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In this paper, the authors introduce a new Weighted Essentially Nonoscillatory (WENO) scheme. This scheme is founded on exponential functions and utilizes arc-length smoothness indicators. The primary purpose of this WENO scheme is to provide accurate approximations for the viscosity numerical solutions of Hamilton–Jacobi equations. The arc-length smoothness indicators are derived from the derivatives of reconstructed polynomials within each sub-stencil. These smoothness indicators play a crucial role in approximating the viscosity numerical solutions of Hamilton–Jacobi equations, ensuring high-resolution results and minimizing absolute truncation errors. Numerous numerical tests have been carried out and presented to demonstrate the performance capabilities and numerical accuracy of the proposed scheme, comparing it to several traditional WENO schemes. [ABSTRACT FROM AUTHOR]

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab's LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems. [ABSTRACT FROM AUTHOR]

This paper provides a qualitative and quantitative study of a second-order Singularly Perturbed Reaction–Diffusion type System of Integro-differential equations with discontinuous source term. To obtain the numerical solution of the problem, an exponentially-fitted method that can be applied to a Shishkin mesh. This method shows that uniform convergence with respect to the perturbation parameter and necessary examples are given. [ABSTRACT FROM AUTHOR]

Boussinesq equations are used as mathematical models for investigating the behaviour of small-amplitude long waves on water surfaces. In this paper, we focus on the logarithmic Boussinesq equation. We first generalize the model to consider different model coefficients. Secondly, we provide analytical and numerical solutions to the direct problem. Thirdly, we applied the modulating function method to solve coefficients inverse problems in the generalized logarithmic Boussinesq equation. Finally, numerical simulations are given to show the stability, accuracy, and effectiveness of the proposed methods for solving the posed direct and inverse problems. [ABSTRACT FROM AUTHOR]

This paper presents an approach for the time domain statistical simulation of field-to-wire coupling model with uncertain wire position in traverse direction. The parameters of governing equations of field-to-wire coupling model about the wire position are expanded to the augmented governing equations by using the generalized polynomial chaos technique with normalized orthogonal basis functions. And the coefficients of the polynomial chaos expansions are solved by combining with the stochastic Galerkin method. However, the solution of augmented governing equations is a complex problem. Therefore, a new finite-difference time-domain (FDTD) algorithm based on the implicit Wendroff difference format is proposed to solve the governing equations, and the general iterative equations are given in this paper. The proposed methods are verified by a field-to-wire coupling model. The effectiveness and advantages of the new FDTD algorithm and the uncertainty analysis method based on generalized polynomial chaos technique are verified, which are achieved by comparing with the conventional leapfrog fashion FDTD algorithm and the Monte Carlo technique, respectively. [ABSTRACT FROM AUTHOR]

Peak‐valley period partition of load curve is a key aspect of time‐of‐use (ToU) tariff to improve power load characteristics, such as shifting peak loads towards valley time periods. Fuzzy clustering algorithm is an effective and popular method commonly used to solve the peak‐valley period partition of load curves, but it still encounters the difficulty of dividing some data within the boundary regions of different time periods. Therefore, this paper presents a peak‐valley period partition and abnormal time correction scheme for ToU tariffs under typical daily load curves based on improved fuzzy C‐means (FCM) clustering algorithm. In order to improve the accuracy of peak‐valley period partition, modified fuzzy membership functions are proposed to improve the initialization of FCM clustering, and a loss function‐based method is presented for calculating the fuzzy parameters of those membership functions. To resolve the problem of abnormal time partitioning within the boundaries of different time periods, an abnormal time period recognition model and a correction model based on fuzzy subsethood are proposed to obtain the final corrected peak‐valley time period partitioning results. On the MATLAB R2020b platform, the effectiveness of the proposed method is verified through two real daily load curves with a time resolution of 5 min. [ABSTRACT FROM AUTHOR]

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used fifth-order pixels Laplace filters including the difference schemes used to derive them (finite difference method - FDM and finite element method - FEM). The results of the research concerning third-order pixels matrices of the convolution Laplace filters used for digital processing of images were presented in our previous paper: The mathematical characteristic of the Laplace contour filters used in digital image processing. The third order filters is presented byWinnicki et al. (2022). As previously, the authors focused on the mathematical properties of the Laplace filters: their transfer functions and modified differential equations (MDE). The relations between the transfer function for the differential Laplace operator and its difference operators are described and presented here in graphical form. The impact of the corner elements of the masks on the results is also discussed. A transfer function, is a function characterizing properties of the difference schemes applied to approximate differential operators. Since they are relations derived in both types of spaces (continuous and discrete), comparing them facilitates the assessment of the applied approximation method. [ABSTRACT FROM AUTHOR]

