Multi-Adaptive Time-Integration

Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps

Time integration methods for reactor kinetics

"December 1971."Includes bibliographical references (pages 104-107)A technique based on the Padé approximations is applied to the solution of the point kinetics equations. The method consists of treating explicitly the roots of the inhour formula which would make the Padé approximations inaccurate. Also, an analytic method is developed which permits a fast inversion of polynomials of the point kinetics matrix and has direct applicability to the Pads approximations. Results are presented for several cases using various options of the method. It is concluded that the technique provides a fast and accurate computational method for the point kinetics equations. Also, an implicit solution method for the time-dependent multigroup diffusion equations known as the "theta method" is studied. Both the usual method and a variation of it, derived from the precursor integrated equations, are considered. Several properties of both versions of the theta method are demonstrated. An attempt is made to find better integration parameters (thetas) for the method, based on corresponding point kinetics calculations. Calculations are done for several test cases, leading to the conclusion that the improvements obtained are of limited value.U.S. Atomic Energy Commission contract ; AT(11-1) -305

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes

Mixed time integration methods for transient thermal analysis of structures, appendix 5

Mixed time integration methods for transient thermal analysis of structures are studied. An efficient solution procedure for predicting the thermal behavior of aerospace vehicle structures was developed. A 2D finite element computer program incorporating these methodologies is being implemented. The performance of these mixed time finite element algorithms can then be evaluated employing the proposed example problem

Integration of Statistical Methods and Judgment for Time Series

We consider how judgment and statistical methods should be integrated for time-series forecasting. Our review of published empirical research identified 47 studies, all but four published since 1985. Five procedures were identified: revising judgment; combining forecasts; revising extrapolations; rule-based forecasting; and econometric forecasting. This literature suggests that integration generally improves accuracy when the experts have domain knowledge and when significant trends are involved. Integration is valuable to the extent that judgments are used as inputs to the statistical methods, that they contain additional relevant information, and that the integration scheme is well structured. The choice of an integration approach can have a substantial impact on the accuracy of the resulting forecasts. Integration harms accuracy when judgment is biased or its use is unstructured. Equal-weights combining should be regarded as the benchmark and it is especially appropriate where series have high uncertainty or high instability. When the historical data involve high uncertainty or high instability, we recommend revising judgment, revising extrapolations, or combining. When good domain knowledge is available for the future as well as for the past, we recommend rule- based forecasting or econometric methods.statistical methods, statistics, time series, forecasting, empirical research

GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units.Comment: 4 pages, 5 figure

Discrete-time port-Hamiltonian systems: A definition based on symplectic integration

We introduce a new definition of discrete-time port-Hamiltonian systems (PHS), which results from structure-preserving discretization of explicit PHS in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a discrete-time energy balance - expressed in terms of the discrete-time Dirac structure - extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order is consistent with the order of the numerical integration scheme. Implicit Gauss-Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments.Comment: 12 pages. Preprint submitted to Systems & Control Letter

On phenomenon of scattering on resonances associated with discretisation of systems with fast rotating phase

Numerical integration of ODEs by standard numerical methods reduces a continuous time problems to discrete time problems. Discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of system with one fast rotating phase lead to a situation of such kind: numerical solution demonstrate phenomenon of scattering on resonances that is absent in the original system.Comment: 10 pages, 5 figure