The numerical approximation of high frequency wave propagation is important in many applications. Examples include the simulation of seismic, acoustic, optical waves and microwaves. When the frequency of the waves is high, this is a difficult multiscale problem. The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive. Fortunately, there are computationally much less costly models, that are good approximations of many wave equations precisely for very high frequencies. Even for linear wave equations these models are often nonlinear. The goal of this paper is to review such mathematical models for high frequency waves, and to survey numerical methods used in simulations. We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations. We will also discuss finite frequency corrections and some other models.

Parallel algebraic multigrid methods in gyrokinetic turbulence simulations are presented. Discretized equations of the elliptic operator −∇²u+αu= f (with both α=0 and α≠0) are ubiquitous in magnetically confined fusion plasma applications. When α is equal to zero a "pure" Laplacian or Poisson equation results and when α is greater than zero a so called Helmholtz equation is produced. Taking a gyrokinetic turbulence simulation model as a testbed, we investigate the performance characteristics of basic classes of linear solvers (direct, one-level iterative, and multilevel iterative methods) on 2D unstructured finite element method (FEM) problems for both the Poisson and the Helmholtz equations.

Ranging from Re=100 to Re=20,000, several computational experiments are conducted, Re being the Reynolds number. The primary vortex stays put, and the long-term dynamic behavior of the small vortices determines the nature of the solutions. For low Reynolds numbers, the solution is stationary; for moderate Reynolds numbers, it is time periodic. For high Reynolds numbers, the solution is neither stationary nor time periodic: the solution becomes chaotic. Of the small vortices, the merging and the splitting, the appearance and the disappearance, and, sometimes, the dragging away from one corner to another and the impeding of the merging—these mark the route to chaos. For high Reynolds numbers, over weak fundamental frequencies appears a very low frequency dominating the spectra—this very low frequency being weaker than clear-cut fundamental frequencies seems an indication that the global attractor has been attained. The global attractor seems reached for Reynolds numbers up to Re=15,000. This is the lid-driven square cavity flow; the motivations for studying this flow are recalled in the Introduction.

This is the continuation of the paper "Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction" by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on non-staggered grids. This takes a new finite volume approach for approximating non-smooth solutions. A critical step for high-order finite volume schemes is to reconstruct a non-oscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th-order schemes without characteristic decomposition.

A modern approach to model reduction in chemical kinetics is often based on the notion of slow invariant manifold. The goal of this paper is to give a comparison of various methods of construction of slow invariant manifolds using a simple Michaelis-Menten catalytic reaction. We explore a recently introduced Method of Invariant Grids (MIG) for iteratively solving the invariance equation. Various initial approximations for the grid are considered such as Quasi Equilibrium Manifold, Spectral Quasi Equilibrium Manifold, Intrinsic Low Dimensional Manifold and Symmetric Entropic Intrinsic Low Dimensional Manifold. Slow invariant manifold was also computed using the Computational Singular Perturbation (CSP) method. A comparison between MIG and CSP is also reported. 

A novel amplitude factorization method is applied to solve a discrete buoyancy wave equation with arbitrary wind and temperature height distribution. The solution is given in the form of a cumulative product of complex factors, which are computed by a nonlinear, inhomogeneous, two-member recurrence formula, initiated from a radiative condition on top. Singularities of the wave equation due to evanescent winds are eliminated by turbulent friction. The method provides an estimation of the minimal vertical resolution, required to attain a stable accurate solution. The areas of application of the developed numerical scheme are high resolution modelling of orographic waves for arbitrary orography in general atmospheric stratification conditions, and testing of adiabatic kernels of numerical weather prediction models. 

Two methods of discrete images are proposed to approximate the reaction field from ionic solvent for a point charge inside a dielectric spherical cavity. Fast and accurate calculation of such a reaction field is needed in hybrid explicit/implicit solvation models of biomolecules. A first- and a second-order image approximation methods, in the order of u=λa (λ – the inverse Debye screening length of the ionic solvent, a – the radius of the spherical cavity), are derived. Each method involves a point image at the conventional Kelvin image point and a line image along the ray from the Kelvin image point to infinity. Based on these results, discrete point images are obtained by using Jacobi-Gauss quadratures. Numerical results demonstrate that two to three point images are sufficient to achieve a 10−3 accuracy in the reaction field with the second-order image approximation.

In this paper we develop a deterministic numerical method for solving the Boltzmann transport equation for semiconductors based on a transport-collision time-splitting method. Transport phases are solved by means of accurate flux-balance methods while collision steps are computed in the original k-grid. Numerical experiments are shown allowing for a discussion of this method with respect to other present in the literature.