Numerical aspects of some time dependent partial differential equations problems
This work is a study of two time dependent partial differential equation problems- the Kuramoto-Sivashinsky (K-S) equation, in one space dimension, (Part I), and the Dirichlet boundary control of the linear wave equation (Control Problem, for short) in two space dimensions (Part II). The K-S equation is a nonlinear partial differential equation showing bifurcation phenomena. Numerical results had been obtained using a sophisticated numerical approach based on spectral methods; this work shows that those results can be obtained with a simpler approach based on classical finite difference methods. Regarding the Control Problem, a numerical implementation for arbitrary domains in two dimensions is given. This numerical approach is based on a general, systematic, and contructive theoretical method for the exact controllability of systems modeled by time dependent partial differential equations. The numerical implementation combines finite difference (in time) and finite element (in space) approximations to a biharmonic Ty-chonoff regularization; the algorithm is implemented for arbitrary domains, and is tested on the unit disk. Both problems are time dependent partial differential equation problems. However, for the K-S equation we work in the classical sense- we look for (chaotic) smooth solutions, using smooth data; for the Control Problem, we work, in general, in the weak (or distributional) sense- we look for solutions not necessarily smooth from not necessarily smooth data. They share, in this work, a common numerical issue- the biharmonic operator is present in both problems (in the Control Problem after regularization); to succeed in the solution they both turn around the difficulty associated to the high condition number of the linear systems to solve (resulting from the discretization of the fourth order derivative) by using exact, or approximate, factorization techniques involving the successive solution of discrete second order elliptic problems.