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used third-order 3>3 pixels Laplace contour filters including the difference schemes used to derive them. The authors focused on the mathematical properties of the Laplace filters. The basic reasons of the differences of the properties were studied and indicated using their transfer functions and modified differential equations. The relations between the transfer function for the differential Laplace operator and its difference operators were described and presented graphically. The impact of the corner elements of the masks on the results was discussed. This is a theoretical work. The basic research conducted here refers to a few practical examples which are illustrations of the derived conclusions. We are aware that unambiguous and even categorical final statements as well as indication of areas of the results application always require numerous experiments and frequent dissemination of the results. Therefore, we present only a concise procedure of determination of the mathematical properties of the Laplace contour filters matrices. In the next paper we shall present the spectral characteristic of the fifth order filters of the Laplace type. [ABSTRACT FROM AUTHOR]

In this paper, we use the well-known Du Fort-Frankel finite difference scheme to solve the oxygen diffusion problem inside one cell which is modeled by an initial moving boundary value problem for one dimensional time-dependent partial differential equation. The main problem consists in tracking the moving boundary that represents the oxygen penetration depth inside the cell. We explore the possibilities of numerical approximation of the problem posed by the different formulations. Some numerical experiments are also provided with comparisons with analytical solution. The theoretical analysis is given for the numerical scheme. It is shown that all the results obtained by this method are compared with earlier authors. [ABSTRACT FROM AUTHOR]

This paper discusses the back-flashover (BFO) phenomena in a high-voltage (HV) vertical double-circuit transmission line by means of numerical simulations. The single- and multiphase BFOs are analyzed considering 24 cases of AC-voltage angles and nonlinear characteristic of flashovers, taking advantage of the circuit analysis method in time domain. The models used in circuit analysis are the TEM-delay model, which can take into account the non-TEM characteristics of tower and line, and the conventional models. Prior to the BFO analysis, the characteristics of circuit models are compared and discussed with the numerical electromagnetic analysis results. In BFO analysis, it is clarified that the occurrence probability of the multiphase BFO is heavily depending on the bias AC-voltages. The relation of measured BFO phases and AC-voltages are well explained by the TEM-delay model with the valid lightning current magnitude. The conventional circuit models could have underestimated the occurrence of BFOs in HV transmission lines. The analysis and discussions shown in this paper can be utilized for advanced evaluation of lightning performance of transmission lines that considers the seriousness of lightning accidents. [ABSTRACT FROM AUTHOR]

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme's error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes. [ABSTRACT FROM AUTHOR]

This paper concentrates on the error estimation of novel time second‐order splitting conservative finite difference method (FDM) for high‐dimensional nonlinear fractional Schrödinger equation rigorously. The discrete preservation property of our scheme is exhibited. By virtue of the cut‐off technique and discrete energy method, it is shown that our scheme possesses the accuracy of O(Δt2+hx2+hy2) in sense of L2‐ norm. Numerical experiments are exhibited to validate the accuracy and conservation property of our scheme. [ABSTRACT FROM AUTHOR]

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values. [ABSTRACT FROM AUTHOR]

Intelligent metasurfaces to create smart electromagnetic environments have attracted a surge of interest. The multiscale property of metasurfaces results in a huge computational challenge, further hindering their engineering development. Our previous paper has developed a fast finite difference time domain (FDTD) algorithm for patchlike metasurfaces analysis using the surface susceptibility model (SSM). This study extends that to efficient fishnetlike metasurfaces analysis using the surface porosity model (SPM), a useful complement to SSM. We introduce the SPM to FDTD, replacing brute force simulations, to accelerate fishnetlike metasurfaces analysis. At last, we conduct a series of numerical experiments, demonstrating that the proposed FDTD-SPM algorithm greatly reduces the number of computational meshing lattices and the consumed time. [ABSTRACT FROM AUTHOR]

