Analysis of coupling interface problems for bi-domain diffusion equations
Taj Munir · Digitalen Hochschulbibliothek Sachsen-Anhalt (Universitäts- und Landesbibliothek Sachsen-Anhalt) · 2020
In this thesis we study various numerical interface coupling conditions for diffusion equations in bio-physics or heat conduction problems. For this we take a one-dimensional case of a 3D model of Falcke [7] with two coupling conditions. The coupling interface conditions are given in Thul [37]. Originally they considered a system that models the intracellular calcium dynamics in a realistic fashion between the cytosolic region and the endoplasmic reticulum (ER) region of a living cell via channels on the membrane which separates both regions. The phenomenon of calcium dynamics is a multi-domain phenomenon. We analyze a numerical mathematical problem related to the calcium transport, i.e. a bi-domain problem with coupling conditions. In order to understand fundamental numerical issues better, we make an analytical and computational study of the one dimensional case derived from the three dimensional model of Falcke [7]. We compare these with other related coupling conditions. The coupling conditions that we consider in this thesis include the well-known Dirichlet-Neumann coupling, the heat ux coupling, a channel pumping, a simplified channel pumping, a membrane pumping and its special cases simplified membrane pumping and linearized membrane pumping conditions. We implemented three coupling algorithms namely, an explicit coupling algorithm (A1), an implicit monolithic coupling algorithm (A2), an implicit partitioned iterative coupling algorithm (A3) for the various coupling conditions with bi-domain diffusion equations. The partitioned iterative approach is a bit more complicated because we have the two unknowns corresponding to the each sub-domain. Despite this problem we manage to achieve a numerical solution via sub-iterations. Such algorithms may be useful for parallel computation. The main emphasis of this work is to study the numerical properties of coupling conditions. We give a detailed account of the Godunov-Ryabenkii stability theory for coupling conditions that was introduced by Giles [10] for this purpose. An important point is to maintain conservativity of the overall scheme. Therefore, we first study this property for the coupling conditions. Unfortunately, Giles neglected to maintain conservativity of his scheme and by using an inconsistent scheme produced artificial instabilities. We show how conservativity is maintained in nodal based as well as finite volume type discretizations. Nodal based schemes need a central difference approximation with respect to the node at the coupling boundary. Finite volume schemes have to use one sided difference with respect to the cell center. It is a central difference with respect to the cell boundary which is also the interface boundary. An analogous result is shown for the homogeneous Neumann outer boundary condition. We then proceed to prove stability for these coupling conditions. For this purpose we prove a lemma that describes in detail properties of the solutions to the normal mode equations that are useful to Godunov-Ryabenkii analysis of the coupling conditions. For comparison we first treat some boundary conditions related to ux coupling conditions. The simplest coupling, which was the one considered by Giles [10], is Dirichlet-Neumann coupling. The more complex couplings considered in this thesis lead to additional conditional stability conditions. The theoretical results on conservativity and stability are confirmed in computations for a variety of test cases.