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On the choice of initial conditions of difference schemes for parabolic equations
Author(s) -
Berikelashvili Givi,
Gupta Murli M.,
Muskhelishvili N.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200701007
Subject(s) - smoothness , mathematics , sobolev space , boundary value problem , convergence (economics) , initial value problem , parabolic partial differential equation , rate of convergence , domain (mathematical analysis) , mathematical analysis , finite difference , class (philosophy) , finite difference method , boundary (topology) , partial differential equation , computer science , channel (broadcasting) , computer network , artificial intelligence , economics , economic growth
We study finite difference schemes to approximate the first initial‐boundary value problem for linear second order parabolic equations and obtain some convergence rate estimates. When difference schemes are constructed for such problems, in the process of obtaining convergence rate estimates compatible with smoothness of the solution, various authors assume that the solution of the problem can be extended to the exterior of the domain of integration, preserving the Sobolev class. Our investigations show that this restriction can be removed if, instead of using the exact initial condition, we use certain approximations of the initial conditions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)