In this paper, the validity of two extended transmission line approaches proposed for calculating the ground-return impedance and admittance of an underground insulated cable is evaluated taking as reference a full-wave approach based on the 3-D finite-difference time-domain (FDTD) method. The transient responses of an underground cable with realistic radius are calculated for a wide range of soil resistivities and cable lengths. It is shown that the extended transmission line approaches lead to transient waveforms in good agreement with the FDTD method even for relatively short cables and high soil resistivity. The limitations of a traditional formulation that neglects the influence of the ground-return admittance on the calculation of the cable parameters are also demonstrated. Overall, the results confirm the accuracy of the newly-proposed expressions for the calculation of underground cable parameters. [ABSTRACT FROM AUTHOR]

A new technique has been recently introduced to define finite difference schemes that preserve local conservation laws. So far, this approach has been applied to find parametric families of numerical methods with polynomial conservation laws. This paper extends the existing approach to preserve non polynomial conservation laws. Although the approach is general, the treatment of the nonlinear terms depends on the problem at hand. New parameter depending families of conservative schemes are here introduced for the sine–Gordon equation and a magma equation. Optimal methods in each family are identified by finding values of the parameters that minimize a defect-based approximation of the local error in the time discretization. • An approach to preserve non polynomial local conservation laws is proposed. • Parametric schemes are found for the sine–Gordon equation and a magma equation. • Optimal methods are identified by minimizing an estimate of the local error in time. • Numerical experiments are proposed to test accuracy and efficiency of the new schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the convergence rates of the finite difference methods (FDMs) for solving the PDEs with spatial delays which arise in the option pricing under regime switching hard-to-borrow models. The PDEs are coupled for different regime states and involve delays in two spatial directions. One of the boundary conditions is implicitly given by an initial-boundary value problem of coupled PDEs which needs to be solved before solving the main equations. This paper proves convergence rates of the FDM based on mesh-dependent expansions for solving the problems. Numerical examples confirm the theory. [ABSTRACT FROM AUTHOR]

Soil conductivity and permittivity have a substantial influence on lightning-electromagnetic transients in power system. In this paper, lightning-induced voltages are computed on a multi-conductor overhead line due to nearby first and subsequent return strokes. Besides, this study considers the longitudinal complex inductance of the line due to the penetration depth of electromagnetic fields through finitely conducting ground as well as the frequency dependent soil parameters using the finite-difference time-domain method. Various case studies in light of variations of return stroke velocity, line height, and soil conductivity are explored for thoughtful investigations. The obtained results reveal that the influence of this complex inductance on peak values of lightning induced voltages changes with varying the return stroke velocity. Also, the frequency dependence of soil parameters markedly impacts the computed lightning-induced voltages particularly for poor soil conductivity. [ABSTRACT FROM AUTHOR]

The accurate assessment of lightning-induced over-voltages is essential for proper insulation coordination studies. In this paper, a novel machine learning based Gaussian process regression model is developed for the prediction of lightning-induced over-voltages considering wide range of ground resistivity and permittivity. The lightning-induced over-voltages have been computed using two-dimensional finite-difference time-domain technique associated with Agrawal field to line coupling model. To account for the input space variability, seven predictor variables namely: resistivity, permittivity, return stroke velocity, distance from flashpoint, return stroke peak current, front-time, and height of an overhead line are considered to determine the response variable, that is, lightning-induced over-voltages. Accordingly, the training of the Gaussian process regression model is carried out using lightning-induced over-voltages obtained from two-dimensional finite-difference time-domain technique for both first and the subsequent strokes. The estimation accuracy of model is appraised using root mean squared error indicating that the exponential kernel function has the least error than other kernel types for describing the covariance. The predictive performance is evaluated by corroborating the predictions from the model against the random test cases generated from two-dimensional finite-difference time-domain technique. The predicted results showed a good agreement with those obtained from finite-difference time-domain technique. [ABSTRACT FROM AUTHOR]

This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes of order two and four that preserve three conservation laws. We show that the new technique is practicable for PDEs with three dependent variables, introducing as an example new families of second-order schemes for the potential Kadomtsev–Petviashvili equation. [ABSTRACT FROM AUTHOR]

