerns the discretization of the initial value problem $u_{t}=f(u)$ under the three structural conditions:\\ (i) $f:{\Bbb C}^{N}\longrightarrow {\Bbb C}^{N},\;\;\;$ $\Re e\langle f(u),\, u\rangle\leq a-b|u|^{2},\;\;\;\;a\geq 0,\;b0$ 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) $f:W\longrightarrow H,\;\;\;$ $\Re e\langle f(w),\,w\rangle_{H} \leq a -b|w|_{H}^{2},\;\;\;\;a\geq 0,\,b0$ 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).