VI PanAmerican Workshop

Javier Salcedo-Ruíz
jasr12@hotmail.com

Universidad Autonoma Metropolitana-Iztapalapa
México, D.F.

Abstract: In this talk we study the application of high order operator decomposition methods to the solution of stiff differential equations. We analyse and compare the relative errors of the results generated when solving stiff reaction-diffusion equations with several high order operator decomposition methods.

A Fully Adaptive Non-Stiff Method for the Cahn-Hilliard Equation

Hector D. Ceniceros
hdc@math.ucsb.edu

Alexandre M. Roma
roma@ime.usp.br

Department of Mathematics
University of California Santa Barbara
Santa Barbara, California, USA

Abstract: We present a non-stiff, fully adaptive mesh refinement-based method for the Cahn-Hilliard equation. The method uses a splitting with linear leading order combined with a robust semi-implicit time discretization. The fully discretized equation is then written as a system which is efficiently solved on the composite adaptive grids using the linear multigrid method without any constraint on the time step size. We demonstrate the efficacy and unconditional stability of the method with numerical examples. We also show numerically that there could be several 2D stationary solutions in the doubly-periodic case.

Webpage: www.math.ucsb.edu/~hdc

Radial Basis Function Methods for the Solution of Convective Diffusive Problems

Pedro Gonzalez Casanova
pedrogc@dgsca2.unam.mx

Unidad de Investigacion en Computo Aplicado
DGSCA, UNAM, Insurgentes Sur No. 3000
Ciudad Universitaria
Mexico, D.F. Mexico

Abstract: The numerical solution of PDE by radial basis function methods is still in its infancy. Since Kansa's works on the unsymmetrical RBF collocation method, a substantial amount of research has been done in this field. Still, many numerical and theoretical problems remain to be solved. In this talk we will discuss the solution of parabolic convective diffusive problems emphasizing the use of multiquadric radial basis function kernels. An h-c algorithm -which is a radial basis function equivalent of the h-p finite element method- will be presented and the role of the shape parameter of the multiquadric kernel on the behavior of the numerical solution will be discussed. Numerical results showing the exponential rate of convergence of this technique will be presented.

A Stabilized Advection-Diffusion Equation Obtained by FIC

Miguel Angel Moreles
moreles@cimat.mx

CIMAT
Guanajuato, Guanajuato, Mexico

Abstract: When advection is dominant in advection-diffusion equations, it is well known that most classical numerical methods lead to non physical solutions. In this talk, we present the so called Finite Increment Calculus (FIC) to derive a stabilized advection-diffusion equation. A crucial step in stabilized equations is the proper choice of the stabilization parameter. In the framework of FEM, we introduce an iterative technique for this choice. We illustrate that with this scheme the problem of spurious solutions is corrected.



















	Mini Workshop: Numerical Computing, Optimization and Control of PDE-I This minisimposium intends to join researchers in the field of partial differential equations. Both novel applications as well as numerical and theoretical presentations on optimization, control and solution of direct problems on PDE. Solving the Reaction-Diffusion Equation with High Order Operator Decomposition Methods Francisco Sánchez-Bernabe fjsb@xanum.uam.mx Javier Salcedo-Ruíz jasr12@hotmail.com Universidad Autonoma Metropolitana-Iztapalapa México, D.F. Abstract: In this talk we study the application of high order operator decomposition methods to the solution of stiff differential equations. We analyse and compare the relative errors of the results generated when solving stiff reaction-diffusion equations with several high order operator decomposition methods. A Fully Adaptive Non-Stiff Method for the Cahn-Hilliard Equation Hector D. Ceniceros hdc@math.ucsb.edu Alexandre M. Roma roma@ime.usp.br Department of Mathematics University of California Santa Barbara Santa Barbara, California, USA Abstract: We present a non-stiff, fully adaptive mesh refinement-based method for the Cahn-Hilliard equation. The method uses a splitting with linear leading order combined with a robust semi-implicit time discretization. The fully discretized equation is then written as a system which is efficiently solved on the composite adaptive grids using the linear multigrid method without any constraint on the time step size. We demonstrate the efficacy and unconditional stability of the method with numerical examples. We also show numerically that there could be several 2D stationary solutions in the doubly-periodic case. Webpage: www.math.ucsb.edu/~hdc Radial Basis Function Methods for the Solution of Convective Diffusive Problems Pedro Gonzalez Casanova pedrogc@dgsca2.unam.mx Unidad de Investigacion en Computo Aplicado DGSCA, UNAM, Insurgentes Sur No. 3000 Ciudad Universitaria Mexico, D.F. Mexico Abstract: The numerical solution of PDE by radial basis function methods is still in its infancy. Since Kansa's works on the unsymmetrical RBF collocation method, a substantial amount of research has been done in this field. Still, many numerical and theoretical problems remain to be solved. In this talk we will discuss the solution of parabolic convective diffusive problems emphasizing the use of multiquadric radial basis function kernels. An h-c algorithm -which is a radial basis function equivalent of the h-p finite element method- will be presented and the role of the shape parameter of the multiquadric kernel on the behavior of the numerical solution will be discussed. Numerical results showing the exponential rate of convergence of this technique will be presented. A Stabilized Advection-Diffusion Equation Obtained by FIC Miguel Angel Moreles moreles@cimat.mx CIMAT Guanajuato, Guanajuato, Mexico Abstract: When advection is dominant in advection-diffusion equations, it is well known that most classical numerical methods lead to non physical solutions. In this talk, we present the so called Finite Increment Calculus (FIC) to derive a stabilized advection-diffusion equation. A crucial step in stabilized equations is the proper choice of the stabilization parameter. In the framework of FEM, we introduce an iterative technique for this choice. We illustrate that with this scheme the problem of spurious solutions is corrected.