Convergence of dynamic iteration methods for initial value problems
SIAM Journal on Scientific and Statistical Computing
The use of Runge-Kutta formulae in waveform relaxation methods
Applied Numerical Mathematics - Special issue: parallel methods for ordinary differential equations
Waveform relaxation with overlapping splittings
SIAM Journal on Scientific Computing
Parallel and sequential methods for ordinary differential equations
Parallel and sequential methods for ordinary differential equations
Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation
SIAM Journal on Scientific Computing
Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems
SIAM Journal on Numerical Analysis
Optimized waveform relaxation methods for longitudinal partitioning of transmission lines
IEEE Transactions on Circuits and Systems Part I: Regular Papers - Special section on 2008 custom integrated circuits conference (CICC 2008)
The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
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Schwarz waveform relaxation algorithms (SWR) are naturally parallel solvers for evolution partial differential equations. They are based on a decomposition of the spatial domain into subdomains, and a partition of the time interval of interest into time windows. On each time window, an iteration, during which subproblems are solved in space-time subdomains, is then used to obtain better and better approximations of the overall solution. The information exchange between subdomains in space-time is performed through classical or optimized transmission conditions (TCs). We analyze in this paper the optimization problem when the time windows are short. We use as our model problem the optimized SWR algorithm with Robin TCs applied to the heat equation. After a general convergence analysis using energy estimates, we prove that in one spatial dimension, the optimized Robin parameter scales like the inverse of the length of the time window, which is fundamentally different from the known scaling on general bounded time windows, which is like the inverse of the square root of the time window length. We illustrate our analysis with a numerical experiment.