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Ringvorlesung Wave Phenomena (Summer Semester 2019)

In the iRTG lecture series four members of CRC 1173 will talk about topics whithin the analysis and numerics of wave phenomena. The lectures are directed to Ph.D. students (in particular of the integrated research training group of CRC 1173) and to advanced master students with a solid background in partial differential equations. The series is organized by Willy Dörfler and Roland Schnaubelt.

Lecture: Monday 14:00-15:30 SR 1.067

Cardiac Depolarization Waves in the Bidomain Setting

Axel Loewe

In this series of lectures, we will explore and discuss cardiac depolarization waves as described by bidomain theory. The series is structured according to the multiscale nature of the problem: starting from the ion channel level, we will work our way up to the cellular and eventually tissue scale.

1) Ion Channel Electrophysiology
We will first introduce the basic biophysical principles governing cardiac electrophysiology on the ion channel level. Starting from the Nernst equilibrium and the gating behavior of Hodgkin-Huxley type ion channels, we will develop suitable models to describe current flow through ion channels in the cell membrane. Aspects of these models will be implemented in Matlab.

2) Cellular Electrophysiology
Based on the concept of ion channel gating, we will move to more comprehensive descriptions of the membrane and intracellular calcium cycling. The interplay of different ionic currents and their coupling via the transmembrane voltage and the intracellular ion concentrations will be considered. The focus will be on aspects which are crucial for the propagation of excitation waves and arrhythmogenesis such as the initiation of action potentials and refractoriness. The resulting system of coupled ODEs will be solved numerically using Matlab.

3) Excitation Propagation - Cardiac Depolarization Waves
We will employ bidomain theory to couple a set of cardiac myocytes to a syncitial tissue. Starting with cable theory in the 1D setting, we will model the spatial distribution of the potential in the intracellular and extracellular domains and the transmembrane currents between them. This concept will be extended to a multidimensional setting yielding the bidomain reaction-diffusion PDEs, which will allow us to study depolarization waves in physiological and arrhythmic scenarios.

The lectures take place on April 29, May 6, and May 13.

Axel Loewe is assistant professor at the Institute of Biomedical Engineering and Head of Computational Cardiac Modeling at KIT.

Quasilinear wave equations with small data

Martin Spitz

In this series of lectures we will investigate the long-time behavior of quasilinear wave equations with small initial data.

1) Energy estimates and local wellposedness
In the first lecture we will recall the basic wellposedness results for the linear wave equation. Based on the corresponding energy estimates, we then prove the local wellposedness of the nonlinear problem.

2) Invariant vector fields
We introduce the invariant vector fields and show the Klainerman-Sobolev inequality. We also discuss the significance of the latter for the long-time behavior of quasilinear wave equations.

3) Global and almost global existence
We prove global existence for solutions of quasilinear wave equations with small intitial data in higher spatial dimensions (d \geq 4). In the other dimensions, we provide lower bounds for the lifespan of the solutions. In particular, we will see that in d = 3 we obtain almost global ones. If time permits, we will also discuss the null condition and global existence in this dimension.

The lectures take place on May 20, May 27, and June 3.

Scattering Problems in Periodic Domains

Ruming Zhang

In this lecture series we will consider the scattering problems in periodic domains.

1) Quasi-periodic scattering problems.
The first lecture introduces the mathematical formulation and regularity results for the quasi-periodic scattering problems. Both the integral equation and the variation formulation will be introduced, and applied to the investigation of the scattering problems.

2) The Floquet-Bloch transform and its application to scattering problems
In the second lecture, first we introduce an important tool, the Floquet-Bloch transform, and some of its important properties. Then we presents its application to the scattering problems with (locally perturbed) periodic structures. We also study the regularity of the transformed scattering problems.

3) Numerical analysis for Floquet-Bloch transform based method
In the third lecture, we consider the numerical solutions of the scattering problems with (locally perturbed) periodic structures, based on the Floquet-Bloch transform. We present the Galerkin discretization of the transformed problem, and then consider the convergence results numerical method.

A. Kirsch, Diffraction by periodic structures, Lecture Notes in Physics 422, Springer, 1993.
A. Lechleiter & R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces. SIAM J. Sci. Comput., 39(5), B819 – B839, 2017.

The lectures take place on June 17, June 24, and July 1.

Error analysis of full discretizations of wave-type problems

Jonas Köhler

In this series of lectures we investigate the error of full discretizations of linear wave-type problems. In particular, we apply a discontinuous Galerkin method (dG) in space and either the Crank-Nicolson or the leapfrog (or Verlet) method in time.

1) Framework: Friedrichs' operators.
We consider linear wave-type problems of the form \partial_t u = \mathcal{L} u + f, where \mathcal{L} is a first-order spatial differential operator belonging to the class of Friedrichs' operators. Well-posedness of such problems (supplied with suitable initial and boundary conditions) is shown by proving that \mathcal{L} is maximal dissipative and therefore generates a strongly continuous semigroup by the Lumer-Phillips theorem.

2) Spatial and temporal discretization.
Next, we discretize the Friedrichs' operator \mathcal{L} using a central fluxes dG method and provide some basic properties of the resulting discrete operator. To discretize in time we consider the Crank-Nicolson and the leapfrog scheme applied to the spatially discrete wave-type equation.

3) Error analysis.
Finally, we analyze the error of the fully discretized schemes obtained in 2). With k denoting the polynomial degree used in the dG method, we show that the discrete solution converges to the exact solution with order k in space and order two in time for both schemes if the exact solution is sufficiently smooth.

The lectures take place on July 8, July 11, and July 15. On Thursday, July 11, the lecture takes place in SR 1.067 from 11:30 to 13:00.