121 research outputs found
Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations
In this work we study the problem of step size selection for numerical
schemes, which guarantees that the numerical solution presents the same
qualitative behavior as the original system of ordinary differential equations,
by means of tools from nonlinear control theory. Lyapunov-based and Small-Gain
feedback stabilization methods are exploited and numerous illustrating
applications are presented for systems with a globally asymptotically stable
equilibrium point. The obtained results can be used for the control of the
global discretization error as well.Comment: 33 pages, 9 figures. Submitted for possible publication to BIT
Numerical Mathematic
Open issues in devising software for the numerical solution of implicit delay differential equations
AbstractWe consider initial value problems for systems of implicit delay differential equations of the formMy′(t)=f(t,y(t),y(α1(t,y(t))),…,y(αm(t,y(t)))),where M is a constant square matrix (with arbitrary rank) and αi(t,y(t))⩽t for all t and i.For a numerical treatment of this kind of problems, a software tool has been recently developed [6]; this code is called RADAR5 and is based on a suitable extension to delay equations of the 3-stage Radau IIA Runge–Kutta method.The aim of this work is that of illustrating some important topics which are being investigated in order to increase the efficiency of the code. They are mainly relevant to(i)the error control strategies in relation to derivative discontinuities arising in the solutions of delay equations;(ii)the integration of problems with unbounded delays (like the pantograph equation);(iii)the applications to problems with special structure (as those arising from spatial discretization of evolutions PDEs with delays).Several numerical examples will also be shown in order to illustrate some of the topics discussed in the paper
A -version of convolution quadrature in wave propagation
We consider a novel way of discretizing wave scattering problems using the
general formalism of convolution quadrature, but instead of reducing the
timestep size (-method), we achieve accuracy by increasing the order of the
method (-method). We base this method on discontinuous Galerkin timestepping
and use the Z-transform. We show that for a certain class of incident waves,
the resulting schemes observes(root)-exponential convergence rate with respect
to the number of boundary integral operators that need to be applied. Numerical
experiments confirm the findings
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