Asymptotic behavior of splitting schemes involving time-subcycling techniques

This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, time-subcycling techniques are widely used every day to improve computational efficiency. These methods rely on a decomposition of the vector field in a fast part and a slow part and take advantage of that decomposition. This way, if an unconditionnally stable (semi-)implicit scheme cannot be easily implemented, one can integrate the fast equations with a much smaller time step than that of the slow equations, instead of having to integrate the whole system with a very small time-step to ensure stability. Then, one can build a numerical integrator using a standard composition method, such as a Lie or a Strang formula for example. Such methods are primarily designed to be convergent in short-time to the solution of the original problems. However, their longtime behavior rises interesting questions, the answers to which are not very well known. In particular, when the solutions of the problems converge in time to an asymptotic equilibrium state, the question of the asymptotic accuracy of the numerical longtime limit of the schemes as well as that of the rate of convergence is certainly of interest. In this context, the asymptotic error is defined as the difference between the exact and numerical asymptotic states. The goal of this paper is to apply that kind of numerical methods based on splitting schemes with subcycling to some simple examples of evolutionary ODEs and PDEs that have attractive equilibrium states, to address the aforementioned questions of asymptotic accuracy, to perform a rigorous analysis, and to compare them with their counterparts without subcycling. Our analysis is developed on simple linear ODE and PDE toy-models and is illustrated with several numerical experiments on these toy-models as well as on more complex systems. Lie andComment: IMA Journal of Numerical Analysis, Oxford University Press (OUP): Policy A - Oxford Open Option A, 201

Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test

Computation of shock profiles in radiative hydrodynamics

International audienceThis article is devoted to the construction of a numerical scheme to solve the equations of radiative hydrodynamics. We use this numerical procedure to compute shock profiles and illustrate some earlier theoretical results about their smoothness and monotonicity properties. We first consider a scalar toy model, then we extend our analysis to a more realistic system for the radiative hydrodynamics that couples the Euler equations and an elliptic equation

Shock profiles for hydrodynamic models for fluid-particles flows in the flowing regime

Starting from coupled fluid-kinetic equations for the modeling of laden flows, we derive relevant viscous corrections to be added to asymptotic hydrodynamic systems, by means of Chapman-Enskog expansions and analyse the shock profile structure for such limiting systems. Our main findings can be summarized as follows. Firstly, we consider simplified models, which are intended to reproduce the main difficulties and features of more intricate systems. However, while they are more easily accessible to analysis, such toy-models should be considered with caution since they might lose many important structural properties of the more realistic systems. Secondly, shock profiles can be identified also in such a case, which can be proven to be stable at least in the regime of small amplitude shocks. Last, but not least, regarding at the temperature of the mixture flow as a parameter of the problem, we show that the zero-temperature model admits viscous shock profiles. Numerical results indicate that a similar conclusion should apply in the regime of small positive temperatures

Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant

In this paper, we study a flower population in which self-reproduction is not permitted. Individuals are diploid, {that is, each cell contains two sets of chromosomes}, and {distylous, that is, two alleles, A and a, can be found at the considered locus S}. Pollen and ovules of flowers with the same genotype at locus S cannot mate. This prevents the pollen of a given flower to fecundate its {own} stigmata. Only genotypes AA and Aa can be maintained in the population, so that the latter can be described by a random walk in the positive quadrant whose components are the number of individuals of each genotype. This random walk is not homogeneous and its transitions depend on the location of the process. We are interested in the computation of the extinction probabilities, {as} extinction happens when one of the axis is reached by the process. These extinction probabilities, which depend on the initial condition, satisfy a doubly-indexed recurrence equation that cannot be solved directly. {Our contribution is twofold : on the one hand, we obtain an explicit, though intricate, solution through the study of the PDE solved by the associated generating function. On the other hand, we provide numerical results comparing stochastic and deterministic approximations of the extinction probabilities.Comment: 23 page

A model describing the growth and the size distribution of multiple metastatic tumors

International audienceCancer is one of the greatest killers in the world, particularly in western countries. A great effort from medical research is devoted to cancer and mathematical modeling must be consider as an additional tool for the physicians and biologists to understand cancermechanisms and to determine the adapted treatments. Metastasis make all the seriousness of cancer. In 2000, Iwata et al. [9] proposed a model which describes the evolution of an untreated metastatic tumors population. We provide here a mathematical analysis of this model which brings us to the determination of a Malthusian rate characterizing the population exponential growth. We provide as well a numerical analysis of the PDE given by the model