This paper studies an implicit-explicit (IMEX) finite difference scheme for solving a system of moving boundary partial integro-differential equations (PIDEs) which arises in Asian option pricing under regime-switching jump-diffusion models. First, the moving boundary PIDEs are recast into a fixed boundary problem of the PIDEs. Then the IMEX scheme is proposed to solve the problem and the second-order convergence rates are proved. Numerical examples are carried out to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In medium-voltage distribution lines, lightning outages associated with indirect strokes are less frequent when surge arresters are installed at close intervals. In this case, the lightning performance is mainly determined by the line response due to direct strikes, usually assessed disregarding the effect of the lightning electromagnetic pulse (LEMP). This paper examines the influence of such an electromagnetic effect on the voltages across the line insulators caused by direct lightnings. The presence of both surge arresters and a shield wire is considered. By means of adequate modeling and numerical calculations, we show that the flashover rate due to the direct strikes is increased by the LEMP. The obtained results indicate that in the statistical assessment of the direct lightning performance of lines equipped by surge arresters and shield wires, the LEMP effect is not negligible and its accurate evaluation is hence required. [ABSTRACT FROM AUTHOR]

This paper concerns with solving nonlinear boundary value problems with Dirichlet boundary conditions by a novel approximation technique based on Bernstein polynomials and meta-heuristic algorithms. A trial solution is expressed as a weighted sum of Bernstein polynomials of degree n . Some weights of this trial solution are determined by enforcing exact satisfaction of the Dirichlet boundary conditions. The remaining weights are determined such that they minimize the sum of squares of the residuals of the differential equation computed at arbitrary set of interior points in the domain of the problem. Error bounds for the approximate solutions for ordinary and partial differential equations are derived. The resulting optimization problem is solved using meta-heuristic algorithms, namely particle swarm optimization (PSO), genetic algorithm (GA), and a hybrid PSO–GA algorithm. The accuracy of the proposed approach is demonstrated by solving some nonlinear boundary value problems including Bratu problems in one- and two-dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we deal with singularly perturbed Fredholm integro differential equation (SPFIDE) with mixed boundary conditions. By using interpolating quadrature rules and exponential basis function, fitted second order difference scheme has been constructed on a Shishkin mesh. The stability and convergence of the difference scheme have been analyzed in the discrete maximum norm. Some numerical examples have been solved and numerical outcomes are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the effect of overhead shielding wires on the lightning-induced voltages of two typical multiconductor lines configurations at distances from 50 to 1000 m from the lightning channel, considering perfect and lossy ground, respectively, by using the 2-D finite-difference time-domain method and Agrawal field-line coupling model. The results show that the effect of overhead shielding wires on the lightning-induced voltages is closely related to the finite ground conductivity (σ) and the observation distance (d) from the lightning channel. For the ground conductivity ranging from 0.1 to 0.001 S/m, the values of shielding factor (SF) range about from 0.7 to 0.8 for vertical configuration, and range about from 0.6 to 0.7 for horizontal configuration, within distances from 50 to 1000 m from the lightning channel. Also, we have compared the SF values predicted by using Rusck simplified formula with ours in this paper, and it is noted that the Rusck simplified formula underestimates the SF values and the errors may be above 10% under poor ground conductivities of 0.001 S/m when distances larger than 500 m. [ABSTRACT FROM AUTHOR]

In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known. [ABSTRACT FROM AUTHOR]

This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated. [ABSTRACT FROM AUTHOR]

This paper is focused on using plasma as element of a frequency selective surface (FSS). FSSs have been used for filtering electromagnetic waves for many years. Conventional FSSs use metal patch pattern as periodic element. This paper takes the plasma tube as a substitution for metal patch. The 3-D finite-difference time-domain method with periodic boundary condition is utilized to simulate the interaction of incident wave and plasma FSS. Numerical calculation results in that electron number density of plasma can dominate the resonance frequency obviously. The resonance frequency increases as the increasing of electron number density of plasma to the limit of that of perfectly electric conductor. Thus, the FSS can be designed to be tunable by changing the ionized electron number density. Both of the noncollision and collisional plasma model are introduced to study the FSS characteristics. The numerical calculation results show that the collision frequency only influences the reflectivity while has no effect on the resonant frequency. The resonant frequency and transmitted power ratio can be tuned by assigning the plasma’s electron number density and collision frequency. Thus, plasma elements offer the possibility of improved shielding effect along with reconfigurability. Plasma FSS can also be made transparent by tuning the plasma off which makes the use of FSS more versatile. [ABSTRACT FROM AUTHOR]

