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The paper analyses the preconditioning of non‐linear nonsymmetric equations with approximations of the discrete Laplace operator. The test problems are non‐linear 2‐D elliptic equations that describe natural convection, Darcy flow, in a porous medium. The standard second order accurate finite difference scheme is used to discretize the models’ equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the preconditioned generalised conjugate gradient (PGCG) methods as inner iteration. Three PGCG algorithms, CGN, CGS and GMRES, are tested. The preconditioning with discrete Laplace operator approximations consists of replacing the solving of the equation with the preconditioner by a few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky (IC) and multigrid (MG). The numerical results show that MG preconditioning leads to mesh independence. CGS is the most robust algorithm but its efficiency is lower than that of GMRES.

The purpose of this paper is to demonstrate new properties of continuous‐ and discrete‐time dynamical systems.

First, definitions of two types of spatial symmetry are introduced. These are used as definitions, which, along with existing knowledge show that it is possible to identify properties of dynamical systems that were previously unknown.

The main result of the paper is a new theorem regarding new properties of continuous‐ and discrete‐time dynamical systems.

The present study provides a starting point for further research on the differences between continuous‐ and discrete‐time dynamical systems. This work builds on the definition of spatial symmetry.

The theorem proved in this paper and the new properties of dynamical systems can be used to introduce new methods of approximating continuous‐time dynamical systems by discrete‐time dynamical systems and vice versa. Such approaches can also be helpful in constructing chaotic sources to model noise.

This paper offers contributions to the broader discussion of differences between continuous‐ and discrete‐time dynamical systems. In particular, the paper supports the statement that many discrete‐time processes cannot be embedded into continuous ones.

Aircraft pitch control is fundamental for the performance of micro aerial vehicles (MAVs). The purpose of this paper is to establish a simple experimental procedure to calibrate pitch instrumentation and classical control algorithms. This includes developing an efficient pitch angle observer with optimal estimation and evaluating controllers under uncertainty and external disturbances.

A wind tunnel test bench is designed to simulate fixed-wing aircraft dynamics. Key elements of the instrumentation commonly found in MAVs are characterized in a gyroscopic test bench. A data fusion algorithm is calibrated to match the gyroscopic test bench measurements and is then integrated into the autopilot platform. The elevator-angle to pitch-angle dynamic model is obtained experimentally. Two different control algorithms, based on model-free and model-based approaches, are designed. These controllers are analyzed in terms of parametric uncertainties due to wind speed variations and external perturbation because of sudden weight distribution changes. A series of experimental tests is performed in wind-tunnel facilities to highlight the main features of each control approach.

With regard to the instrumentation algorithms, a simple experimental methodology for the design of optimal pitch angle observer is presented and validated experimentally. In the context of the platform design and identification, the similitude among the theoretical and experimental responses shows that the platform is suitable for typical pitch control assessment. The wind tunnel experiments show that a fixed linear controller, designed using classical frequency domain concepts, is able to provide adequate responses in scenarios that approximate the operation of MAVs.

The aircraft orientation observer can be used for both pitch and roll angles. However, for simultaneousyaw angle estimation the proposed design method requires further research. The model analysis considers a wind speed range of 6-18 m/s, with a nominal operation of 12 m/s. The maximum experimentally tested reference for the pitch angle controller was 20°. Further operating conditions may require more complex control approaches (e.g. scheduling, non-linear, etc.). However, this operating range is enough for typical MAV missions.

The study shows the design of an effective pitch angle observer, based on a simple experimental approach, which achieved locally optimum estimates at the test conditions. Additionally, the instrumentation and design of a test bench for typical pitch control assessment in wind tunnel facilities is presented. Finally, the study presents the development of a simple controller that provides adequate responses in scenarios that approximate the operation of MAVs, including perturbations that resemble package delivery and parametric uncertainty due to wind speed variations.

The authors employ single‐factor models to estimate daily temperature variations for the valuation of weather derivatives. Classical financial models are adapted to fit temperature seasonality to a time series. As an example, Monte Carlo simulations of heating and cooling degree‐days are used as the underlying for weather derivatives that reference temperatures in regions of Spain. The article also discusses potential applications to hedging energy‐related risks.

