Mini
Workshop:
Mathematical and
Computational Approaches
to Complex Nonlinear Physical Phenomena
This workshop is aimed at novel approaches to simulation
and understanding of complex nonlinear physical phenomena.
Computational
Combustion: Recent Advances in 2Dimensional Simulations of the PatternForming
KuramotoSivashinsky Equation
Peter Blomgren
blomgren@terminus.sdsu.edu
Department of Mathematics and Statistics
San Diego State University
Abstract: We present
an overview of recent advances in numerical simulations of the twodimensional
KuramotoSivashinsky equation, describing the flamefront deformation
in a combustion experiment. Algorithmic development includes a a second
order unconditionally Astable CrankNicolson scheme, using distributed
approximating functionals (DAFs) for welltempered, highly accurate, representation
of the physical quantity and its derivatives. The simulator reproduces
a multitude of patterns observed in experimentsinthewild, including
rotating 2cell, 3cell, hopping 3cell, stationary 2, 3, 4, 5cell, stationary
5/1, 6/1, 7/1, 8/2 tworing patterns, etc. The numerical observation of
hopping flame patternscharacterized by nonuniform rotations of a ring
of cells, in which individual cells make abrupt changes in their angular
positions while they rotate around the ring  is the first outside of
physical experiments. We show modal decomposition analysis of the simulated
patterns, via the singular value decomposition (SVD), which exposes the
spatiotemporal behavior in which the overall temporal dynamics is similar
to that of equivalent experimental states. Symmetrybased arguments are
used to derive normal form equations for the temporal behavior, and a
bifurcation analysis of the associated normal form equations quantifies
the complexity of hopping patterns. Conditions for their existence and
their stability are also derived from the bifurcation analysis. In addition
we briefly discuss the effects on the pattern formation due to "other
physics" in the system, including noise (more fully developed in
a related talk by Scott Gasner), thermal backconduction, and acoustic
waves in the combustion chamber.
NoiseInduced
Cellular Patterns on Circular Domains
Scott Gasner
sgasner@yahoo.com
Computational Science Research Center
San Diego State University
Abstract: We study
the effects of thermal noise in a stochasticformulation of a generic example,
the KuramotoSivashinsky equation, of a patternforming dynamical system
with twodimensional circular domain. Numerical integration reveals that
the presence of noise increases the propensity of dynamic cellular states,
which seems to explain the generic behavior of related laboratory experiments.
Most importantly, we also report on observations of certain dynamic states,
homoclinic intermittent states, previously only observed in physical experiments.
Investigation
of the Effect of NonDimensional Numbers for the Mixing and Reacting
Flow Inside a Burner
Álvaro L. de Bortoli
dbortoli@mat.ufrgs.br
Federal University of Rio Grande do
Sul
Department of Pure and Applied Mathematics, Brazil
Abstract: The
aim of this work is the investigation of the effect of nondimensional
numbers for the mixing and reacting flow inside a burner. The model
considers the overall, single step, irreversible reaction between two
species, where the reaction rate is governed by the Arrhenius law. Numerical
tests, for governing equations discretized by the finite difference
explicit RungeKutta fivestage scheme, are carried out for Reynolds
1000 and 10000, Damköhler 30 and 300, Zel'dovich 8 and 10, Heat
Release 10 and 20 and Prandtl/Schmidt 0.7 and 0.9, which are reasonable
values for gaseous hydrocarbon chemistry. The results contribute to
obtain a better understanding of the mixing and reacting flow inside
the burner.
Investigation
of the Effect of NonDimensional Numbers for the Mixing and Reacting
Flow Inside a Burner
Álvaro L. de Bortoli
dbortoli@mat.ufrgs.br
Federal University of Rio Grande do
Sul
Department of Pure and Applied Mathematics, Brazil
Abstract: The
aim of this work is the investigation of the effect of nondimensional
numbers for the mixing and reacting flow inside a burner. The model
considers the overall, single step, irreversible reaction between two
species, where the reaction rate is governed by the Arrhenius law. Numerical
tests, for governing equations discretized by the finite difference
explicit RungeKutta fivestage scheme, are carried out for Reynolds
1000 and 10000, Damköhler 30 and 300, Zel'dovich 8 and 10, Heat
Release 10 and 20 and Prandtl/Schmidt 0.7 and 0.9, which are reasonable
values for gaseous hydrocarbon chemistry. The results contribute to
obtain a better understanding of the mixing and reacting flow inside
the burner.
Numerical
Simulation of Solitary Waves in Nonlinear Dispersive Equations
Angel Durán
angel@mac.uva.es
Departamento de Matemática Aplicada
Escuela Técnica Superior de Ingenieros de Telecomunicación
Universidad de Valladolid
Campus Miguel Delibes
Valladolid, Spain
Abstract: Many
physical and mathematical studies concerning solitary waves require
the use of numerical experiments. A correct simulation is an important
guide in questions of stability, solitary wave interactions or resolution
of initial data into a train of solitary waves. This fact motivates
an appropriate selection of the numerical integrator used for the experiments.
In this work it is analyzed which properties of the numerical method
are needed for a good simulation of solitary waves in a wide class of
nonlinear dispersive equations. It is also explained the influence of
these properties when studying topics such that the stability of solitary
waves or solitary wave collisions.
Fully
Adaptive Multiresolution Schemes for Strongly Degenerate Parabolic Equations
Ricardo Ruiz Baier
email: rruiz@ingmat.udec.cl
Abstract:
We present an adaptive multiresolution scheme to quasilinear strongly
degenerate parabolic equations. The numerical scheme is based on a finite
volume discretization using EngquistOsher's technique for flux evaluation
and explicit time stepping. An adaptive multiresolution scheme with
cell averages is then used to speed up the CPU time and the memory requirements
of the underlying finite volume scheme. Using the convergence properties
of the finite volume discretization we use the optimal choice of the
threshold in the adaptive multiresolution method. Applications to mathematical
models of sedimentationconsolidation processes and traffic flow with
driver reaction, which involve different type of boundary conditions,
illustrate the computational efficiency together with the convergence
properties of the new method.
