A fundamental issue in CFD is the role of coordinates and, in particular, the search for "optimal" coordinates. This paper reviews and generalizes the recently developed unified coordinate system (UC). For one-dimensional flow, UC uses a material coordinate and thus coincides with Lagrangian system. For two-dimensional flow it uses a material coordinate, with the other coordinate determined so as to preserve mesh othorgonality (or the Jacobian), whereas for three-dimensional flow it uses two material coordinates, with the third one determined so as to preserve mesh skewness (or the Jacobian). The unified coordinate system combines the advantages of both Eulerian and the Lagrangian system and beyond. Specifically, the followings are shown in this paper. (a) For 1-D flow, Lagrangian system plus shock-adaptive Godunov scheme is superior to Eulerian system. (b) The governing equations in any moving multi-dimensional coordinates can be written as a system of closed conservation partial differential equations (PDE) by appending the time evolution equations – called geometric conservation laws – of the coefficients of the transformation (from Cartesian to the moving coordinates) to the physical conservation laws; consequently, effects of coordinate movement on the flow are fully accounted for. (c) The system of Lagrangian gas dynamics equations is written in conservation PDE form, thus providing a foundation for developing Lagrangian schemes as moving mesh schemes. (d) The Lagrangian system of gas dynamics equations in two- and three-dimension are shown to be only weakly hyperbolic, in direct contrast to the Eulerian system which is fully hyperbolic; hence the two systems are not equivalent to each other. (e) The unified coordinate system possesses the advantages of the Lagrangian system in that contact discontinuities (including material interfaces and free surfaces) are resolved sharply. (f) In using the UC, there is no need to generate a body-fitted mesh prior to computing flow past a body; the mesh is automatically generated by the flow. Numerical examples are given to confirm these properties. Relations of the UC approach with the Arbitrary-Lagrangian-Eulerian (ALE) approach and with various moving coordinates approaches are also clarified.

We investigate the behavior and sensitivity of the frozen mode phenomenon in finite structures with anisotropic materials, including both magnetic materials and non-normal incidence. The studies are done by using a high-order accurate discontinuous Galerkin method for solving Maxwell's equations in the time domain. We confirm the existence of the phenomenon also in the time-domain and study carefully the impact of the finite crystal on the frozen mode. This sets the stage for a thorough study of the robustness of the frozen mode phenomenon, resulting in guidelines for which design parameters are most sensitive and acceptable tolerances.

Liquid metal droplets are accelerated by electrostatic forces in a process known as field emission. In this study, we simulate the emission of charged indium droplets on a needle in 2D cylindrical coordinates. The boundary element method is used to rapidly and accurately calculate the electric field on the fluid surface, which is then advected forward in time using level sets. This is the first time these methods have been combined, and this combination addresses difficult detachable surface tracking issues successfully. A histogram of droplet charge-to-mass ratio is generated in which it is predicted that smaller satellite droplets are more densely charged. In addition, our model is compared with independent pre- and post-snap off data and produces good agreement with the result.

In this paper, based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nyström (SRKN) methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs, explicit multi-symplectic schemes are constructed and investigated, where the nonlinear wave equation is taken as a model problem. Numerical comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.

A novel numerical method, based on physical intuition, for particle-in-cell simulations of electromagnetic plasma microturbulence with fully kinetic ion and electron dynamics is presented. The method is based on the observation that, for low-frequency modes of interest [ω/ω_ci≪1, ω is the typical mode frequency and ω_ci is the ion cyclotron frequency] the impact of particles that have velocities larger than the resonant velocity, vr∼ω/kk (kk is the typical parallel wavenumber) is negligibly small (this is especially true for the electrons). Therefore it is natural to analytically segregate the electron response into an adiabatic response and a nonadiabatic response and to numerically resolve only the latter: this approach is termed the splitting scheme. However, the exact separation between adiabatic and nonadiabatic responses implies that a set of coupled, nonlinear elliptic equations has to be solved; in this paper an iterative technique based on the multigrid method is used to resolve the apparent numerical difficulty. It is shown that the splitting scheme allows for clean, noise-free simulations of electromagnetic drift waves and ion temperature gradient (ITG) modes. It is also shown that the advantage of noise-free kinetic simulations translates into better energy conservation properties.

