08:30
MS32: Recent advances in modelling and numerics of wave phenomena (Part 1)
Chair: Vanja Nikolic
08:30
25 mins
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Controllability results for the Moore-Gibson-Thompson equation arising in nonlinear acoustics
Sebastián Zamorano, Carlos Lizama
Abstract: we show that the Moore-Gibson-Thomson equation
$$
\tau \partial_{ttt} y +\alpha \partial_{tt} y -c^2\Delta y - b\Delta \partial_t y = k \partial_{tt} {(y^2) }+ \chi_{\omega(t)}u,
$$
is controlled by a force that is supported on an moving subset $\omega(t)$ of the domain, satisfying a geometrical condition. Using the concept of approximately outer invertible map, a generalized implicit function theorem and assuming that $\gamma := \alpha -\frac{\tau c^2}{b} > 0$, the local null controllability in the nonlinear case is established. Moreover, the analysis of the critical value $\gamma = 0$ for the linear equation is included.
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08:55
25 mins
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Error analysis for a discretization of quasilinear wave-type equations
Marlis Hochbruck, Bernhard Maier
Abstract: We consider the discretization of a class of quasilinear wave-type equations on a bounded domain with smooth boundary, where the Lipschitz continuous nonlinearities are local in time and where the differential operator is skew-adjoint. The well-posedness of problems falling in this class is for example considered the framework by Kato (1985) and the refined framework by Müller (2014) under suitable assumptions on the operators and the initial value.
In the first part of this talk, we present a space discretization for quasilinear wave-type problems. Due to the assumption on the smoothness of the domain, it would be very restrictive to assume the discretized domain to coincide with the domain of the continuous problem. Hence, we consider nonconforming space discretizations.
The well-posedness analysis presented in this talk consists of the following three steps.
1. Show well-posedness with the Picard-Lindelöf theorem (on arbitrary short time intervals depending on the space discretization parameter).
2. Bound the error in the energy norm based on the unified error analysis proposed by Hipp, Hochbruck and Stohrer (2018) for linear problems.
3. Use an inverse estimate to show well-posedness on the whole time interval of existence of the solution to the continuous problem.
In the second part of this talk, we combine these results with time discretizations based on implicit Runge-Kutta schemes.
D. Hipp, M. Hochbruck, and C. Stohrer, Unified error analysis for nonconforming space discretizations of wave-type equations, IMA Journal of Numerical Analysis, online first (2018).
T. Kato, Abstract differential equations and nonlinear mixed problems, Fermi Lectures, Scuola Normale Superiore, Pisa; Accademia Nazionale dei Lincei, Rome, p. 89 (1985)
D. Müller, Well-posedness for a general class of quasilinear evolution equations - with applications to Maxwell’s equations, PhD thesis, KIT (2014).
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09:20
25 mins
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Iterative Coupling for Fully Dynamic Poroelasticity
Markus Bause, Jakub W. Both, Florin A. Radu
Abstract: We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge–Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.
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09:45
25 mins
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Optimal Control of Waves and Geometric Inverse Problems
Stephan Schmidt, Marc Herrmann, Roland Herzog, Jose Vidal-Nuñez
Abstract: The primary concern of the presentation is geometric inverse problems governed by waves and hyperbolic partial differential equations, meaning we are interested in reconstructing geometric objects such that they reproduce a measured echo of a scanning wave. There are wide applications for problems of this type, including CFD, computational acoustics, Electrodynamics and mathematical imaging and each will be considered in the presentation.
We also study approaches that are robust with respect to non-smoothness, which arises naturally when objects with kinks are to be reconstructed. To this end, we consider optimization strategies for Total Variation denoising of surfaces.
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