When establishing a mathematical model to simulate solid mechanics, considering realistic geometries, special tools are needed to translate measured data, possibly with noise, into idealized geometrical entities. As an engineering application, wheel-rail contact interactions are fundamental in the dynamic modeling of railway vehicles. Many approaches used to solve the contact problem require a continuous parametric description of the geometries involved. However, measured wheel and rail profiles are often available as sets of discrete points. A reconstruction method is needed to transform discrete data into a continuous geometry.

The authors present an approximation method based on optimization to solve the problem of fitting a set of points with an arc spline. It consists of an initial guess based on a curvature function estimated from the data, followed by a least-squares optimization to improve the approximation. The authors also present a segmentation scheme that allows the method to increment the number of segments of the spline, trying to keep it at a minimal value, to satisfy a given error tolerance.

The paper provides a better understanding of arc splines and how they can be deformed. Examples with parametric curves and slightly noisy data from realistic wheel and rail profiles show that the approach is successful.

The developed methods have theoretical value. Furthermore, they have practical value since the approximation approach is better suited to deal with the reconstruction of wheel/rail profiles than interpolation, which most methods use to some degree.

This paper seeks to review the literature on methods for solving the radiative transfer equation (RTE) and integrating the radiant energy quantities over the spectrum required to predict the flow, the flame and the thermal structures in chemically reacting and radiating combustion systems.

The focus is on methods that are fast and compatible with the numerical algorithms for solving the transport equations using the computational fluid dynamics techniques. In the methods discussed, the interaction of turbulence and radiation is ignored.

The overview is limited to four methods (differential approximation, discrete ordinates, discrete transfer, and finite volume) for predicting radiative transfer in multidimensional geometries that meet the desired requirements. Greater detail in the radiative transfer model is required to predict the local flame structure and transport quantities than the global (total) radiation heat transfer rate at the walls of the combustion chamber.

The RTE solution methods and integration of radiant energy quantities over the spectrum are assessed for combustion systems containing only the infra‐red radiating gases and gas particle mixtures. For strongly radiating (i.e. highly sooting) and turbulent flows the neglect of turbulence/radiation interaction may not be justified.

Methods of choice for solving the RTE and obtaining total radiant energy quantities for practical combustion devices are discussed.

The paper has identified relevant references that describe methods capable of accounting for radiative transfer to simulate processes arising in combustion systems.

In this study, we assume that stock prices follow piecewise geometric Brownian motion, a variant of geometric Brownian motion except the ex-dividend date, and find pricing formulas of American call options. While piecewise geometric Brownian motion can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our work differs from other studies using the same assumption in at least three respects. First, we investigate the analytical approximations of American call options and examine European call options as a special case, while most analytical approximations in the literature cover only European options. Second, we provide both the upper and the lower bounds of option prices. Third, our solutions are equal to the exact price when the size of the dividend is proportional to the stock price, while binomial tree results never match the exact option price in any circumstance. The numerical analysis therefore demonstrates the efficiency of our method. Especially, the lower bound formula is accurate, and it can be further improved by considering second order approximations although it requires more computing time.

The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.

The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.

Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the study of the time‐space structure of the numerical solutions.

The approach can be used for the numerical simulation of shallow‐water flows in domains of both Cartesian and spherical geometries, providing the solution adequate from the physical and mathematical standpoints in the sense of keeping its mass and total energy constant even when fully discrete shallow‐water models are applied.

A new algorithm is presented for the discretization of semiconductor models in one space dimension plus time. A complete error analysis is given, showing that the discretization errors do not depend on any derivatives of ill‐behaved quantities such as carrier densities. In this algorithm, the electrostatic potential is updated from a discretization of the equation of total current continuity, and parabolic equations for the current densities are discretized, rather than those for the carrier densities. Projection methods, e.g. simple finite‐element methods, are used for the space discretization. The equations for the current densities are similar to the familiar Scharfetter‐Gummel expressions in the stationary limit. However, the discrete time‐dependent current densities are required here to be H1 functions of x, obtained in a space with at least second order approximation property in L2. This method is fully compatible with recently developed methods for uncoupling the discrete systems to be solved at each time step, for an individual device or when a given problem involves multiple, coupled devices.