Low-order Equations Research Articles

Design and Implementation of an Simulation of Water Wave Transformation Using Higher Order Mild-Slope Equation

A higher-order mild-slope equation (HOMSE) is derived based on theory of Hsu et al. [14]. Depth function is approximated up to third-order accurate in terms of both wave nonlinearity and bottom slope. Classical Galerkin method is used to solve HOMSE. Developed model is verified with a series of benchmark tests, including propagation of a sinusoidal wave past a submerged bar, wave propagating on a sloping bed, wave propagating over an elliptic shoal on a uniform slope, and wave propagating through a semicircular slope bottom, respectively. Computed results are compared with experiment data and prediction of low-order mild-slope equation model as well as Boussinesq equations model, and show good agreement.

Görtler instability of the axisymmetric stagnation-point flow along a concave wall

Axisymmetric stagnation-point flow, if it is along a concave wall, is susceptible to the Görtler instability, which is governed by partial differential equations with respect to both the normal-to-wall coordinate and the local Reynolds number. A series solution of the exact disturbance equations is obtained in the form expanded into inverse powers of the Reynolds number. The lowest-order equations concerning the leading terms of the series construct a non-parallel version of Görtler's eigenvalue problem and present a neutral stability curve with the critical point in a finite range of wave numbers for disturbances of the steady longitudinal-vortex type. Higher-order terms of the series solution indicate the dependence of the Görtler instability on the local Reynolds number and suggest a quantitative limit of validity of the Görtler approximation based on the lowest-order equation system alone. In the case of the concave wall that slightly rotates around the axis of symmetry, the instability induces oblique traveling waves instead of steady longitudinal vortices.

An integral perturbation model of flow and momentum transport in rotating microchannels with smooth or microstructured wall surfaces

This paper summarizes the development of an integral perturbation solution of the equations governing flow momentum transport and energy conversion in microchannels between disks of multiple-disk drag turbines such as Tesla turbines. Analysis of this type of flow problem is a key element in optimal design of Tesla drag-type turbines for geothermal or solar alternative energy technologies. In multiple-disk turbines, high speed flow enters tangentially at the outer radius of cylindrical microchannels formed by closely spaced parallel disks, spiraling through the channel to an exhaust at a small radius, or at the center of the disk. Previous investigations have generally developed models based on simplifying idealizations of the flow in these circumstances. Here, beginning with the momentum and continuity equations for incompressible and steady flow in cylindrical coordinates, an integral solution scheme is developed that leads to a dimensionless perturbation series solution that retains the full complement of momentum and viscous effects to consistent levels of approximation in the series solution. This more rigorous approach indicates all dimensionless parameters that affect flow and transport and allows a direct assessment of the relative importance of viscous, pressure, and momentum effects in different directions in the flow. The resulting lowest-order equations are solved explicitly and higher order terms in the series solutions are determined numerically. Enhancement of rotor drag in this type of turbine enhances energy conversion efficiency. We also developed a modified version of the integral perturbation analysis that incorporates the effects of enhanced drag due to surface microstructuring. Results of the model analysis for smooth disk walls are shown to agree well with experimental performance data for a prototype Tesla turbine and predictions of performance models developed in earlier investigations. Model predictions indicate that enhancement of disk drag by strategic microstructuring of the disk surfaces can significantly increase turbine efficiency. Exploratory calculations with the model indicate that turbine efficiencies exceeding 75% can be achieved by designing for optimal ranges of the governing dimensionless parameters.

High-Order Spectral Filter for the Spherical-Surface Limited Area

Abstract A high-order spectral filter for the spherical-surface limited-area domain, either window or sector type, is presented, where the window domain is finite both in longitude and latitude and the sector domain is finite in longitude, but is ranged from Pole to Pole in latitude. The data given in the physical domain are extended to either extended window or sector domain by padding artificial data that are appropriate for spectral decomposition with half-ranged Fourier series. The high-order filter equation of Laplacian operator type was split into first- or second-order spherical elliptic equations as in the global domain high-order spectral filter. Each low-order elliptic equation is discretized using half-ranged Fourier series both in longitudinal and latitudinal direction. Since the domain is of the spherical geometry, the window domain spectral filter consists of full matrices for each zonal wavenumber and thus performs filtering with O(N3) operation for N × N grids. On the other hand, the sector domain filter performs with efficient O(N2 logN) operation for the same grids, because the matrices can be constructed in tridiagonal form. The error of the filter was analyzed using the eigenfunctions of the filter equation to get insights on data-extension parameters. The limited-area domain filters were tested with localized analytic function and the midlatitude observed data of meteorological variables, and the results were compared to those of the global domain filter. Accuracies of window and sector domain filters were found almost the same, and they turned out to vary with the filter parameters and the location and size of the domain. Application of the limited domain filters to the tropical cyclone initialization scheme showed very close performances to each other for the track and intensity forecasts.

