954 research outputs found
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
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Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs
Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising
On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation
International audienceThis paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided
A New Domain Decomposition Method for the Compressible Euler Equations
In this work we design a new domain decomposition method for the Euler
equations in 2 dimensions. The basis is the equivalence via the Smith
factorization with a third order scalar equation to whom we can apply an
algorithm inspired from the Robin-Robin preconditioner for the
convection-diffusion equation. Afterwards we translate it into an algorithm for
the initial system and prove that at the continuous level and for a
decomposition into 2 sub-domains, it converges in 2 iterations. This property
cannot be preserved strictly at discrete level and for arbitrary domain
decompositions but we still have numerical results which confirm a very good
stability with respect to the various parameters of the problem (mesh size,
Mach number, ....).Comment: Submitte
PARAOPT: A parareal algorithm for optimality systems
The time parallel solution of optimality systems arising in PDE constraint optimization could be achieved by simply applying any time parallel algorithm, such as Parareal, to solve the forward and backward evolution problems arising in the optimization loop. We propose here a different strategy by devising directly a new time parallel algorithm, which we call ParaOpt, for the coupled forward and backward non-linear partial differential equations. ParaOpt is inspired by the Parareal algorithm for evolution equations, and thus is automatically a two-level method. We provide a detailed convergence analysis for the case of linear parabolic PDE constraints. We illustrate the performance of ParaOpt with numerical experiments both for linear and nonlinear optimality systems
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