In this paper I review three key topics in CFD that have kept researchers busy for half a century. First, the concept of upwind differencing, evident for 1-D linear advection. Second, its implementation for nonlinear systems in the form of high-resolution schemes, now regarded as classical. Third, its genuinely multidimensional implementation in the form of residual-distribution schemes, the most recent addition. This lecture focuses on historical developments; it is not intended as a technical review of methods, hence the lack of formulas and absence of figures.

In this paper, we introduce a three-dimensional numerical method for computing the wake behind a flat plate advancing perpendicular to the flow. Our numerical method is inspired by the panel method of J. Katz and A. Plotkin [J. Katz and A. Plotkin, Low-speed Aerodynamics, 2001] and the 2D vortex blob method of Krasny [R. Krasny, Lectures in Appl. Math., 28 (1991), pp. 385–402]. The accuracy of the method will be demonstrated by comparing the 3D computation at the center section of a very high aspect ratio plate with the corresponding two-dimensional computation. Furthermore, we compare the numerical results obtained by our 3D numerical method with the corresponding experimental results obtained recently by Ringuette [M. J. Ringuette, Ph.D. Thesis, 2004] in the towing tank. Our numerical results are shown to be in excellent agreement with the experimental results up to the so-called formation time.

The dyadic Green's function in multi-layer structures for Maxwell equations is a key component for the integral equation method, but time consuming to calculate. A novel algorithm, the Fast Interpolation and Filtering Algorithm (FIFA), for the calculation of the dyadic Green's function in multi-layer structures is proposed in this paper. We discuss in specific details, ready for use in practical calculations of scattering in layer media, how to apply FIFA to calculate various components of the dyadic Green's function. The algorithm is based on two techniques: interpolation of Green's function both in the spectral domain and spatial domain, and low pass filter window based acceleration. Compared to the popular Complex Image Method (CIM), FIFA provides the same speed and overcomes several difficulties associated with CIM while being more general and robust. Specifically, there are no limitations on the frequency range, the number of layers in the structure and the type of Green's functions to be calculated, and moreover, no need to extract surface wave poles from the spectral form of the Green's function. Numerical results are given to demonstrate the efficiency and robustness of the proposed method.

In this paper, we present solutions for the one-dimensional coupled nonlinear Schrödinger (CNLS) equations by the Constrained Interpolation Profile-Basis Set (CIP-BS) method. This method uses a simple polynomial basis set, by which physical quantities are approximated with their values and derivatives associated with grid points. Nonlinear operations on functions are carried out in the framework of differential algebra. Then, by introducing scalar products and requiring the residue to be orthogonal to the basis, the linear and nonlinear partial differential equations are reduced to ordinary differential equations for values and spatial derivatives. The method gives stable, less diffusive, and accurate results for the CNLS equations.

Mesh adaptation is studied from the mesh control point of view. Two principles, equidistribution and alignment, are obtained and found to be necessary and sufficient for a complete control of the size, shape, and orientation of mesh elements. A key component in these principles is the monitor function, a symmetric and positive definite matrix used for specifying the mesh information. A monitor function is defined based on interpolation error in a way with which an error bound is minimized on a mesh satisfying the equidistribution and alignment conditions. Algorithms for generating meshes satisfying the conditions are developed and two-dimensional numerical results are presented.

A new multi-dimensional scheme for the Maxwell equations is established by the CIP method in combination with the method of characteristics (CIP-MOC). In addition, the CIP-MOC can be extended to arbitrary grid system by the Soroban grid without losing the third-order accuracy. With the accuracy fixed, the grid points required for the CIP are 40 times less than the conventional schemes like the FDTD in three dimensions. Numerical solutions obtained by the CIP-MOC are compared with analytical solution and the FDTD in plane-wave scattering by a perfectly-conducting circular cylinder, and the CIP-MOC agrees very well with analytical solutions. The Soroban grid is also applied to the Vlasov equation that describes the kinematics of plasmas that is frequently combined with the Maxwell equation. The adaptively moving points in velocity space are similar to the particle codes but can provide accurate solutions.

Direct numerical simulation of vertical rotating open-channel flow with heat transfer has been carried out for the rotation number N_τ from 0 to 0.1, the Prandtl number 1, and the Reynolds number 180 based on the friction velocity of non-rotating flow and the height of the channel. The ob jective of this study is to reveal the effect of rotation on the characteristics of turbulent flow and heat transfer, in particular near the free surface and the wall of the open-channel. Statistical quantities, e.g., the mean velocity, temperature and their fluctuations, turbulent heat fluxes, and turbulence structures, are analyzed. The depth of surface-influenced layer decreases with the increase of the rotation rate. In the free surface-influenced layer, the turbulence and thermal statistics are suppressed due to the effect of rotation. In the wall-influenced region, two typical rotation regimes are identified. In the weak rotation regime with 0 < N_τ < 0.06 approximately, the turbulence and thermal statistics correlated with the spanwise velocity fluctuation are enhanced since the shear rate of spanwise mean flow induced by Coriolis force increases; however, the other statistics are suppressed. In the strong rotation regime with Nτ > 0.06, the turbulence and thermal statistics are suppressed significantly because the Coriolis force effect plays a dominant role in the rotating flow. To elucidate the effect of rotation on turbulent flow and heat transfer, the budget terms in the transport equations of Reynolds stresses and turbulent heat fluxes are investigated. Remarkable change of the direction of streak structures based on the velocity and temperature fluctuations is discussed.

The electromagnetic backscattering of a crosscut of a cruise missile coated by a thin homogeneous layer made of radar absorbent material is modeled using a finite element method. Based on the radar cross section and a reflection coefficient, optimization problems are formulated for evaders and interrogators leading to optimal material parameters for the coating and optimal monostatic radar operating frequencies, respectively. Optimal coating materials are constructed for several radar frequencies. Tuning only dielectric permittivity gives a narrow frequency range of high absorption while also tuning magnetic permeability widens it significantly. However the coating layers considered do not provide substantial reduction of backscattering in the entire frequency range from 0.2 to 1.6 GHz. Computational experiments also demonstrate that the reflection coefficient based on a simple planar geometry can predict well the strength of radar cross section in the sector of interest with a substantially reduced computational burden.