Hairer and Iserles (2016) presented a detailed study of skew-symmetric matrix approximation to a first derivative which is proved to be fundamental in ensuring stability of discretisation for evolutional partial differential equations with variable coefficients. An open problem is proposed in that paper which concerns about the existence and construction of the perturbed grid that supports high-order skew-symmetric differentiation matrix for a given grid and only the case p = 2 for this problem have been solved. This paper is an attempt to solve the problem for any p ⩾ 3 . We focus ourselves on the symmetric grid and prove the existence of the perturbed grid for arbitrarily high order p and give in detail the construction of the perturbed grid. Numerical experiments are carried out to illustrate our theory. [ABSTRACT FROM AUTHOR]

The Landau-Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint |m| = 1. A crucial issue in designing efficient numerical schemes is to preserve this constraint in the discrete level. A simple and frequently used one is the projection method, which projects the numerical solution onto a unit sphere at each time step. The method has been used in many areas in the past several decades, while analysis has not been explored. In this paper, we present optimal error analysis of a backward Euler and a Crank-Nicolson semi-implicit projection finite difference scheme for the Landau-Lifshitz equation. The analysis is based on new and precise estimates of the difference between the errors of projected and unprojected solutions in both L2 and H1 norms. Some numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and accurate computational scheme based on the Chebyshev collocation method and finite difference approximation is proposed to solve the time‐fractional convection‐diffusion equation (TFCDE) on a finite domain. The time fractional‐order derivative μ ∈ (0, 1] is considered in the Caputo sense. The finite‐difference approximation is used in time direction while the Chebyshev collocation method is used in space direction to reduce the TFCDE into a system of algebraic equations. We also illustrate the error and convergence analysis of the proposed scheme. The proposed method is very convenient for solving such problems since the initial and boundary conditions are automatically taken into account. The efficiency and accuracy of the proposed algorithm are examined through some examples and comparisons with existing methods. [ABSTRACT FROM AUTHOR]

While the particle-in-cell (PIC) method is quite mature, verification and validation of both newly developed methods and individual codes have largely focused on an idiosyncratic choice of a few test cases. Many of these test cases involve either 1-D or 2-D simulations. This is either due to the availability of (quasi-) analytic solutions or historical reasons. In addition, tests often focus on the investigation of particular physics problems, such as particle emission or collisions, and do not necessarily study the combined impact of the suite of algorithms necessary for a full-featured PIC code. As 3-D codes become the norm, there is a lack of benchmark tests that can establish the validity of these codes; existing papers either do not delve into the details of the numerical experiment or provide other measurable numeric metrics (such as noise) that are outcomes of the simulation. This article seeks to provide several test cases that can be used for validation and bench-marking of PIC codes in 3-D. We focus on examples that are collisionless and can be run with reasonable computational resources. Three test cases are presented in significant detail; these include basic particle motion, beam expansion, and adiabatic expansion of plasma. All presented cases are compared either against existing analytical data or other codes. We anticipate that these cases should help fill the void of bench-marking and validation problems and help the development of new PIC codes. [ABSTRACT FROM AUTHOR]

Voltage and current waveforms of transmission lines struck by lightning are calculated for use in back flashover analysis. Calculation by using equivalent circuits is the general practice, however, there are limited experimental results for reference to evaluate validity of developed equivalent circuits under various line conditions. Calculation results by numerical electromagnetic analysis can be used as references if they are proved reliable. In this paper, the voltage rises at a model of a 154 kV transmission line hit by lightning is calculated by Method of Moments (MoM) and FDTD for comparison. They solve Maxwell's equations by using different algorithms, so, if the calculated results by these methods reasonably agree under various line conditions, the results of these numerical analyses are regarded reliable. The calculated voltage and current waveforms by the two numerical methods agree well under a variety of footing resistance of transmission towers, thus the authors conclude that the calculated results by FDTD or MoM can be employed as references to evaluate validity of circuit models for back flashover analyses. Thus, the results calculated by circuit analysis are compared with those calculated by FDTD. The multistory tower model (Ishii model) better reproduces the voltage waveforms in a wide range of the footing resistance than simple circuit models of a tower, and can be easily tuned for further accuracy. [ABSTRACT FROM AUTHOR]

