Gutierrez, Angel Enrique Ramirez; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4659022Z4

### Abstract:

Some parabolic problems can be rewritten in complementarity form and appear in many
applications such as liquid flows within a domain, diffusion, heat flow involving phase
change and chemical reactions. These types of problems have many analytical and numerical
difficulties, usually due to the evolution in time or moving boundary. Since the
analytical solution is very difficult to obtain, so it is important to study numerical methods
that allow the search for an approximate solution of the exact solution for these types
of problems. We study the conservation laws and the types of solutions associated with
the Riemann Problem, these types of laws are essentially balance laws that express the
fact that some substance is balanced. The study of this theory is important because the
conservation laws often appear when the parabolic problems are neglected the diffusive
terms of second order. It also presents a numerical method which is a variation of the
Newton’s method for solving nonlinear systems, the method is based on an implicit finite
difference scheme and an algorithm complementary non-linear FDA–NCP. The given
method has the advantage of providing a global convergence with respect to the finite
difference method with Newton’s method which has only local convergence. The theory
is applied to the model difussion in tissue of oxygen and oxygen combustion model in
situ, this two models are linear and nonlinear parabolics problems respectively and which
can be rewritten in the form of complementarity problem. The first model that can be
applied in the treatment of cancer cells leads to a free boundary problem, while the second
model, consider a one-dimensional process of air injection inside a porous medium initially
containing solid fuel and where combustion occurs gas - solid thus the model involves the
heat balance law, law and the fuel molar ideal gas law, in addition, studies the thermal
wave and the wave associated fuel.