VI PanAmerican Workshop

Mini Workshop:
Mathematical and Computational Approaches
to Complex Nonlinear Physical Phenomena

Abstract: We present an overview of recent advances in numerical simulations of the two-dimensional Kuramoto-Sivashinsky equation, describing the flame-front deformation in a combustion experiment. Algorithmic development includes a a second order unconditionally A-stable Crank-Nicolson scheme, using distributed approximating functionals (DAFs) for well-tempered, highly accurate, representation of the physical quantity and its derivatives. The simulator reproduces a multitude of patterns observed in experiments-in-the-wild, including rotating 2-cell, 3-cell, hopping 3-cell, stationary 2, 3, 4, 5-cell, stationary 5/1, 6/1, 7/1, 8/2 two-ring patterns, etc. The numerical observation of hopping flame patterns---characterized 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 spatio-temporal behavior in which the overall temporal dynamics is similar to that of equivalent experimental states. Symmetry-based 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 back-conduction, and acoustic waves in the combustion chamber.

Abstract: We study the effects of thermal noise in a stochasticformulation of a generic example, the Kuramoto-Sivashinsky equation, of a pattern-forming dynamical system with two-dimensional 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 Non-Dimensional 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 non-dimensional 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 Runge-Kutta five-stage 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 Non-Dimensional 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

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@ing-mat.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 Engquist-Osher'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 sedimentation-consolidation 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.