This paper presents a new narrow-stencil finite difference method for approximating viscosity solutions of Hamilton--Jacobi--Bellman equations. The proposed finite difference scheme naturally extends the Lax--Friedrichs scheme for first order fully nonlinear PDEs to second order fully nonlinear PDEs which are approximated by Lax--Friedrichs-like numerical operators. The crux for constructing such a numerical operator is to introduce a stabilization term, which is called a "numerical moment" and corresponds to the numerical viscosity term in the original Lax--Friedrichs scheme for first order PDEs. It is proved that the proposed Lax--Friedrichs-like scheme has a unique solution and is stable in both the ℓ²-norm and the ℓ∞-norm. Moreover, the convergence of the proposed finite difference scheme to the viscosity solution of the underlying Hamilton--Jacobi--Bellman equation is also established using a novel discrete comparison argument. [ABSTRACT FROM AUTHOR]

Recently, many numerical techniques were presented to solve the fractional anomalous sub‐diffusion equations, and the results were excellent. In this paper, we study a simple numerical technique to solve two important types of fractional anomalous sub‐diffusion equations that appear strongly in chemical reactions and spiny neuronal dendrites, which are the two‐dimensional fractional Cable equation and the two‐dimensional fractional reaction sub‐diffusion equation. The proposed technique is a simple one which is an extension of the weighted average finite difference technique. The stability analysis of the proposed method is studied by means of John von Neumann stability analysis technique. An accurate stability criterion which is valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced. Four numerical examples are presented (two for the Cable equation and two for the reaction sub‐diffusion equation) to demonstrate the effectiveness and the accuracy of the presented method. [ABSTRACT FROM AUTHOR]

Dividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options. [ABSTRACT FROM AUTHOR]

This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices. [ABSTRACT FROM AUTHOR]

• Unconditionally convergent energy dissipative numerical methods are developed and analyzed for nonlinear wave equations. • Two novel efficient energy dissipative difference schemes for the wave equations are first set forth and analyzed. • The discrete energy dissipation properties, solvability, unconditional convergence and stability results are proven rigidly. • Some numerical results are provided to illustrate the computational accuracy and efficiency of the proposed schemes. In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete energy dissipation properties, solvability, unconditional convergence and stability of the proposed schemes are exhibited rigidly. By the discrete energy analysis method, it is rigidly shown that the proposed schemes achieve the unconditional convergence rates of O (Δ t 2 + h 2) in the discrete L ∞ -norm for the associated numerical solutions. At last, some numerical results are provided to illustrate the dynamical behaviors of the damping terms and unconditional energy stability of the suggested schemes, and testify the efficiency of theoretical results. [ABSTRACT FROM AUTHOR]

The electromagnetic fields, which are produced by lightning currents flowing into a transmission tower and into overhead ground wires, spherically expand with time, and propagate as non-transverse electromagnetic (TEM) waves at least within the round trip time of the traveling wave along the tower. In this paper, a distributed-parameter line model of a vertical double-circuit transmission tower with an overhead line struck by lightning is composed considering the non-TEM characteristics. The surge impedances of the tower, ground wires, and phase conductors increase with time and reach their TEM-mode or steady state values with time constants determined from the configuration. In addition, the significant wave attenuation along the tower is included in the circuit model according to the propagation characteristic determined by the tower configuration. The virtual inductive grounding impedance or damping circuits for approximately representing the wave attenuation is no longer needed. The proposed model is applied to four types of transmission towers. Waveforms of voltages across insulator strings computed by the finite-difference time-domain method, which are regarded as the reference, are well reproduced by the proposed model. Computation efficiency is drastically improved by the proposed model comparing to the numerical electromagnetic field analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two‐factor interest rate model with stochastic volatility, and we assume that the instantaneous interest rate follows a jump‐diffusion process. In this kind of problems, a two‐dimensional partial integro‐differential equation is derived for the values of zero‐coupon bonds. To apply standard numerical methods to this equation, it is customary to consider a bounded domain and incorporate suitable boundary conditions. However, for these two‐dimensional interest rate models, there are not well‐known boundary conditions, in general. Here, in order to approximate bond prices, we propose new boundary conditions, which maintain the discount function property of the zero‐coupon bond price. Then, we illustrate the numerical approximation of the corresponding boundary value problem by means of an alternative direction implicit method, which has been already applied for pricing options. We test these boundary conditions with several interest rate pricing models. [ABSTRACT FROM AUTHOR]

