Métodos miméticos de pasos fraccionarios - E-Archivo - Universidad ...

4 introduction Ch. 1
of K methods, since they define the flux as u = −Kδψ. That is to say, there
exist explicit expressions for fluxes in terms of pressure drops. Recall that the
expanded MFE method adds a negative gradient variable to the standard MFE
scheme, thus involving an inner product containing the conductivity tensor. In
this way, K methods are more easily viewed as extensions of finite volume discretizations.
An insightful comparison of the preceding and some other methods
is given by Klausen and Russell (2004).
The spatial semidiscretization derived from any of the preceding techniques
gives rise to stiff systems of nonlinear ordinary differential equations (ODEs).
Thus, the unknown ψ in (1.1) is suitably approximated by Ψ h satisfying:
Ψ h t + A h (Ψ h ) = R h (Ψ h ) + F h . (1.2)
In this case, A h (Ψ h ) and R h (Ψ h ) are semidiscrete approximations to the diffusion
and reaction terms, respectively, and F h approximates f.
Plenty of methods have been studied for the time integration of problems of
type (1.2). Owing to the stiffness of the semidiscrete diffusion term, A h (Ψ h ),
the use of explicit schemes usually requires excessively small time steps (mainly
when fine spatial meshes are used). This fact can lead to computations which are
prohibitively expensive, especially when dealing with multi-dimensional problems.
On the other hand, fully implicit methods require the implicit treatment
of both A h (Ψ h ) and R h (Ψ h ). Hence, one or several nonlinear systems need to
be solved at each time step. The so-called splitting methods provide an efficient
alternative to these classical time integrators. Broadly speaking, they combine
a suitable splitting function with a splitting formula (cf. van der Houwen and
Verwer (1979)).
In order to describe such methods, we first consider the case in which A h (Ψ h )
can be expressed as A h Ψ h . Then, both A h and F h are decomposed into a
number of simpler split terms, that is, A h = A h 1 + A h 2 + . . . + A h m and F h =
F h 1 + F h 2 + . . . + F h m. In this framework, two different types of partitioning
techniques can be defined. If the splitting is based on different spatial variables,
it is referred to as a dimensional or component-wise splitting. Otherwise, if it is
related to a suitable decomposition of the spatial domain, it is called a domain
decomposition splitting.
The first type of splitting reformulates the original multi-dimensional problem
as a set of essentially one-dimensional problems, following the idea of the
well-known ADI methods (cf. Marchuk (1990); Hundsdorfer and Verwer (2003)).
These methods combine certain advantages of classical explicit and implicit
schemes. In particular, they preserve a stable behaviour independently of the
time step and spatial mesh sizes, while showing the same computational complexity
per time step as explicit methods. However, although a variety of stable
ADI methods are available for multi-dimensional parabolic equations in the absence
of mixed derivatives, it is well known that cross terms are difficult to be
handled implicitly using the ADI technique. So far, several approaches have been
proposed for the case in which mixed derivative terms are present (cf. McKee
et al. (1996); in ’t Hout and Welfert (2007, 2009) and references therein). In