129,536 research outputs found
Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
We consider time discretization methods for abstract parabolic problems with
inhomogeneous linear constraints. Prototype examples that fit into the general
framework are the heat equation with inhomogeneous (time dependent) Dirichlet
boundary conditions and the time dependent Stokes equation with an
inhomogeneous divergence constraint. Two common ways of treating such linear
constraints, namely explicit or implicit (via Lagrange multipliers) are
studied. These different treatments lead to different variational formulations
of the parabolic problem. For these formulations we introduce a modification of
the standard discontinuous Galerkin (DG) time discretization method in which an
appropriate projection is used in the discretization of the constraint. For
these discretizations (optimal) error bounds, including superconvergence
results, are derived. Discretization error bounds for the Lagrange multiplier
are presented. Results of experiments confirm the theoretically predicted
optimal convergence rates and show that without the modification the (standard)
DG method has sub-optimal convergence behavior.Comment: 35 page
The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces
We introduce two general hybrid iterative approximation methods (one implicit and one explicit)
for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly
positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a
reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve
and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009)
Adaptive wavelet methods for a class of stochastic partial differential equations
An abstract interpretation of Rothe’s method for the discretization of evolution equations
is derived. The error propagation is analyzed and condition on the tolerances
are proven, which ensure convergence in the case of inexact operator evaluations. Substantiating
the abstract analysis, the linearly implicit Euler scheme on a uniform time
discretization is applied to a class of semi-linear parabolic stochastic partial differential
equations. Using the existence of asymptotically optimal adaptive solver for the elliptic
subproblems, sufficient conditions for convergence with corresponding convergence
orders also in the case of inexact operator evaluations are shown. Upper complexity
bounds are proven in the deterministic case.
The stochastic Poisson equation with random right hand sides is used as model
equation for the elliptic subproblems. The random right hand sides are introduced
based on wavelet decompositions and a stochastic model that, as is shown, provides
an explicit regularity control of their realizations and induces sparsity of the wavelet
coefficients. For this class of equations, upper error bounds for best N-term wavelet
approximation on different bounded domains are proven. They show that the use
of nonlinear (adaptive) methods over uniform linear methods is justified whenever
sparsity is present, which in particularly holds true on Lipschitz domains of two or
three dimensions.
By providing sparse variants of general Gaussian random functions, the class of
random functions derived from the stochastic model is interesting on its own. The
regularity of the random functions is analyzed in certain smoothness spaces, as well as
linear and nonlinear approximation results are proven, which clarify their applicability
for numerical experiments
Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
In this paper, we extend the implicit-explicit (IMEX) methods of Peer type
recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203--215, 2017]
to a broader class of two-step methods that allow the construction of
super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of
explicit methods to efficiently solve systems of ordinary differential
equations with both stiff and non-stiff parts included in the source term. To
construct super-convergent IMEX-Peer methods with favourable stability
properties, we derive necessary and sufficient conditions on the coefficient
matrices and apply an extrapolation approach based on already computed stage
values. Optimised super-convergent IMEX-Peer methods of order s+1 for s=2,3,4
stages are given as result of a search algorithm carefully designed to balance
the size of the stability regions and the extrapolation errors. Numerical
experiments and a comparison to other IMEX-Peer methods are included.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1610.0051
Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation
We consider the development of high order space and time numerical methods
based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic
systems with relaxation. More specifically, we consider hyperbolic balance laws
in which the convection and the source term may have very different time and
space scales. As a consequence the nature of the asymptotic limit changes
completely, passing from a hyperbolic to a parabolic system. From the
computational point of view, standard numerical methods designed for the
fluid-dynamic scaling of hyperbolic systems with relaxation present several
drawbacks and typically lose efficiency in describing the parabolic limit
regime. In this work, in the context of Implicit-Explicit linear multistep
methods we construct high order space-time discretizations which are able to
handle all the different scales and to capture the correct asymptotic behavior,
independently from its nature, without time step restrictions imposed by the
fast scales. Several numerical examples confirm the theoretical analysis
A class of implicit-explicit two-step Runge-Kutta methods
This work develops implicit-explicit time integrators based on two-step Runge-Kutta methods.
The class of schemes of interest is characterized by linear invariant
preservation and high stage orders. Theoretical consistency and stability analyses are performed to reveal the properties of these methods. The new framework offers extreme flexibility
in the construction of partitioned integrators, since no coupling conditions are necessary.
Moreover, the methods are not plagued by severe order reduction, due to their high stage orders.
Two practical schemes of orders four and six are constructed, and are used to solve several test problems.
Numerical results confirm the theoretical findings
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