On Extrapolated Multirate Methods

In this manuscript we construct extrapolated multirate discretization methods that allow to efficiently solve problems that have components with different dynamics. This approach is suited for the time integration of multiscale ordinary and partial differential equations and provides highly accurate discretizations. We analyze the linear stability properties of the multirate explicit and linearly implicit extrapolated methods. Numerical results with multiscale ODEs illustrate the theoretical findings

Achieving Very High Order for Implicit Explicit Time Stepping: Extrapolation Methods

In this paper we construct extrapolated implicit-explicit time stepping methods that allow to efficiently solve problems with both stiff and non-stiff components. The proposed methods can provide very high order discretizations of ODEs, index-1 DAEs, and PDEs in the method of lines framework. These methods are simple to construct, easy to implement and parallelize. We establish the existence of perturbed asymptotic expansions of global errors, explain the convergence orders of these methods, and explore their linear stability properties. Numerical results with stiff ODEs, DAEs, and PDEs illustrate the theoretical findings and the potential of these methods to solve multiphysics multiscale problems

Efficient Predictor for Co-Simulation with Multistep Sub-System Solvers

An industrial model of a dynamic system is usually not just a set of differential equations. External inputs acting on the system are common, such as an external force acting on a body or wind pressing on a car. Update of these inputs needs to be handled by the numerical solver in an efficient way.In dynamical simulation, multistep methods are commonly used. A multistep method uses the solution history in order to predict the future solution. When an input is changed, the history is no longer a good approximation for the future solution which may result in order reductions and simulation failure.In this paper, a modification of the predictor is presented. Modifying the predictor, instead of restarting the method, results in an increased performance of the method. The cost of the modification must be weighed with the cost of restarting the method. Experiments show that the benefit of modifying the predictor outweighs the cost of a restart

Multistep integration formulas for the numerical integration of the satellite problem

The use of two Class 2/fixed mesh/fixed order/multistep integration packages of the PECE type for the numerical integration of the second order, nonlinear, ordinary differential equation of the satellite orbit problem. These two methods are referred to as the general and the second sum formulations. The derivation of the basic equations which characterize each formulation and the role of the basic equations in the PECE algorithm are discussed. Possible starting procedures are examined which may be used to supply the initial set of values required by the fixed mesh/multistep integrators. The results of the general and second sum integrators are compared to the results of various fixed step and variable step integrators

A conservative implicit multirate method for hyperbolic problems

This work focuses on the development of a self adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations, that could benefit from different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach, that can be generalized to various implicit time discretization methods. It is based on flux partitioning, so that flux exchanges between a cell and its neighbors are balanced. A number of numerical experiments on both non-linear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach

Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection–diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank–Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the t2+hp+12 error estimates for the L2-norm under either the standard hyperbolic CFL condition, when piecewise affine (p=1) approximation is used, or in the case of finite element approximation of order p≥1, a stronger, so-called 4/3-CFL, i.e. t≤Ch4/3. The theory is illustrated with some numerical examples

A Software Framework for Implementation and Evaluation of Co-Simulation Algorithms

In this thesis, simulation of coupled dynamic models, denoted sub-systems, is analyzed and described in a co-simulation context. This means that the respective coupled systems contain their own internal integrator, hidden from the coupling interface. Co-Simulation is an interesting and active research field where industry is a driving force. The problems where co-simulation is an interesting approach is two-fold. On one-hand, there is the coupling of sub-systems between tools. Consider the case where tools use different representation of the sub-systems and the problem presented by the coupling of the two. On the other hand, there is the performance issue. There is a potential performance increase for the overall system simulation when using a tailored integrator for each sub-system compared to using a general integrator for the monolithic system. The aim of this thesis is to develop a testing framework for currently used co-simulation approaches and to describe the state of the art in co-simulation. Additionally, the aim is to be able to test the approaches on industrially relevant models and academic test models. Using co-simulation for simulation of coupled systems may result in stability problems depending on the approach used, and the intention here is to describe when it occurs and how to handle it. The commonly used methods use fixed step-size for determining when information between the models are to be exchanged. A recent development for co-simulation of coupled systems using a variable step-size method is described together with the requirements for performing such a simulation. Attaining the goals of the thesis has required a substantial effort in software development to create a foundation in terms of a testing framework. For gaining access to models from industry, the newly defined Functional Mock-up Interface has been used and a tool for working with these type of models, called PyFMI, has been developed. Another part is access to integrators, necessary for evaluating the impact of the internal integrator in the sub-systems. A tool providing a unified high-level interface to various integrators has been developed and is called Assimulo. The key component is the algorithm for performing the co-simulation. It has been developed to be easily extensible and to support the currently used co-simulation methods. The developed framework has been proven to be successful in evaluating co-simulation approaches on both academic test examples and on industrially relevant models as will be shown in the thesis