Convergence and stability constant of the theta-method

Faragó, István. "Convergence and stability constant of the theta-method." Applications of Mathematics 2013. Prague: Institute of Mathematics AS CR, 2013. 42-51. <http://eudml.org/doc/287863>.

@inProceedings{Faragó2013,
abstract = {The Euler methods are the most popular, simplest and widely used methods for the solution of the Cauchy problem for the first order ODE. The simplest and usual generalization of these methods are the so called theta-methods (notated also as $\theta $-methods), which are, in fact, the convex linear combination of the two basic variants of the Euler methods, namely of the explicit Euler method (EEM) and of the implicit Euler method (IEM). This family of the methods is well-known and it is introduced almost in any arbitrary textbook of the numerical analysis, and their consistency is given. However, in its qualitative investigation the convergence is proven for the EEM, only, almost everywhere. At the same time, for the rest of the methods it is usually missed (e.g. [1,2,7,8]). While the consistency is investigated, the stability (and hence, the convergence) property is usually shown as a consequence of some more general theory. In this communication we will present an easy and elementary prove for the convergence of the general methods for the scalar ODE problem. This proof is direct and it is available for the non-specialists, too.},
author = {Faragó, István},
booktitle = {Applications of Mathematics 2013},
keywords = {initial value problems; convergence; stability; error constants; theta-method; Euler formula; implicit methods; explicit method},
location = {Prague},
pages = {42-51},
publisher = {Institute of Mathematics AS CR},
title = {Convergence and stability constant of the theta-method},
url = {http://eudml.org/doc/287863},
year = {2013},
}

TY - CLSWK
AU - Faragó, István
TI - Convergence and stability constant of the theta-method
T2 - Applications of Mathematics 2013
PY - 2013
CY - Prague
PB - Institute of Mathematics AS CR
SP - 42
EP - 51
AB - The Euler methods are the most popular, simplest and widely used methods for the solution of the Cauchy problem for the first order ODE. The simplest and usual generalization of these methods are the so called theta-methods (notated also as $\theta $-methods), which are, in fact, the convex linear combination of the two basic variants of the Euler methods, namely of the explicit Euler method (EEM) and of the implicit Euler method (IEM). This family of the methods is well-known and it is introduced almost in any arbitrary textbook of the numerical analysis, and their consistency is given. However, in its qualitative investigation the convergence is proven for the EEM, only, almost everywhere. At the same time, for the rest of the methods it is usually missed (e.g. [1,2,7,8]). While the consistency is investigated, the stability (and hence, the convergence) property is usually shown as a consequence of some more general theory. In this communication we will present an easy and elementary prove for the convergence of the general methods for the scalar ODE problem. This proof is direct and it is available for the non-specialists, too.
KW - initial value problems; convergence; stability; error constants; theta-method; Euler formula; implicit methods; explicit method
UR - http://eudml.org/doc/287863
ER -