DAE-formulation for optimal solutions of a multirate model

A dynamical system including frequency modulated signals can be transformed into multirate partial differential algebraic equations. Optimal solutions are determined by a necessary condition. A method of lines yields a semi-discretisation in the case of initial-boundary value problems. We show that the resulting system can be written in a standard formulation of differential algebraic equations. Hence appropriate time integration schemes are available for a numerical solution. We present results for a test example modelling the electric circuit of a ring oscillator. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multirate finite step methods for linear acoustics

Many processes contain phenomena on different time scales, leading to model equations with fast and small parts. There are several approaches to solve these equations, like additive Runge Kutta methods or multirate infinitesimal steps methods (MIS). Both methods make use of the additive splitting of the ODE in fast and small parts. The multiple infinitesimal step method integrates the slow part with a large macro stepsize, whereas the fast terms are solved with several smaller steps of a simpler method. The order conditions of a MIS method are derived under the assumption of the exact integration of the fast parts. We develop the multirate finite step methods (MFS). These methods are derived from the MIS methods, by taking a simple small scale integrator for the fast terms. This small scale integrator uses the same number of steps in each stage. With these assumptions, we derive the order conditions, such that the order is independent in the number of small steps. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computing time investigations for variational multirate integration

Consider a mechanical system that contains slow and fast dynamics. Let it be possible, to split the potential energy into a slow and a fast potential and the configuration vector into slow and fast variables. For such systems, multirate schemes simulate the different parts using different time steps with the goal to save computing time. For the proposed multirate scheme, a time grid consisting of micro and macro nodes is used and the integrator is derived from a discrete variational principle. Variational integrators conserve properties like symplecticity and momentum maps and have good energy behavior. To solve the resulting system of coupled nonlinear equations, a Newton-Raphson iteration with an analytical Jacobian is used. It is demonstrated that the multirate approach leads to less computing time compared to singlerate simulation by means of three example systems, the Fermi-Pasta-Ulam problem, a triple spherical pendulum and a simple atomistic model, where the latter two are subject to constraints. Computing times are compared for different numbers of micro and macro nodes for dynamic simulations during a certain time interval. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of a multirate RODAS method for stiff ODEs

Multirate time stepping is a numerical technique for efficiently solving large-scale ordinary differential equations (ODEs) with widely different time scales localized over the components. This technique enables one to use large time steps for slowly varying components, and small steps for rapidly varying ones. Multirate methods found in the literature are normally of low order, one or two. Focusing on stiff ODEs, in this paper we discuss the construction of a multirate method based on the fourth-order RODAS method. Special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction. For stiff, linear systems containing a stiff source term, we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method.     

Multidimensional models for analysing frequency modulated signals

In radio frequency (RF) applications, slowly varying signals often modulate the amplitude and frequency of fast carrier waves. Thus a numerical simulation of the differential algebraic equations (DAEs) modelling the electric circuit becomes tedious. Alternative models are required to achieve efficient simulations. A multivariate formulation of signals yields a suitable representation via decoupling the widely separated time scales. Consequently, the circuit's DAEs change into warped multirate partial DAEs. On the other hand, the transient behaviour of the circuit can also be approximated by a parameter-dependent DAE model including a multivariate structure. The properties of this alternative strategy are investigated. In particular, the two multidimensional approaches are compared with respect to the simulation of RF signals.     

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are di?cult to mimic using conventional time integrators that are used to solve systems of ordinary di?erential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.     

Variational multirate integration in multi-body dynamics

For systems that contain slow and fast dynamics, variational multirate integration schemes are used. These schemes split the system into parts which are simulated using two time grids consisting of micro and macro nodes. This formulation can be extended for multi-body systems. The rigid multi-body system is described by the so called director formulation and constraints describing the joints connecting the bodies. With the Lagrange multiplier method, the constraints are introduced into the equations of motion. A way to implement the null space method into the variational multirate framework is shown and the influence on the number of unknowns is investigated. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multirate finite step methods with varying step sizes

