A monolithic multi-time-step computational framework for first-order transient systems with disparate scales

Abstract

A monolithic multi-time-step computational framework for first-order transient systems with disparate scalesAn e-print of the paper is available on arXiv: http://arxiv.org/abs/1405.3230. Authored by S. KarimiGraduate Student, University of Houston. Abstract. Developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering. A desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time-steps (multi-time-step coupling), and different time integrators and different numerical formulations (mixed methods) in different regions of the computational domain. To this end, we present two new monolithic multi-time-step mixed coupling methods for first-order transient systems. We shall employ unsteady advection-diffusion-reaction equation with linear decay as the model problem, which offers several unique challenges in terms of non-self-adjoint spatial operator and rich features in the solutions. We shall employ the dual Schur domain decomposition technique to split the computational domain into an arbitrary number of subdomains. It will be shown that the governing equations of the decomposed problem, after spatial discretization, will be differential/algebraic equations. This is a crucial observation to obtain stable numerical results. Two different methods of enforcing compatibility along the subdomain interface will be used in the time discrete setting. A systematic theoretical analysis (which includes numerical stability, influence of perturbations, bounds on drift along the subdomain interface) will be performed. The first coupling method ensures that there is no drift along the subdomain interface, but does not facilitate explicit/implicit coupling. The second coupling method allows explicit/implicit coupling with controlled (but non-zero) drift in the solution along the subdomain interface. Several canonical problems will be solved to numerically verify the theoretical predictions, and to illustrate the overall performance of the proposed coupling methods. Finally, we shall illustrate the robustness of the proposed coupling methods using a multi-time-step transient simulation of a fast bimolecular advective-diffusive-reactive system. K. B. Nakshatrala

Convergence results of a heterogeneous asynchronous newmark time integrators

“…(4. 19) shows that the constraint relation in Eq. (1.1) is also satisfied, namely, L 1 U1 +L 2 U2 = 0.…”

A novel space/time integration technology via altogether different space and time stepping methods for nonlinear first-order systems

“…Karimi and Nakshatrala (2014) developed a useful subdomain differential algebraic equation (DAE) framework for the Newmark family of algorithm that couples multiple subdomains using DAEs, rather than ordinary differential equations. However, the application was limited to using and coupling implicit–explicit algorithms, which is rather straightforward, and not general implicit–implicit couplings because of limitations with current state of the art and was later applied to the framework for first-order systems (Karimi and Nakshatrala, 2015).…”

Numerical time integration of lumped parameter systems governed by implicit constitutive relations

“…In fact, a numerical time-integrator can be successful in approximating the solution to ODEs, but to fail in estimating the solution to a DAE. Considering the importance of DAEs to mathematical modeling of mechanical systems such as multi-body dynamics [Schiehlen, 1997, Brüls andCardona, 2010], domain decomposition for PDEs [Bursi et al, 2013, Karimi and Nakshatrala, 2014, 2015, as well as implicit constitutive relations in mechanical systems [Rajagopal, 2010, Darbha et al, 2010, Pražák and Rajagopal, 2012 and many others, many research endeavors have been centered around development of reliable numerical integration of DAEs for various applications. It is worthy to note that the algebraic constraints or relations in a DAE can be of different types.…”