20Decomposition Methods for Differential Equations Theory and Applications
2.1.4 Numerical Analysis of the Decomposition Methods
The underlying model equations for the multi-physics problems are evolu-
tion systems of partial differential equations. To solve such equations, nu-
merical methods as discretization and solver methods have to be studied. We
concentrate on the coupling of the equation systems and study decoupled
equations with respect to large time-steps.
The numerical analysis for the partial differential equations is studied in
[65], [200], and [201].
The large time-steps specialization for evolution equations is accomplished
with mixed discretization or splitting methods. This specialization is stud-
ied in [124] for convection-diffusion equations, and in [150] for hyperbolic
problems, in [118] for Hamiltonian problems. Decompositions of the equa-
tion systems in space and time are studied using domain decomposition in
[179], [189], and [197]; time decomposition in [184] and [194]; and the classical
splitting method in [154] and [185].
To obtain large time-steps of evolution equations, the application of implicit
time discretization methods, as discussed in the numerical methods for ordi-
nary differential equations, see [116] and [117], are proposed. Mixed methods
of implicit and explicit discretization methods can also be applied as needed
for decoupling stiff and nonstiff operators, as in references [171] and [172].
We further treat spatial adaptive methods, which refine the domain regions,
and apply standard implicit discretization methods, see, for example, [10],
[28], [29], [39], and [64]. The spatial error estimates decrease the local errors;
therefore, the stiffness of the underlying operators decreases, too, see [200]
and [201].
2.1.5 Decoupling Index
The benefit, of the splitting methods were fine in the 1960s, when computer
power was limited to one-dimensional problems and splitting was done with
respect to decoupling into one-dimensional problems. One of the first studies
can be found in [185], and [154] describes the decoupling with respect to
higher-order ideas.
Nowadays faster computers and fast solver methods exist (e.g., fast-block
ILU solvers [196], multigrid methods [112] and [115]), and such methods can-
not become more efficient due to the complexity of the underlying equation
systems. So the splitting methods can be addressed in the context of accel-
erating a method (e.g., with Newton method [173], a solver method, or by
decoupling multiscales for special equations), see [168].
Therefore, it is important to define an index to classify the decomposition.
One idea is to define the decoupling index as a spectrum of the different
operators of the equation, see Section 2.1.3.
In our applications, we deal with the more physically oriented decompo-
sition that allows us to decouple the equation on the level of the partial
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