Double Interpolation Formulae and Partial Derivatives in Terms of Finite Differences

1955 ◽  
Vol 9 (50) ◽  
pp. 52
Author(s):  
Kuo-Chu Ho
Keyword(s):  
Finite Differences ◽  
Partial Derivatives

Implicit Co-Simulation and Solver-Coupling: Efficient Calculation of Interface-Jacobian and Coupling Sensitivities/Gradients

2021 ◽  
Author(s):  
Jan Kraft ◽  
Stefan Klimmek ◽  
Tobias Meyer ◽  
Bernhard Schweizer
Keyword(s):  
Finite Differences ◽  
Weak Sense ◽  
Analytical Approximation ◽  
State Variables ◽  
Partial Derivatives ◽  
Time Points ◽  
Solver Coupling ◽  
Approximation Formulas ◽  
Ill Conditioning ◽  
Coupling Variables

Abstract We consider implicit co-simulation and solver-coupling methods, where different subsystems are coupled in time domain in a weak sense. Within such weak coupling approaches, a macro-time grid is introduced. Between the macro-time points, the subsystems are integrated independently. The subsystems only exchange information at the macro-time points. To describe the connection between the subsystems, coupling variables have to be defined. For many implicit co-simulation and solver-coupling approaches an Interface-Jacobian is required. The Interface-Jacobian describes, how certain subsystem state variables at the interface depend on the coupling variables. Concretely, the Interface-Jacobian contains partial derivatives of the state variables of the coupling bodies with respect to the coupling variables. Usually, these partial derivatives are calculated numerically by means of a finite difference approach. A calculation of the coupling gradients based on finite differences may entail problems with respect to the proper choice of the perturbation parameters and may therefore cause problems due to ill-conditioning. A second drawback is that additional subsystem integrations with perturbed coupling variables have to be carried out. In this manuscript, analytical approximation formulas for the Interface-Jacobian are derived, which may be used alternatively to numerically calculated gradients based on finite differences. Applying these approximation formulas, numerical problems with ill-conditioning can be circumvented. Moreover, efficiency of the implementation may be increased, since parallel simulations with perturbed coupling variables can be omitted. The derived approximation formulas converge to the exact gradients for small macro-step sizes.

On the motion of comet P/d’Arrest

1974 ◽  
Vol 22 ◽  
pp. 145-148
Author(s):  
W. J. Klepczynski
Keyword(s):  
Equations Of Motion ◽  
Variational Equations ◽  
Partial Derivatives

AbstractThe differences between numerically approximated partial derivatives and partial derivatives obtained by integrating the variational equations are computed for Comet P/d’Arrest. The effect of errors in the IAU adopted system of masses, normally used in the integration of the equations of motion of comets of this type, is investigated. It is concluded that the resulting effects are negligible when compared with the observed discrepancies in the motion of this comet.

THE METHOD OF FINITE DIFFERENCES APPLIED TO DESCRIPITION OF A METALIC COMPONENT

2017 ◽  
Author(s):  
Lisiane Trevisan ◽  
Juliane Donadel ◽  
Bianca de Castro
Keyword(s):  
Finite Differences

Finite differences with exponential filtering in the calculation of reactivity

Kerntechnik ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. 210-213 ◽  
Author(s):  
D. Suescún Díaz ◽  
A. Senra Martinez
Keyword(s):  
Finite Differences

Umbral Calculus

10.37236/24 ◽  
2002 ◽  
Vol 1000 ◽  
Author(s):  
A. Di Bucchianico ◽  
D. Loeb
Keyword(s):  
19Th Century ◽  
Finite Differences ◽  
State Of The Art ◽  
Umbral Calculus ◽  
Current State ◽  
Logical Foundation ◽  
Complete Bibliography ◽  
The 19Th Century ◽  
Mathematical Literature

We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of “magic rules” for lowering and raising indices, through its rebirth in the 1970’s as Rota’s school set it on a firm logical foundation using operator methods, to the current state of the art with numerous generalizations and applications. The survey itself is complemented by a fairly complete bibliography (over 500 references) which we expect to update regularly.

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