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Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs
Surana, Karan S. Euler, L. Reddy, J. N. Romkes, A.
Surana, Karan S.
Euler, L.
Reddy, J. N.
Romkes, A.
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Abstract
The present study considers mathematical classification of the time differential operators and then applies
methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form
(/GMWF ), Petrov-Galerkin method (PGM), weighted residual method (WRY ), and least squares method
or process (LSM or LSP ) to construct finite element approximations in time. A correspondence is estab-
lished between these integral forms and the elements of the calculus of variations: 1) to determine which
methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) com-
putational processes for which types of operators and, 2) to establish which integral forms do not yield un-
conditionally stable computations (variationally inconsistent integral forms, VIC). It is shown that varia-
tionally consistent time integral forms in hpk framework yield computational processes for ODEs in
time that are unconditionally stable, provide a mechanism of higher order global differentiability approxima-
tions as well as higher degree local approximations in time, provide control over approximation error when
used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical
studies are presented using standard model problems from the literature and the results are compared with
Wilson’s method as well as Newmark method to demonstrate highly meritorious features of the pro-
posed methodology.
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This is the published version. Copyright © 2011 Scientific Research Publishing
Date
2011-06
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Scientific Research Publishing
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Keywords
Finite Element Approximations, Numerical Studies, Time Approximation, Variationally Consistent Integral Forms
Citation
Surana, K.s., L. Euler, J.n. Reddy, and A. Romkes. "Methods of Approximation in Hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs." American Journal of Computational Mathematics AJCM 01.02 (2011): 83-103. http://dx.doi.org/10.4236/ajcm.2011.12009