Comparison of the asymptotic stability properties for two multirate strategies

This paper contains a comparison of the asymptotic stability properties for two multirate strategies. For each strategy, the asymptotic stability regions are presented for a 2 x 2 test problem and the differences between the results are discussed. The considered multirate schemes use Rosenbrock type methods as the main time integration method and have one level of temporal local refinement. Some remarks on the relevance of the results for 2 x 2 test problems are presented

A multirate approach for time domain simulation of very large power systems

The time evolution of power systems is modeled by systems of differential and algebraic equations (DAEs) [8]. The variables involved in these DAEs may exhibit different time scales. Some of the variables can be highly active while other variables can stay constant during the entire time integration period. In standard numerical time integration methods for DAEs the most active variables impose the time step for the whole system. We present a strategy, which allows the use of different, local time steps over the variables. The partitioning of the components of the system in different classes of activity is performed automatically based on the topology of the power system. The performance of the multirate approach for two case studies is presented

Analysis of a multirate theta-method for stiff ODEs

This paper contains a study of a simple multirate scheme, consisting of the ¿-method with one level of temporal local refinement. Issues of interest are local accuracy, propagation of interpolation errors and stability. The theoretical results are illustrated by numerical experiments, including results for more levels of refinement with automatic partitionin

Construction of high-order multirate Rosenbrock methods 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 multirate methods based on the higher-order, stiff Rosenbrock integrators. 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

Analysis, numerics, and optimization of algae growth

We extend the mathematical model for algae growth as described in [11] to include new effects. The roles of light, nutrients and acidity of the water body are taken into account. Important properties of the model such as existence and uniqueness of solution, as well as boundedness and positivity are investigated. We also discuss the numerical integration of the resulting system of ordinary differential equations and derive a condition which guarantees positivity of the numerical solution. An optimization problem is formulated which demonstrates an application of the model

A multirate time stepping strategy for stiff ODEs

To solve ODEs with different time scales which are localized over the components, multirate time stepping is examined. In this paper we introduce a self-adjusting 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 for the whole system. We primarily consider implicit time stepping methods, suitable for stiff or mildly stiff ODEs. Numerical results for 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

Analysis of explicit multirate and partitioned Runge-Kutta schemes for conservation laws

Multirate schemes for conservation laws or convection-dominated problems seem to come in two flavors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods

A multirate time stepping strategy for parabolic PDE.

To solve PDE problems with different time scales that are localized in space, multirate time stepping is examined. We introduce a self-adjusting multirate time stepping strategy, in which the step size at a particular grid point is determined by the local temporal variation of the solution, instead of using a minimal single step size for the whole spatial domain. The approach is based on the `method of lines', where first a spatial discretization is performed, together with local error estimates for the resulting semi-discret system. We will primarily consider implicit time stepping methods, suitable for parabolic problems. Our multirate strategy is tested on several parabolic problems in one spatial dimension (1D