2,256 research outputs found
On convergence conditions of waveform relaxation methods for linear differential-algebraic equations
AbstractFor linear constant-coefficient differential-algebraic equations, we study the waveform relaxation methods without demanding the boundedness of the solutions based on infinite time interval. In particular, we derive explicit expression and obtain asymptotic convergence rate of this class of iteration schemes under weaker assumptions, which may have wider and more useful application extent. Numerical simulations demonstrate the validity of the theory
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
Waveform Relaxation for the Computational Homogenization of Multiscale Magnetoquasistatic Problems
This paper proposes the application of the waveform relaxation method to the
homogenization of multiscale magnetoquasistatic problems. In the monolithic
heterogeneous multiscale method, the nonlinear macroscale problem is solved
using the Newton--Raphson scheme. The resolution of many mesoscale problems per
Gauss point allows to compute the homogenized constitutive law and its
derivative by finite differences. In the proposed approach, the macroscale
problem and the mesoscale problems are weakly coupled and solved separately
using the finite element method on time intervals for several waveform
relaxation iterations. The exchange of information between both problems is
still carried out using the heterogeneous multiscale method. However, the
partial derivatives can now be evaluated exactly by solving only one mesoscale
problem per Gauss point.Comment: submitted to JC
Application of the Waveform Relaxation Technique to the Co-Simulation of Power Converter Controller and Electrical Circuit Models
In this paper we present the co-simulation of a PID class power converter
controller and an electrical circuit by means of the waveform relaxation
technique. The simulation of the controller model is characterized by a
fixed-time stepping scheme reflecting its digital implementation, whereas a
circuit simulation usually employs an adaptive time stepping scheme in order to
account for a wide range of time constants within the circuit model. In order
to maintain the characteristic of both models as well as to facilitate model
replacement, we treat them separately by means of input/output relations and
propose an application of a waveform relaxation algorithm. Furthermore, the
maximum and minimum number of iterations of the proposed algorithm are
mathematically analyzed. The concept of controller/circuit coupling is
illustrated by an example of the co-simulation of a PI power converter
controller and a model of the main dipole circuit of the Large Hadron Collider
An efficient algorithm for the parallel solution of high-dimensional differential equations
The study of high-dimensional differential equations is challenging and
difficult due to the analytical and computational intractability. Here, we
improve the speed of waveform relaxation (WR), a method to simulate
high-dimensional differential-algebraic equations. This new method termed
adaptive waveform relaxation (AWR) is tested on a communication network
example. Further we propose different heuristics for computing graph partitions
tailored to adaptive waveform relaxation. We find that AWR coupled with
appropriate graph partitioning methods provides a speedup by a factor between 3
and 16
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