Accelerating waveform relaxation methods with application to parallel semiconductor device simulation

In this paper we apply a Galerkin method to solving the system of second-kmd Volterra integral equations that characterize waveform relaxation, or dynamic iteration, methods for solving linear time-varying initial-value problems. It is shown that the Galerkin approximations can be computed iteratively using Krylovsubspace algorithms. The resulticg iterative methods are combined with an operator Newton method and applied to solving the nonhear differential-algebraic system generated by spatial discretization of the timedependent semiconductor device equations. Experimental results are included to demonstrate that waveform Krylov-subspace methods converge significantly faster than classical waveform relaxation, and are better able to exploit the parallelism available in loosely coupled parallel machines than parallel versions of standard point-wise iterative schemes.

[1]  R. Temam,et al.  Asymptotic numerical analysis for the Navier-Stokes equations, 1 , 1982 .

[2]  P. Holmes,et al.  Regularity, approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation , 1993 .

[3]  O. Ladyzhenskaya,et al.  A dynamical system generated by the Navier-Stokes equations , 1975 .

[4]  Jean-Michel Ghidaglia,et al.  Dimension of the attractors associated to the Ginzburg-Landau partial differential equation , 1987 .

[5]  R. Temam,et al.  Determination of the solutions of the Navier-Stokes equations by a set of nodal values , 1984 .

[6]  R. Temam,et al.  On the dimension of the attractors in two-dimensional turbulence , 1988 .

[7]  C. Foiaș,et al.  Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$ , 1967 .

[8]  R. Temam,et al.  Determining modes and fractal dimension of turbulent flows , 1985, Journal of Fluid Mechanics.

[9]  E. Titi,et al.  Parameterizing the global attractor of the navier-stokes equations by nodal values , 1995 .

[10]  Edriss S. Titi,et al.  Determining nodes, finite difference schemes and inertial manifolds , 1991 .

[11]  G. Sell,et al.  Inertial manifolds for nonlinear evolutionary equations , 1988 .

[12]  R. Temam,et al.  Inertial Forms of Navier-Stokes Equations on the Sphere , 1993, chao-dyn/9304004.

[13]  R. Téman,et al.  Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations , 1988 .

[14]  Darryl D. Holm,et al.  Low-dimensional behaviour in the complex Ginzburg-Landau equation , 1988 .

[15]  Donald A. Jones,et al.  Determining finite volume elements for the 2D Navier-Stokes equations , 1992 .

[16]  Dean A. Jones,et al.  On the number of determining nodes for the 2D Navier-Stokes equations , 1992 .

[17]  Igor Kukavica,et al.  On the number of determining nodes for the Ginzburg-Landau equation , 1992 .

[18]  R. Temam,et al.  Structure of the set of stationary solutions of the navier-stokes equations , 1977 .

[19]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[20]  W. M. Oliva,et al.  An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory , 1983 .

[21]  George R. Sell,et al.  Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations , 1989 .

[22]  O. Ladyzhenskaya.  Finite-dimensionality of bounded invariant sets for Navier-stokes systems and other dissipative systems , 1985 .

[23]  R. Temam,et al.  Attractors Representing Turbulent Flows , 1985 .

[24]  O. P. Manley,et al.  Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Bénard convection , 1982 .

[25]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[26]  Minimum number of modes in approximate solutions to equations of hydrodynamics , 1981 .