Solutions of Linear and Nonlinear ODEs in Time Resulting from Decoupling of Space and Time in IVPs

Abstract

This thesis presents considerations for determining a meritorious time integration strategy for a system of linear and nonlinear ordinary differential equations (ODEs) in time resulting from decoupling space and time in initial value problems (IVPs) using GM/WF for the spatial discretization. It is shown that Wilson's $\theta$ and Newmark's methods are meritorious over the others. It is further established that Newmark's method is meritorious over Wilson's $\theta$ method for integrating ODEs in time resulting from non-structural applications such as the mathematical models in Eulerian description for fluid mechanics. Newmark's linear method is considered for integrating linear and nonlinear first order and second order ODEs in time resulting from decoupling space and time for IVPs derived in Eulerian and Lagrangian descriptions of the deforming continua. The assumption of isothermal physics reduces the mathematical models of the IVPs to the Balance of linear momenta (BLM) in Eulerian and Lagrangian descriptions. In the present work we consider a space-time decoupled finite element method in which the space-time approximations of the dependent variables use approximation functions in space while the degrees of freedom are functions of time. This assumption induces space-time decoupling. Galerkin method with Weak form (GM/WF) for spatial discretization of the spatial domain is the preferred method of constructing the integral form in space. Integration over the spatial discretization yields a first order or second order system of linear or nonlinear ODEs in time. Though decoupling of space and time may introduce irreversible damage, the benefits in terms of simplicity of implementation and the speed of the calculations outweigh the shortcomings, particularly for IVPs in $\mathbb{R}^3$ for which a space-time coupled approach is almost prohibitive. Discretization and $p$-level in space are based on converged solutions of the corresponding boundary value problems (BVPs). Integration time step, $\Delta t$, is based on smoothness of the evolution and consideration of the stationary state of the evolution being the same as the solution of the corresponding BVP. The work presented here needs to be augmented with stability analyses in order to realize its full potential as it is only then we have a criteria for the choices of $\Delta x$ and $\Dt$. A priori and a posteriori analyses in conjunction with stability analyses will enable optimal choices of discretization $\Delta x$, $p$-levels in space, and time integration step, $\Delta t$.