Many partial differential equations consist of slow and fast scales. Often, the right hand side of semidiscretized PDEs can be split additively in corresponding fast and slow parts. Many methods utilise the additive splitting of these equations, like generalized additive Runge-Kutta (GARK) methods or multirate infinitesimal step methods. The latter one treat the slow part with macro step sizes, whereas the fast part is integrated a ODE solver. The corresponding order conditions assume the exact solution of the auxiliary ODE, i.e. assume an infinite number of small steps. We extend the MIS approach by fixing the number of steps, which leads to the multirate finite steps (MFS) method. The order conditions are derived, such that the order is independent in the number of small steps in each stage. Finally, we confirm the theoretical results by numerical experiments. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multirate models for simulating a colpitts oscillator

A model based on multirate partial differential algebraic equations yields an efficient numerical simulation of electric circuits in radio frequency applications. Considering frequency modulation, free parameters of the model are determined appropriately by a minimisation strategy. We apply the multirate approach to simulate a modified version of a Colpitts oscillator, which exhibits frequency modulation at widely separated time scales. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multirate W-method for electrical networks in state–space formulation

Subunits of coupled technical systems typically behave on differing time scales, which are often separated by several orders of magnitude. An ordinary integration scheme is limited by the fastest changing component, whereas so-called multirate methods employ an inherent step size for each subsystem to exploit these settings. However, the realization of the coupling terms is crucial for any convergence. Thus the approach to return to one-step methods within the multirate concept is promising. This paper introduces the multirate W-method for ordinary differential equations and gives a theoretical discussion in the context of partitioned Rosenbrock–Wanner methods. Finally, the MATLAB implementation of an embedded scheme of order (3)2 is tested for a multirate version of Prothero–Robinson's equation and the inverter-chain-benchmark.     

A multirate time stepping strategy for stiff ordinary differential equations

To solve ODE systems with different time scales which are localized over the components, multirate time stepping is examined. In this paper we introduce a self-adjusting multirate time stepping strategy, in which the step size for a particular component is determined by its own local temporal variation, instead of using a single step size for the whole system. We primarily consider implicit time stepping methods, suitable for stiff or mildly stiff ODEs. Numerical results with our multirate strategy are presented for several test problems. Comparisons with the corresponding single-rate schemes show that substantial gains in computational work and CPU times can be obtained. AMS subject classification (2000)  65L05, 65L06, 65L50     

From SOI to Abstract Electric-Thermal-1D Multiscale Modeling for First Order Thermal Effects

Self-heating occurs in integrated circuits, specially for SOI-based devices. Naturally, the heat distribution affects the circuit’s functionality. For reliable designs in SOI-chip technology, and other applications, the thermal aspects have to be addressed. Therefore we develop a model, which is based on distributed 1D and lumped 0D elements, and takes into account that heat is stored and slowly conducted between elements. The emerging coupled multiscale system of heat evolution and electric network consists of parabolic partial-differential (thermal part) and differential-algebraic equations (electric network part). For the thermal model, we verify properties as positivity and strict passivity. Since time scales differ largely, the coupled problem exhibits multirate potential.     

Limitations of computing time savings in variational multirate integrations

For systems that contain slow and fast dynamics, variational multirate integration schemes are used. These schemes split the system into parts which are simulated using a time grid consisting of micro and macro nodes. This leads to computing time savings, however not unlimited, for a certain number of micro steps per macro step the computing time is minimal. To find a relation between this minimum computing time and the number of variables in the system, the computing time for the Fermi-Pasta-Ulam problem (FPU) is measured for different numbers of masses and different numbers of micro steps. In addition, the numerical convergence of the variational multirate integration is shown for the FPU. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wavelet-based Adaptive Grids for the Simulation of Multirate Partial Differential-Algebraic Equations

We present an adaptive method to solve biperiodic problems in chip design with widely separated time scales and steep gradients due to digital-like signal structures. Motivated from the example of the switched capacitor filter, we explain the multidimensional signal model and the transformation of network equations to the hyperbolic multirate differential-algebraic system. Then adapted hat-wavelets are introduced to detect the steep gradients. In this way, an adaptive grid is defined for the efficient application of a finite difference method to solve the limit cycle problem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multirate infinitesimal step methods for atmospheric flow simulation

