7 research outputs found
Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations
Domain decomposition based time integrators allow the usage of parallel and
distributed hardware, making them well-suited for the temporal discretization
of parabolic systems, in general, and degenerate parabolic problems, in
particular. The latter is due to the degenerate equations' finite speed of
propagation. In this study, a rigours convergence analysis is given for such
integrators without assuming any restrictive regularity on the solutions or the
domains. The analysis is conducted by first deriving a new variational
framework for the domain decomposition, which is applicable to the two standard
degenerate examples. That is, the -Laplace and the porous medium type vector
fields. Secondly, the decomposed vector fields are restricted to the underlying
pivot space and the time integration of the parabolic problem can then be
interpreted as an operators splitting applied to a dissipative evolution
equation. The convergence results then follow by employing elements of the
approximation theory for nonlinear semigroups
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
Error Analysis of Explicit Partitioned Runge-Kutta Schemes for Conservation Laws
An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate
Modified Douglas splitting methods for reaction–diffusion equations
We present modifications of the second-order Douglas stabilizing corrections method, which is a splitting method based on the implicit trapezoidal rule. Inclusion of an explicit term in a forward Euler way is straightforward, but this will lower the order of convergence. In the modifications considered here, explicit terms are included in a second-order fashion. For these modified methods, results on linear stability and convergence are derived. Stability holds for important classes of reaction–diffusion equations, and for such problems the modified Douglas methods are seen to be often more efficient than related methods from the literature