Nonlinear Weighted Flux Methods for Particle Transport Problems in Two-Dimensional Cartesian Geometry

A family of nonlinear weighted flux (NWF) methods for solving the transport equation in two-dimensional (2-D) Cartesian geometry is considered. The low-order equations of these methods are defined by means of special linear-fractional factors that are determined by the high-order transport solution. An asymptotic diffusion limit analysis is performed on methods with a general weight function. The analysis revealed conditions on the weight necessary for an accurate approximation of the diffusion equation in this limit. We study methods with weights defined by linear and bilinear functions of directional cosines. As a result, we developed 2-D NWF methods formulated with the low-order equations that give rise to the diffusion equation in optically thick diffusive regions if their factors are calculated by means of the leading-order transport solution. The inherent asymptotic boundary conditions for the NWF methods are analyzed. Numerical results are presented to confirm theoretical results and demonstrate performance of the proposed methods.

Explicit and implicit finite difference schemes for fractional Cattaneo equation

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor–corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor–corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.

Fourier Analysis of Inexact Parallel Block-Jacobi Splitting with Transport Synthetic Acceleration

A Fourier analysis is conducted for the discrete ordinates, or SN, approximation of the neutron transport problem solved with Richardson iteration (source iteration) and Richardson iteration preconditioned with transport synthetic acceleration (TSA), using the inexact parallel block-Jacobi (IPBJ) algorithm both in slab and two-dimensional Cartesian geometry. Both traditional, or “beta,” TSA (TTSA) and a modified TSA (MTSA), in which only the scattering in the low-order equations is reduced by some nonnegative factor β < 1, are considered.The results for the unaccelerated algorithm show that convergence of IPBJ can degrade, leading in particular to stagnation of the generalized minimum residual method with restart parameter m, GMRES(m), in problems containing optically thin subdomains. The IPBJ algorithm preconditioned with TTSA can be effective, provided the β parameter is properly tuned for a given scattering ratio c, but is potentially unstable. Compared to TTSA, MTSA is less sensitive to the choice of β, more effective for the same computational effort, measured in terms of the effective scattering ratio c′, and it is unconditionally stable.

A Multiderivative Collocation Method for 5th Order Ordinary Differential Equations

Problem statement: The conventional methods of solving higher order differential equations have been by reducing them to systems of first order equations. This approach is cumbersome and increases computational time. Approach: To address this problem, a numerical algorithm for direct solution of 5th order initial value problems in ordinary differential equations (odes), using power series as basis function, is proposed in this research. Collocation of the differential system is taken at selected grid points to reduce the number of functions to be evaluated per iteration. A number of predictors and their derivatives having the same order of accuracy with the main method are proposed. Results: The approach yields a multiderivative method of order six. Numerical examples solved show increased efficiency of the method with increased number of iterations, converging to the theoretical solutions. Conclusion/Recommendations: The new mutiderivative method is efficient to solve linear and nonlinear fifth order odes without reduction to system of lower order equations.

Oscillations of a higher order nonlinear differential equations with impulses

AbstractImpulsive differential equations are important in simulation of processes with jump conditions. Although there are quite a few studies on the oscillation properties of low order equations, there are not too many studies of higher order equations. In this paper, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.

온도변화에 강인한 EPB 시스템의 퍼지모델 기반 고장검출 방법

In this paper, a robust fault detection method for varying temperature based on fuzzy model is proposed. To develop a robust force estimation model, it needs temperature information because the output of force sensor is affected by a temperature variation. The nonlinear dynamic system, such as the parking force of the EPB (Electronic Parking Brake) system is necessary to have a higher order equation model. But, because of the calculation time, the higher order equation model is hard to be used in real application. In case of the lower order equation model, the result is not as accurate as acceptable. To solve this problem, the robust fuzzy model-based fault detection is developed. A proposed fault detection method for varying temperature is verified by HILS (hardware in the loop simulation).

A new method in highway route design: joining circular arcs by a single C-Bézier curve with shape parameter

We constructed a single C-Bezier curve with a shape parameter for G2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the Bezier curves, has no curvature extrema, while a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms were presented in detail, strict mathematical proofs were given, and the effectiveness of the method was shown by examples. This method has the following three advantages: (1) the pattern is unified; (2) the parameter able to adjust the shape of the transition curve is available; (3) the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. These advantages make the method simple and easy to implement.

