Sequential Methods for Coupled Geomechanics and Multiphase Flow

1.4. CONVERGENCE OF SEQUENTIAL SCHEMES 7
are necessary (Hundsdorfer and Splijer, 1981; Hairer and T ´ ’urke, 1984; Hairer, 1986; Simo,
1991; Araújo, 2004).
1. Determine whether the problem is contractive, or dissipative. An appropriate norm,
or functional, is defined at this step to show the contractivity property, or energy
decay.
2. Show that the operator split corresponding to the sequential method honors, at the
continuum level, the contractivity property relative to the norm defined in the previous
step. If the operator split is not contractive, then it is not possible to obtain an
unconditionally stable solution scheme (Richtmyer and Morton, 1967).
3. When the operator split is contractive at the continuum level, one must then show
contractivity at the discrete time level (B-stability) for the individual subproblems
for a specific time discretization scheme (e.g., backward Euler, or midpoint rule).
The algorithmic (discrete) stability properties of the individual subproblems (i.e.,
uncoupled) are not necessarily applicable to the B-stability of the coupled problem.
This is because the natural norms associated with the subproblems may be different
from the appropriate norm for the coupled problem.
1.4 Convergence of Sequential Schemes
For nonlinear problems, stability, in general, does not guarantee convergence. The fully
coupled method provides first-order convergence with respect to time. On the other hand,
the convergence properties of sequential schemes depend strongly on the details of the split-
ting strategy, the specific form and discretization schemes used for the various subproblems,
and how the subproblems communicate during a time step.
For example, when an original operator A can be additively split as
˙y(t) = Ay(t) = (A1 + A2)y(t), (1.1)