Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

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 ( / GM WF ), 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 established 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 ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally 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 approximations 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 proposed methodology.