The numerical solution of the Euler equations requires the treatment of processes in different temporal scales. Sound waves propagate fast compared to advective processes. Based on a spatial discretisation on staggered grids, a multirate time integration procedure is presented here generalising split-explicit Runge-Kutta methods. The advective terms are integrated by a Runge-Kutta method with a macro stepsize restricted by the CFL number. Sound wave terms are treated by small time steps respecting the CFL restriction dictated by the speed of sound.Split-explicit Runge-Kutta methods are generalised by the inclusion of fixed tendencies of previous stages. The stability barrier for the acoustics equation is relaxed by a factor of two.Asymptotic order conditions for the low Mach case are given. The relation to commutator-free exponential integrators is discussed. Stability is analysed for the linear acoustic equation. Numerical tests are executed for the linear acoustics and the nonlinear Euler equations.     

Multirate Runge–Kutta schemes for advection equations

Explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena. The wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step. Multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps, resulting in a moderate to substantial reduction of the computational cost, depending on the scenario to simulate [S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput. 41 (1983) 321–336; H. Tang, G. Warnecke, A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and pace grids, SIAM J. Sci. Comput. 26 (4) (2005) 1415–1431; E. Constantinescu, A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, SIAM J. Sci. Comput. 33 (3) (2007) 239–278]. In air pollution modeling the advection part is usually integrated explicitly in time, where the time step is constrained by a locally varying Courant–Friedrichs–Lewy (CFL) number. Multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction. Therefore it is of major interest to apply multirate schemes to the advection equation. We introduce a generic recursive multirate Runge–Kutta scheme that can be easily adapted to an arbitrary number of refinement levels. It preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit Runge–Kutta methods as base method.     

Wavelet-based adaptive grids for multirate partial differential-algebraic equations

The mathematical model of electric circuits yields systems of differential-algebraic equations (DAEs). In radio frequency applications, a multivariate model of oscillatory signals transforms the DAEs into a system of multirate partial differential-algebraic equations (MPDAEs). Considering quasiperiodic signals, an approach based on a method of characteristics yields efficient numerical schemes for the MPDAEs in time domain. If additionally digital signal structures occur, an adaptive grid is required to achieve the efficiency of the technique. We present a strategy applying a wavelet transformation to construct a mesh for resolving steep gradients in respective signals. Consequently, we employ finite difference methods to determine an approximative solution of characteristic systems in according grid points. Numerical simulations demonstrate the performance of the adaptive grid generation, where radio frequency signals with digital structures are resolved.     

A new algorithm for computing modified z-transforms

A new method of computing modified z-tranforms is introduced,which is easy to implement and use. The new algorithm is explainedin detail, and a compact listing is included. Six examples aregiven to demonstrate different uses of the algorithm. Assumptionsused to develop the basic algorithm are explained, and proceduresto overcome the restrictions of these assumptions are explainedand demonstrated. How to solve multirate digital systems problemsusing the introduced algorithm is demonstrated with an example.Time-saving and memory-saving modifications of the algorithmfor multirate problems are also explained.     

Multirate multicast service provisioning II: a tâtonnement process for rate allocation

In multirate multicast different users in the same multicast group can receive services at different rates depending on their own requirements and the congestion level of the network. In this two-part paper we present a general framework for addressing the optimal rate control problem in multirate multicast where the objective is the maximization of a social welfare function expressed by the sum of the users’ utility functions. In Part II we present a market based mechanism and an adjustment process that have the following features. They satisfy the informational constraints imposed by the nature of multirate multicast; and when they are combined with the results of Part I they result in an optimal solution of the corresponding centralized multirate multicast problem.     

An Algebraic Method for Pole Placement in Multivariable Systems

This paper considers the pole placement in multivariable systems involving known delays by using dynamic controllers subject to multirate sampling. The controller parameterizations are calculated from algebraic equations which are solved by using the Kronecker product of matrices. It is pointed out that the sampling periods can be selected in a convenient way for the solvability of such equations under rather weak conditions provided that the continuous plant is spectrally controllable. Some overview about the use of nonuniform sampling is also given in order to improve the system's performance.