Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations

Domain decomposition methods in science and engineering XIX, LNCSE, Springer Verlag, 2010.Schwarz waveform relaxation methods have been studied for a wide range of scalar linear partial differential equations (PDEs) of parabolic and hyperbolic type. They are based on a space-time decomposition of the computational domain and the subdomain iteration uses an overlapping decomposition in space. There are only few convergence studies for non-linear PDEs. We analyze in this paper the convergence of Schwarz waveform relaxation applied to systems of semi-linear reaction-diffusion equations. We show that the algorithm converges linearly under certain conditions over long time intervals. We illustrate our results, and further possible convergence behavior, with numerical experiments

Optimized schwarz methods for Maxwell equations with discontinuous coefficients

We study non-overlapping Schwarz methods for solving time-harmonic Maxwell’s equations in heterogeneous media. We show that the classical Schwarz algorithm is always divergent when coefficient jumps are present along the interface. In the case of transverse magnetic or transverse electric two dimensional formulations, convergence can be achieved in specific configurations only. We then develop optimized Schwarz methods which can take coefficient jumps into account in their transmission conditions. These methods exhibit rapid convergence, and sometimes converge independently of the mesh parameter, even without overlap. We illustrate our analysis with numerical experiments

A Time-Dependent Dirichlet-Neumann Method for the Heat Equation

We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using a Laplace transform argument, we show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann method converges, similar to the case of Schwarz waveform relaxation algorithms. The convergence rate depends on the length of the subdomains as well as the size of the time window. In this discussion, we only stick to the linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and Engineering, Vol. 98, Springer-Verlag 201

Importance of single or blended polymer types for controlled in vitro release and plasma levels of a somatostatin analogue entrapped in PLA/PLGA microspheres.

The aim of the work was to develop biodegradable microspheres for controlled delivery of the somatostatin analogue vapreotide and maintenance of sustained plasma levels over 2–4 weeks after a single injection in rats. Vapreotide was microencapsulated into end-group capped and uncapped low molecular weight poly(lactide) (PLA) and poly(lactide-co-glycolide) (PLGA) by spray-drying and coacervation. Microspheres were prepared from single and blended (1:1) polymer types. The microparticles were characterized for peptide loading, in vitro release and pharmocokinetics in rats. Spray-drying and coacervation produced microspheres in the size range of 1–15 and 10–70 μm, respectively, and with encapsulation efficiencies varying between 46% and 87%. In vitro release of vapreotide followed a regular pattern and lasted more than 4 weeks, time at which 40–80% of the total dose were released. Microspheres made of 14-kDa end-group uncapped PLGA50:50 or 1:1 blends of this polymer with 35 kDa end-group uncapped PLGA50:50 gave the best release profiles and yielded the most sustained plasma levels above a pre-defined 1 ng/ml over approximately 14 days. In vitro/in vivo correlation analyses showed for several microsphere formulations a linear correlation between the mean residence time in vivo and the mean dissolution time (r=0.958) and also between the amount released between 6 h and 14 days and the AUC6h–14d (r=0.932). For several other parameters or time periods, no in vitro/in vivo correlation was found. This study demonstrates that controlled release of the vapreotide is possible in vivo for a duration of a least 2 weeks when administered i.m. to rats. These results constitute a step forward towards a twice-a-month or once-a-month microsphere-formulation for the treatment of acromegaly and neuroendocrine tumors

In vitro and in vivo evaluation of a somatostatin analogue released from PLGA microspheres

The purpose of this study was to design poly(lactide-co-glycolide) (PLGA) microspheres for the continuous delivery of the somatostatin analogue, vapreotide, over 2–4 weeks. The microspheres were produced by spray-drying and the desired characteristics, i.e. high encapsulation efficiency and controlled release over 2–4 weeks, achieved through optimizing the type of polymer, processing solvent, and co-encapsulated additive. The in vitro release was tested in fetal bovine serum preserved with 0.02% of thiomersal. Furthermore, formulations were injected intramuscularly into rats to obtain pharmacokinetic profiles. Encapsulation efficiency was between 34 and 91%, depending on the particular formulation. The initial peptide release (within 6 h) was lowest, i.e. 1 ng/ml) over 21–28 days in rats was the one made with end-group uncapped PLGA 50:50, the solvent acetic acid and the additive polyethyleneglycol. In conclusion, the optimization of formulation parameters allowed us to produce vapreotide-loaded PLGA microspheres of suitable characteristics for therapeutic use

Convergence of Parareal for the Navier-Stokes Equations Depending on the Reynolds Number

The paper presents first a linear stability analysis for the time-parallel Parareal method, using an IMEX Euler as coarse and a Runge-Kutta-3 method as fine propagator, confirming that dominant imaginary eigenvalues negatively affect Parareal’s convergence. This suggests that when Parareal is applied to the nonlinear Navier-Stokes equations, problems for small viscosities could arise. Numerical results for a driven cavity benchmark are presented, confirming that Parareal’s convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection. The effect is found to strongly depend on the spatial resolution

Optimized schwarz methods and model adaptivity in electrocardiology simulations

[No abstract available

Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow

In this paper we combine the Parareal parallel-in-time method together with spatial parallelization and investigate this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations. Parallelization of time stepping provides a new direction of parallelization and allows to employ additional cores to further speed up simulations after spatial parallelization has saturated. We report on numerical experiments performed on a Cray XE6, simulating a driven cavity flow with and without obstacles. Distributed memory parallelization is used in both space and time, featuring up to 2,048 cores in total. It is confirmed that the space-time-parallel method can provide speedup beyond the saturation of the spatial parallelization

Multicore optical fibre sensors for differential strain measurement

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