Asymptotic-induced numerical methods for conservation laws

Asymptotic-induced methods are presented for the numerical solution of hyperbolic conservation laws with or without viscosity. The methods consist of multiple stages. The first stage is to obtain a first approximation by using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problems identified by using techniques derived via asymptotics. Finally, a residual correction increases the accuracy of the scheme. The method is derived and justified with singular perturbation techniques

Stabilization of Unstable Procedures: The Recursive Projection Method

Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a “black-box” time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the symplectic structure in the case $\epsilon=0$. In the case $0<\epsilon\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order

Exact regularized point particle method for multi-phase flows in the two-way coupling regime

Particulate flows have been largely studied under the simplifying assumptions of one-way coupling regime where the disperse phase do not react-back on the carrier fluid. In the context of turbulent flows, many non trivial phenomena such as small scales particles clustering or preferential spatial accumulation have been explained and understood. A more complete view of multiphase flows can be gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange between the carrier and the suspended phase, certainly relevant at increasing mass loading. In such regime, partially investigated in the past by the so-called Particle In Cell (PIC) method, much is still to be learned about the dynamics of the disperse phase and the ensuing alteration of the carrier flow. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. In fact, the momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere where the transient disturbance produced by each particle is evaluated in a closed form. The particles are described as lumped, point masses which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the physically relevant interactions between particles and fluid which avoids any "ah hoc" assumption. The approach is suited for high efficiency implementation on massively parallel machines since the transient disturbance produced by the particles is strongly localized in space around the actual particle position. As will be shown, hundred thousands particles can therefore be handled at an affordable computational cost as demonstrated by a preliminary application to a particle laden turbulent shear flow.Comment: Submitted to Journal of Fluid Mechanics, 56 pages, 15 figure