Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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.
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