A new multiscale scheme for computing statistical moments in single phase flow in heterogeneous porous media

In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order

We consider systems of linear differential-algebraic equations of order ≥ 2 and present conditions under which there exist nonsingular transformations splitting the system into systems of linear ordinary differential equations of lower order and, possibly, an algebraic part.

New simple algebraic root locus method for design of feedback control systems

New concept of algebraic characteristic equation decomposition method is presented to simplify the design of closed-loop systems for practical applications. The method consists of two decompositions. The first one, decomposition of the characteristic equation into two lower order equations, was performed in order to simplify the analysis and design of closed loop systems. The second is the decomposition of Laplace variable, s, into two variables, damping coefficient, ?, and natural frequency, ? n. Those two decompositions reduce the design of any order feedback systems to setting of two complex dominant poles in the desired position. In the paper, we derived explicit equations for six cases: first, second and third order system with P and PI. We got the analytical solutions for the case of fourth and fifth order characteristic equations with the P and PI controller; one may obtain a complete analytical solution of controller gain as a function of the desired damping coefficient. The complete derivation is given for the third order equation with P and PI controller. We can extend the number of specified poles to the highest order of the characteristic equation working in a similar way, so we can specify the position of each pole. The concept is similar to the root locus but root locus is implicit, which makes it more complicated and this is simpler explicit root locus. Standard procedures, root locus and Bode diagrams or Nichol Charts, are neither algebraic nor explicit. We basically change controller parameters and observe the change of some function until we get the desired specifications. The derived method has three important advantage over the standard procedures. It is general, algebraic and explicit. Those are the best poles design results possible; it is not possible to get better controller design results.

Innovative method for modelling tree structured interconnects

A new technique to generate macromodels of tree structured interconnects including the effect of inductances and frequency dependent per unit length parameters is presented. The proposed method allows the derivation of the exact transfer function of an interconnect tree with distributed RLC models. It presents some advantages over other techniques: first, it enables to easily incorporate frequency dependent phenomena such as skin effect and dielectric losses in a macromodel of the entire interconnect tree; second, a model order reduction can be efficiently performed at the level of each single interconnect, thus leading to a reduced-order model of the entire interconnect; third, the knowledge of all the poles of the tree is made possible by solving low-order algebraic equations at a reduced computational cost. The examples confirm that the proposed method is able to capture the physics of the interconnect trees, being accurate and efficient at the same time.

Boundary value problems for the inhomogeneous polyanalytic equation II

SummaryFurther mixed boundary value problems are studied for the inhomogeneous polyanalytic equation in the unit disc. As in part I, see [4], the boundary conditions are some combinations of Schwarz, Dirichlet and Neumann conditions. The method applied is based on an iteration process leading from lower order equations to higher ones, see [1–3]. The method can be used to solve all kind of such mixed problems also for other kinds of complex model equations.

Homogenization Method for the Two-Dimensional Low-Order Quasi-Diffusion Equations for Reactor Core Calculations

In this paper, we develop a homogenization methodology for the two-dimensional low-order quasi-diffusion equations for full-core reactor calculations that is based on a family of spatially consistent coarse-mesh discretization methods. The coarse-mesh solution generated by these methods preserves a number of spatial moments of the fine-mesh transport solution over each assembly. The proposed method reproduces accurately the complicated large-scale behavior of the transport solution within assemblies. To demonstrate the performance of the developed methodology, we present the numerical results of several test problems that simulate mixed-oxide-uranium and assembly-reflector interfacial effects.

Accurate higher-order analytical approximate solutions to nonconservative nonlinear oscillators and application to van der Pol damped oscillators

In general, this paper deals with general nonlinear oscillations of a nonconservative and single degree-of-freedom system with odd nonlinearity and, in particular, it presents accurate higher-order analytical approximate solutions to van der Pol damped nonlinear oscillators having odd nonlinearity and the Rayleigh equation. By combining the linearization of the governing equation with harmonic balancing and the method of averaging, we establish accurate analytical approximate solutions for the general weakly damped nonlinear systems. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. The combination of these two methods results in very accurate transient response of the periodic solution. In addition and for the first time, this paper also presents a method for deducing fourth-, fifth- and higher-order linearized governing equations from the lower-order equations without the requirement of formulating the problem from the first principle. Three examples including the van der Pol damped nonlinear oscillator are presented to illustrate the excellent agreement with approximate solution using the exact frequency.

Two-dimensional equations for electromagnetic waves in multi-layered thin dielectric films

Two-dimensional equations for electromagnetic fields in a multi-layered thin dielectric film are derived from the three-dimensional equations of electrodynamics by expanding the vector potential of the electromagnetic fields into trigonometric series expansions of the film thickness coordinate. The lower order equations are examined. It is shown that they can describe certain long waves in the film. The equations are useful for modeling thin film devices.