Projective integration for hyperbolic conservation laws and multiscale kinetic equations

Virtually all applications in science and engineering exhibit a multiscale structure, in which the constituting processes evolve on different temporal or spatial scales. In most cases, however, the engineer or scientist is only interested in extracting information on the system on a macroscopic (human-level) scale. In this regard, throughout the years, a lot of effort has been put in the development of phenomenological, macroscopic models that describe the system on a scale of human interest while ignoring the underlying microscopic world. Frequently, such macroscopic descriptions are accompanied by a set of constitutive relations that capture important microscopic mechanisms. Regrettably, in a myriad of situations, the assumptions underpinning the constitutive relations or the macroscopic model itself are not well founded, or it can be simply impossible to construct a macroscopic model altogether from first principles. In such cases, one needs to resort to a full multiscale representation of the problem, which subsequently requires efficient multiscale simulation routines.

This thesis encloses three main, multiscale-flavored blocks. The first block aims at designing a general solver for systems of hyperbolic conservation laws. Although intrinsically single-scaled, the hyperbolic problem is replaced with an artificial, two-scale kinetic model, which is easier to handle numerically. The link between these two models, which guarantees that the replacement kinetic problem produces the same dynamics as the original hyperbolic problem, is established by appropriately designing the kinetic equilibrium distribution. To efficiently treat the introduced multiscale nature, we employ a projective integration method, which first takes a number of small steps with an inner integrator, driving the kinetic model towards its particular equilibrium, followed by an extrapolation forward into time over a much larger time step by an outer integrator. The flexibility, generality and high order of accuracy in time of the method are illustrated by several benchmark tests.

In the second block, we step away from artificial kinetic equations and shift our attention to kinetic models describing actual particle systems. The two main models under consideration are the BGK equation, which we regard both in a two-scale as well as a multi-scale setting, and the Boltzmann equation, which is inherently a multi-scale problem. To deal with the multiple scales, we use a telescopic projective integration method, which generalizes the projective integration idea by building a hierarchy of projective levels each using a number of inner integrator steps followed by an (outer) extrapolation step. We apply the method to a number of applications in one and two spatial dimensions.

We then switch gears in the third block, which targets two-scale, slow-fast stochastic systems. Under the assumption of ergodicity of the fast equation, it is known that, for a sufficiently large time scale separation between slow and fast scales, a reduced description only in terms of the slow variable exists. In this description, the time derivative is computed by averaging the original slow equation over the invariant measure produced by the fast dynamics. However, in general, we cannot compute the invariant measure nor the averaging operation analytically. For this purpose, we use the heterogeneous multiscale method (HMM), which combines forward Euler time stepping of the reduced equation with a procedure to estimate the unknown time derivative. Unfortunately, the uncertainty of the HMM estimator, essentially being a Markov chain Monte Carlo method, can be large. Therefore, we propose a novel control variables variance reduction technique based on correlating HMM estimators on consecutive macroscopic time steps and apply the method to a linear and nonlinear model problem.