Extrapolation-based implicit-explicit general linear methods

For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order. We look for methods with large absolute stability region, assuming that the implicit part of the method is A- or L-stable. We provide examples of IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings

An asynchronous leapfrog method II

A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. It is obtained by natural modifications of the well-known leapfrog method, which is a second order, two-step, explicit method. According to the latter method, the input data for an integration step are two system states, which refer to different times. The usage of two states instead of a single one can be seen as the reason for the robustness of the method. Since the time step size thus is part of the step input data, it is complicated to change this size during the computation of a discrete trajectory. This is a serious drawback when one needs to implement automatic time step control. The proposed modification transforms one of the two input states into a velocity and thus gets rid of the time step dependency in the step input data. For these new step input data, the leapfrog method gives a unique prescription how to evolve them stepwise. The stability properties of this modified method are the same as for the original one: the set of absolute stability is the interval [-i,+i] on the imaginary axis. This implies exponential growth of trajectories in situations where the exact trajectory has an asymptote. By considering new evolution steps that are composed of two consecutive old evolution steps we can average over the velocities of the sub-steps and get an integrator with a much larger set of absolute stability, which is immune to the asymptote problem. The method is exemplified with the equation of motion of a one-dimensional non-linear oscillator describing the radial motion in the Kepler problem.Comment: 41 pages, 25 figure

A BEM-BDF scheme for curvature driven moving Stokes flows

A backward differences formulae (BDF) scheme, is proposed to simulate the deformation of a viscous incompressible Newtonian fluid domain in time, which is driven solely by the boundary curvature. The boundary velocity field of the fluid domain is obtained by writing the governing Stokes equations in terms of an integral formulation that is solved by a boundary element method. The motion of the boundary is modelled by considering the boundary curve as material points. The trajectories of those points are followed by applying the Lagrangian representation for the velocity. Substituting this representation into the discretized version of the integral equation yields a system of non-linear ODEs. Here the numerical integration of this system of ODEs is outlined. It is shown that, depending on the geometrical shape, the system can be stiff. Hence, a BDF-scheme is applied to solve those equations. Some important features with respect to the numerical implementation of this method are high-lighted, like the approximation of the Jacobian matrix and the continuation of integration after a mesh redistribution. The usefulness of the method for both two-dimensional and axisymmetric problems is demonstrated

Optimization-based search for nordsieck methods of high order with quadratic stability polynomials

We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7

Extrapolated Implicit–Explicit Runge–Kutta Methods

We investigate a new class of implicit–explicit singly diagonally implicit Runge–Kutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicit–explicit general linear methods

Efficient General Linear Methods of High Order with Inherent Quadratic Stability

We search for general linear methods with s internal stages and r = s + 1 external stages of order p = s + 1 and stage order q = s. We require that stability function of these methods has only two non-zero roots. This is achieved by imposing the so-called inherent quadratic stability conditions. Examples of such general linear methods which are A- and L-stable up to the order p = 8 and stage order q = p - 1 are derived

Construction of Nordsieck Second Derivative General Linear Methods with Inherent Quadratic Stability

This paper describes the construction of second derivative general linear methods in Nordsieck form with stability properties determined by quadratic stability functions. This is achieved by imposing the so–called inherent quadratic stability conditions. After satisfying order and inherent quadratic stability conditions, the remaining free parameters are used to find the methods with L–stable property. Examples of methods with p = q = s = r − 1 up to order four are given

Probabilistic Numerics and Uncertainty in Computations

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.Comment: Author Generated Postprint. 17 pages, 4 Figures, 1 Tabl

Dust and gas mixtures with multiple grain species - a one-fluid approach

GL acknowledges funding from the European Research Council for the FP7 ERC advanced grant project ECOGAL. DJP is very grateful for funding via an ARC Future Fellowship, FT130100034, and Discovery Project grant DP130102078.We derive the single-fluid evolution equations describing a mixture made of a gas phase and an arbitrary number of dust phases, generalizing the approach developed by Laibe & Price. A generalization for continuous dust distributions as well as analytic approximations for strong drag regimes is also provided. This formalism lays the foundation for numerical simulations of dust populations in a wide range of astrophysical systems while avoiding limitations associated with a multiple-fluid treatment. The usefulness of the formalism is illustrated on a series of analytical problems, namely the DUSTYBOX, DUSTYSHOCK and DUSTYWAVE problems as well as the radial drift of grains and the streaming instability in protoplanetary discs. We find physical effects specific to the presence of several dust phases and multiple drag time-scales, including non-monotonic evolution of the differential velocity between phases and increased efficiency of the linear growth of the streaming instability. Interestingly, it is found that under certain conditions, large grains can migrate outwards in protoplanetary discs. This may explain the presence of small pebbles at several hundreds of astronomical units from their central star.Publisher PDFPeer reviewe