Mathematical modeling of many physical processes such as diffusion, viscosity
of fluids and combustion involves differential equations with small
coefficients of higher derivatives. These may be small diffusion coefficients
for modeling the spreading of impurities, small coefficients of viscosity in
fluid flow simulation etc. The difficulty with solving such problem is that if
you set the small parameter at higher derivatives to zero, the solution of the
degenerate problem doesn't correctly approximate the original problem, even if
the small parameter approaches zero; the solution of the original problem
exhibits the emergency of a boundary layer. As a result, the application of
classical difference schemes for solving such equations produces great
inaccuracies. Therefore, numerical solution of differential equations with
small coefficients at higher derivatives demands special difference schemes
exhibiting uniform convergence with respect to the small parameters involved.
In this article author investigates two nonlinear boundary value problems on a
finite interval, resulting in exponential and power-law boundary layers.
martes, 16 de febrero de 2010
Construction Of Difference Schemes For Nonlinear Singular Perturbed Equations By Approximation Of Coefficients. (arXiv:1002.2925v1 [math.NA])
Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points. (arXiv:1002.2727v1 [math.NA])
Hamiltonian Boundary Value Methods are a new class of energy preserving one
step methods for the solution of polynomial Hamiltonian dynamical systems. They
can be thought of as a generalization of collocation methods in that they may
be defined by imposing a suitable set of extended collocation conditions. In
particular, in the way they are described in this note, they are related to
Gauss collocation methods with the difference that they are able to precisely
conserve the Hamiltonian function in the case where this is a polynomial of any
high degree in the momenta and in the generalized coordinates. A description of
these new formulas is followed by a few test problems showing how, in many
relevant situations, the precise conservation of the Hamiltonian is crucial to
simulate on a computer the correct behavior of the theoretical solutions.
The HBVMs Homepage. (arXiv:1002.2757v1 [math.NA])
Hamiltonian Boundary Value Methods (in short, HBVMs) is a new class of
numerical methods for the efficient numerical solution of canonical Hamiltonian
systems. In particular, their main feature is that of exactly preserving, for
the numerical solution, the value of the Hamiltonian function, when the latter
is a polynomial of arbitrarily high degree. Clearly, this fact implies a
practical conservation of any analytical Hamiltonian function. In this notes,
we collect the introductory material on HBVMs contained in the HBVMs Homepage,
available at this http URL