The Second-order in Time Continuous Newton Method

2001 ◽  
pp. 25-36 ◽  
Author(s):  
H. Attouch ◽  
P. Redont
Keyword(s):  
Newton Method ◽  
Second Order ◽  
Second Order In Time ◽  
Continuous Newton Method

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

scholarly journals Linear second-order IMEX-type integrator for the (eddy current) Landau–Lifshitz–Gilbert equation

2019 ◽  
Vol 40 (4) ◽  
pp. 2802-2838 ◽  
Author(s):  
Giovanni Di Fratta ◽  
Carl-Martin Pfeiler ◽  
Dirk Praetorius ◽  
Michele Ruggeri ◽  
Bernhard Stiftner
Keyword(s):  
Eddy Current ◽  
Maxwell Equations ◽  
Coupled System ◽  
Second Order ◽  
Stray Field ◽  
Time Step ◽  
Finite Element Scheme ◽  
Second Order In Time ◽  
Current Approximation ◽  
Numer Math

Abstract Combining ideas from Alouges et al. (2014, A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation. Numer. Math., 128, 407–430) and Praetorius et al. (2018, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics. Comput. Math. Appl., 75, 1719–1738) we propose a numerical algorithm for the integration of the nonlinear and time-dependent Landau–Lifshitz–Gilbert (LLG) equation, which is unconditionally convergent, formally (almost) second-order in time, and requires the solution of only one linear system per time step. Only the exchange contribution is integrated implicitly in time, while the lower-order contributions like the computationally expensive stray field are treated explicitly in time. Then we extend the scheme to the coupled system of the LLG equation with the eddy current approximation of Maxwell equations. Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires the solution of only two linear systems per time step.

On the Convergence Rate of the Continuous Newton Method

2019 ◽  
Vol 239 (6) ◽  
pp. 867-879
Author(s):  
A. Gibali ◽  
D. Shoikhet ◽  
N. Tarkhanov
Keyword(s):  
Convergence Rate ◽  
Newton Method ◽  
Continuous Newton Method

A perfect memory makes the continuous Newton method look ahead

2017 ◽  
Vol 92 (8) ◽  
pp. 085201
Author(s):  
M B Kim ◽  
J W Neuberger ◽  
W P Schleich
Keyword(s):  
Newton Method ◽  
Look Ahead ◽  
Continuous Newton Method ◽  
Perfect Memory

A Low-Dispersion 3-D Second-Order in Time Fourth-Order in Space FDTD Scheme (<tex>$ M3d_24 $</tex>)

2004 ◽  
Vol 52 (7) ◽  
pp. 1638-1646 ◽  
Author(s):  
H.E.A. El-Raouf ◽  
E.A. El-Diwani ◽  
A.E.-H. Ammar ◽  
F.M. El-Hefnawi
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Second Order In Time ◽  
Low Dispersion
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