The Brinkman equation is used to model the isothermal flow of the Newtonian fluids through highly permeable porous media. Due to the multiscale behaviour of this flow regime the standard Galerkin finite element schemes for the Brinkman equation require excessive mesh refinement at least in the vicinity of domain walls to yield stable and accurate results. To avoid this, a multiscale finite element method is developed using bubble functions. It is shown that by using bubble enriched shape functions the standard Galerkin method can generate stable solutions without excessive near wall mesh refinements. In this paper the performances of different types of bubble functions are evaluated. These functions are used in conjunction with bilinear Lagrangian elements to solve the Brinkman equation via a penalty finite element scheme.

In this paper, a divergence-free and pressure-oscillation-free projection method for solving the incompressible Navier-Stokes equations on the non-staggered grid is presented. The exact discrete projection method is used to compute the velocity field, which ensures the discrete divergence of the velocity field is zero. In order to eliminate the odd-even decoupling in the pressure field, a filtering procedure is proposed and applied to the pressure field. We have shown this filter recovers the grid scale ellipticity in the pressure field and the odd-even decoupling can be removed effectively. The proposed numerical scheme is further verified through numerical experiments.

A new numerical scheme to solve the Boltzmann equation in phase space for rarefied gas is described on the basis of the Cubic Interpolated Propagation (CIP) method. The CIP procedure is extended to adaptive unstructured grid system by the Soroban grid. The grid points in velocity space can move dynamically following the spread of velocity space in a spatially localized manner. Such adaptively moving points in velocity space are similar to the particle codes but can provide higher-order-accurate solutions. Numerical solutions obtained by the Soroban-grid CIP are examined and the validity is discussed. 

The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10¹² and more. The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion. It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary: (1) one-class-case algorithm for waves interacting through scales, and (2) two-class-case algorithm for waves interacting through phases. In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-class-case generic algorithm.

Ab initio calculations of dielectronic recombination (DR) processes from the ground state 1s² of He-like argon ion through doubly excited states 1s2snl, 1s2pnl(n=2 to 9) of Li-like argon ions are performed using the multi-configuration Hatree-Fock method with relativistic correction. The theoretical method and its corresponding computation will be outlined. For higher doubly excited states with n>9, the scaling law is used to extrapolate the Auger and radiative transition rates. The total and state-to-state cross sections with corresponding rate coefficients in the temperature from 10²eV to 10⁶eV are presented, as well as the DR strengths for all the separate resonances. Moreover, peculiarities of the DR from doubly excited 1s2s3l ' configurations are analyzed and the contributions of two-electron-one-photo (TEOP) radiative transitions to the DR cross sections are also investigated, such as 1s2s² → 1s²2p, due to the strong configuration interactions. Our theoretical results appear to be in excellent agreement with the previous and recent experimental measurements.

We consider the anisotropic uniaxial formulation of the perfectly matched layer (UPML) model for Maxwell's equations in the time domain. We present and analyze a mixed finite element method for the discretization of the UPML in the time domain to simulate wave propagation on unbounded domains in two dimensions. On rectangles the spatial discretization uses bilinear finite elements for the electric field and the lowest order Raviart-Thomas divergence conforming elements for the magnetic field. We use a centered finite difference method for the time discretization. We compare the finite element technique presented to the finite difference time domain method (FDTD) via a numerical reflection coefficient analysis. We derive the numerical reflection coefficient for the case of a semi-infinite PML layer to show consistency between the numerical and continuous models, and in the case of a finite PML to study the effects of terminating the absorbing layer. Finally, we demonstrate the effectiveness of the mixed finite element scheme for the UPML by a numerical example and provide comparisons with the split field PML discretized by the FDTD method. In conclusion, we observe that the mixed finite element scheme for the UPML model has absorbing properties that are comparable to the FDTD method.