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.
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
A Fully Adaptive Non-Stiff Method for the Cahn-Hilliard Equation
Hector D. Ceniceros
Alexandre M. Roma
Department of Mathematics
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.
Radial Basis Function Methods for the Solution of Convective Diffusive Problems
Pedro Gonzalez Casanova
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
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.