238 research outputs found
Stable L\'{e}vy diffusion and related model fitting
A fractional advection-dispersion equation (fADE) has been advocated for
heavy-tailed flows where the usual Brownian diffusion models fail. A stochastic
differential equation (SDE) driven by a stable L\'{e}vy process gives a forward
equation that matches the space-fractional advection-dispersion equation and
thus gives the stochastic framework of particle tracking for heavy-tailed
flows. For constant advection and dispersion coefficient functions, the
solution to such SDE itself is a stable process and can be derived easily by
least square parameter fitting from the observed flow concentration data.
However, in a more generalized scenario, a closed form for the solution to a
stable SDE may not exist. We propose a numerical method for solving/generating
a stable SDE in a general set-up. The method incorporates a discretized finite
volume scheme with the characteristic line to solve the fADE or the forward
equation for the Markov process that solves the stable SDE. Then we use a
numerical scheme to generate the solution to the governing SDE using the fADE
solution. Also, often the functional form of the advection or dispersion
coefficients are not known for a given plume concentration data to start with.
We use a Levenberg--Marquardt (L-M) regularization method to estimate advection
and dispersion coefficient function from the observed data (we present the case
for a linear advection) and proceed with the SDE solution construction
described above.Comment: Published at https://doi.org/10.15559/18-VMSTA106 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes
It is known that the solution of a conservative steady-state two-sided
fractional diffusion problem can exhibit singularities near the boundaries. As
consequence of this, and due to the conservative nature of the problem, we
adopt a finite volume elements discretization approach over a generic
non-uniform mesh. We focus on grids mapped by a smooth function which consist
in a combination of a graded mesh near the singularity and a uniform mesh where
the solution is smooth. Such a choice gives rise to Toeplitz-like
discretization matrices and thus allows a low computational cost of the
matrix-vector product and a detailed spectral analysis. The obtained spectral
information is used to develop an ad-hoc parameter free multigrid
preconditioner for GMRES, which is numerically shown to yield good convergence
results in presence of graded meshes mapped by power functions that accumulate
points near the singularity. The approximation order of the considered graded
meshes is numerically compared with the one of a certain composite mesh given
in literature that still leads to Toeplitz-like linear systems and is then
still well-suited for our multigrid method. Several numerical tests confirm
that power graded meshes result in lower approximation errors than composite
ones and that our solver has a wide range of applicability
Adaptive Coupling of Finite Element Methods for Simulation of Hydrodynamics and Pollutant Transport in Lakes
Gegenstand dieser Arbeit ist die Entwicklung neuer numerischer Methoden zur Lösung von Problemen der Hydrodynamik in Seen. Für die Berechnung von Transportprozessen von Schadstoffen ist es wichtig Fronten scharf aufzulösen. Dies erfordert eine hohe Genauigkeit in bestimmten Bereichen des Gebiets. Um die erforderliche Genauigkeit zu erreichen und gleichzeitig die Kosten bei der Berechnung moderat zu halten, lösen wir das dreidimensioale Gebiet nicht überall komplett auf. In den Teilen des Gebiet, in denen nur geringe Genauigkeit gefordert wird, genügt eine zweidimensioale Lösung. Für die Bereiche in unserem Gebiet, in denen wir bessere Genauigkeit erzielen wollen, addieren wir zu der zweidimensionalen Lösung eine dreidimensionale Korrektur. Auf diese Weise erreichen wir in gewissen Teilen des Gebiets eine genauere, dreidimensionale Lösung bei moderatem Mehraufwand. Die Gleichungen, die durch diese Kopplung entstehen, werden hergeleitet. Für die Vorkonditionierung des gekoppelten Systems verwenden wir einen Block-Vorkonditionierer. Für die einzelnen Blöcke haben wir einen Mehrgitter-Vorkonditionierer für stetige Finite Elemente auf adaptiv verfeinerten Gittern entwickelt. Dabei geschieht die Glättung nur lokal. Anhand von numerischen Beispielen zeigen wir die Effizienz für Elemente höherer Ordnung
Schnelle Löser für Partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
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Final report on the Copper Mountain conference on multigrid methods
The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners
The Sixth Copper Mountain Conference on Multigrid Methods, part 1
The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth
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