Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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$Dr. Kai Zehmisch SS 2013 Dipl.-Math. Hella Timmermann 3 ...$

$Prof. Dr. W. Wefelmeyer WS 2004/05 Dipl.-Math. K. Tang Übungen ...$

DMV Tagung 2011 - Köln, 19. - 22. September Flavius Guia¸s TU Dortmund A time-adaptive numerical scheme for evolution partial differential equations based on path simulation of Markov jump processes We present a new general scheme of stochastic type for approximating ordinary differential equations, which turns out to be efficient especially for spatially discretized partial differential equations. It is a further development of the method introduced in Guia¸s (2010) which is based on the direct simulation of paths of suitable Markov jump processes. This initial explicit scheme turns out to be stable even on non-uniform grids. In this talk we present the results of exploiting further the full path simulation and improve the convergence order by performing periodically Picard iterations and/or Runge-Kutta steps based on the computed trajectories. We apply the method on a model problem where a very precise time resolution is needed (a reaction-diffusion PDE which exhibits steep and fast traveling fronts) and its efficiency turns out to be superior to standard deterministic methods. We use cartesian grids in 1D and 2D and also moving grids for the 1D problem. The mathematical algorithm behind this scheme is simple and flexible and can be formulated for general evolution partial differential equations. The main challenge in improving the efficiency of the method is shifted towards an optimal ’tuning’ and to implementation aspects such as sampling, data structures and computational features. By taking the mean over several independent paths this scheme is also suitable for using the facilities of parallel computing. Literatur Guia¸s, F. (2010). Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection-diffusion-reaction problems Math. Comput. Simulation, 81, 820 - 836. Guia¸s, F., Eremeev, P. (2011). A time-adaptive numerical scheme for evolution partial differential equations based on path simulation of Markov jump processes (submitted) Lennart Jansen Universität zu Köln Semiexplicit methods for differential-algebraic equations By their definition differential-algebraic equations (DAEs) are a mixture of differential and algebraic equations. With the help of projector functions the solutions of a DAE can also be seperated into a differential and an algebraic part. In the case of an ordinary differential equation (ODE) the whole solution belongs to the differential part. Therefore an ODE can be solved with explicit solver methods which may lead to a reduction of the calculation time, because an explicit method does not need to solve a non-linear system in every time step, in contrast to an implicit method. With the seperation of the solution into differential and algebraic parts, it is possible to formulate an explicit method on the differential solution parts while solving the algebraic ones with an implicit method. For this reason in every time step one has only to solve a non-linear system of the size of the algebraic solution parts. As an application we will use DAEs generated by a simulation of an electric circuit with electromagnetic (EM) devices. Only one third of the solution components of these DAEs belong to the algebraic part. Simulation results of these electric circuits will be shown. 80
DMV Tagung 2011 - Köln, 19. - 22. September René Lamour, Roswitha März Humboldt-Universität zu Berlin Computational aspects of detecting DAE structures II A regularity region describes the local characteristics of a Differential-Algebraic Equation (DAE). We determine regularity regions of DAEs by means of sequences of continuous matrix functions. The matrix sequence is built step-by-step by certains admissible projector functions starting with the Jacobian matrices of the DAE data. For time-dependent and nonlinear DAEs the sequence contains a differentiation of a projector function. The matrix functions are constructed pointwise. Beside common linear algebra tools such as matrix factorizations and generalized inverses, widely orthogonal projector functions are applied and algorithmic differentiation technices are used to realize the differentiation. Several constant-ranks of related matrix functions determine the structure of a regularity region. This also allows for the detection of critical points, which marks the border of different regularity regions. A comparison with other approaches, the discussion of numerical experiments and open problems complete the paper. Roswitha März, René Lamour Humboldt-Universität zu Berlin Computational aspects of detecting DAE structures I We present a different view on DAEs which relies on the decomposition of the DAE definition domains into several so-called regularity regions. We consider the regularity regions as prior in the analysis against obvious and hidden constraints. Determinig regularity regions we do not suppose any knowledge concerning solutions and constraints. On each regularity region the DAE has uniform structure. The structure may be different on different regions. The commonly applied understanding of a DAE having index µ corresponds to the specific case if all solutions remain in exact one regularity region. However, more generally, the DAE solutions may shuttle between different regularity regions and stay on borderlines. If solutions cross borders of regularity regions, a critical flow behaviour must be expected. We demonstrate this phenomenon by examples. We determine regularity regions by means of sequences of continuous matrix functions built with certains admissible projector functions. We benefit from several constant-rank conditions which offer the DAE structures, which is useful also for detecting critical points. 81
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Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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