An analysis on the asymptotic behavior of multistep linearly implicit schemes for the Duffing equation
نویسندگان
چکیده
We consider the discrete gradient method for dissipative linear-gradient systems, which strictly replicates the dissipation property, yielding a remarkable stability. However, it also replicates the nonlinearity of an original equation. To overcome this, we can employ multistep linearly implicit schemes as a relaxation; however, it can in turn destroy the originally aimed stability. Matsuo–Furihata (2014) introduced a dynamical systems viewpoint to understand the behavior for a toy scalar problem. In this letter, we show that their method can work also for the two-dimensional Duffing equation. There a new concept of semi-strong Lyapunov functionals is required.
منابع مشابه
A new two-step Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrodinger equation and related IVPs with oscillating solutions
A new two-step implicit linear Obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrodinger equation and related problems. This algorithm belongs in the category of the multist...
متن کاملLinearly implicit methods for nonlinear evolution equations
We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit–explicit multistep schemes as well as the combination of implicit Runge–Kutta schemes and ext...
متن کاملLinearly implicit methods for nonlinear parabolic equations
We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes...
متن کاملThe new implicit finite difference method for the solution of time fractional advection-dispersion equation
In this paper, a numerical solution of time fractional advection-dispersion equations are presented.The new implicit nite dierence methods for solving these equations are studied. We examinepractical numerical methods to solve a class of initial-boundary value fractional partial dierentialequations with variable coecients on a nite domain. Stability, consistency, and (therefore) convergenceof t...
متن کاملSuperlinearly convergent exact penalty projected structured Hessian updating schemes for constrained nonlinear least squares: asymptotic analysis
We present a structured algorithm for solving constrained nonlinear least squares problems, and establish its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of N...
متن کامل