Methods of approximation in hpk framework for ODEs in time resulting from decoupling of space and time in IVPs

In the initial value problems (IVPs) describing evolutions, the dependent variables naturally exhibit simultaneous dependence on spatial coordinates and time. Thus, space-time coupled methods of approximation are rather natural for obtaining numerical solutions of the mathematical models (IVPs) describing the evolutions. If we decouple space and time, i.e. if we assume the time derivatives to be constant for an instant of time, then we can proceed with the treatment in the spatial direction (finite difference, finite volume or finite element discretizations) which yields a system of ordinary differential equations in time. This work is concerned with the most appropriate choice of a method of approximation for obtaining numerical solutions of the ODEs in time. Since in this case we have ODEs, the finite difference and finite volume methods as methods of approximation are ruled out for the same reasons as they are in the case of ODEs describing boundary value problems [1, 2]. This only leaves us with the finite element method as a method of approximation for obtaining the numerical solutions of the ODEs in time.;The work presented in this paper 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 (WRM), and least squares method or process (LSM or LSP) to construct finite element approximations in time. These methods result in integral forms in time. A correspondence is established between these integral forms and the elements of the calculus of variations: (i) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC) computational processes for which types of operators and, (ii) 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 commonly used in the literature and the results are compared with Wilson's theta method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.