This paper presents the finite-difference time-domain (FDTD) thin-wire models of lossy wire structures with arbitrary inclination for transient analysis, which is difficult to address using the traditional FDTD methods. The frequency-dependent losses of the conductors are fully taken into account, and the vector fitting technique is applied to deal with frequency-dependent parameters for time-domain analysis. The bidirectional coupling within the lossy coaxial conductors is modeled. The currents in inner and outer conductors are not necessarily balanced. Three cases are presented for the investigation of wave propagation velocity, wave attenuation, and current distribution. These data are compared with analytical results and numerical results using other models. It is found that the proposed thin-wire models can depict the transient behaviors in the lossy inclined conductors with a velocity error of less than 1%, and an attenuation error of less than 1.5%. [ABSTRACT FROM AUTHOR]

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic–numeric approach feasible. To illustrate the simplified approach we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg–de Vries equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the multiresolution time domain (MRTD) method with its unique features is tailored for modeling interconnects. To build further credence to this and its profound existence in the recent state-of-the-art, simulations for inclusive crosstalk noise, on complementary metal-oxide semiconductor gate-driven mutually coupled multi-walled carbon nanotube interconnect lines, using MRTD method and conventional finite-difference time-domain (FDTD) model for 32-nm technology are executed. The results demonstrate the dominance of MRTD model over conventional FDTD in terms of accuracy with respect to recursive simulations of Synopsys HSPICE tool. An average error of less than 0.2% is observed in the estimation of dynamic crosstalk noise analysis. The proposed method is used to model interconnects with two and three mutually coupled lines and can be extended for N-coupled lines. The results of the transient analysis prove the efficiency of MRTD method over HSPICE with respect to computational time. The proposed method can also be used to address the issues of electromagnetic compatibility and electromagnetic interference of on-chip interconnects. [ABSTRACT FROM AUTHOR]

This paper presents the Greene second-order parabolic equation (PE) solution to model radio propagation over irregular terrain. This solution is implemented by finite-difference and shift-map technique. The second-order finite-difference approach performs well for slopes up to about 15°, and discontinuous slope changes up to about 30°, which is better than can be achieved using the split-step/Fourier approach. The results prove that the Greene PE solution has greater propagation angles than does the Claerbout PE solution. Thus, the Greene second-order solution can certainly give better results for large angle propagation over complicated terrain boundaries. Besides, we treat the important problem of accuracy of the different approximations of the PE-based propagation models, and derive the general PE solutions with respect to the terrain slope for the different approximations. It is useful to have collected in one paper, the most important PE approximations for radio propagation. [ABSTRACT FROM AUTHOR]

At present, the existing electromagnetic interference (EMI) analysis methods for the cavity with transmission lines (TLs) often need a large number of meshes to satisfy certain accuracy, which would seriously affect the calculation time, memory, and efficiency. Therefore, this paper presents a time-domain parallel method consisting of finite-difference time-domain method, TL equations, and message passing interface library. This method has two obvious advantages: one is that it does not need to mesh the TLs, which can reduce a lot of computational memory; and the other is that it can run on multiple processors or computers to save a lot of computational time. In order to verify the accuracy and validity of this method, the intracavity transmission of multiconductor TLs and coaxial cable under ambient wave excitation is simulated numerically and compared with the traditional method. The results show that the method can be well applied for the EMI analysis of the TLs in cavity under ambient excitation. [ABSTRACT FROM AUTHOR]

This paper investigates lightning surges on a distribution line reported using a finite-difference time-domain (FDTD) method. FDTD simulation results show a satisfactory agreement with measured lightning currents when lightning strikes ground nearby a distribution line. The current and voltage characteristics along the line, which were not measured in the test, are calculated by the FDTD simulation. Parametric analysis shows that lightning currents and voltages are dependent on the lightning position, distance from the lightning to the nearest pole, and pole grounding resistance. A large portion of phase-wire voltage is induced by the lightning channel current, and correspondingly, a large portion of phase-wire current is produced by the induced voltage. When the lightning channel is inclined, the peak current and voltage vary by +10% to −10% depending on the inclined angle. The effect of neutral wire position in the line is also investigated, and the induced voltage at the nearest pole to the lightning can be effectively suppressed by installing two neutral wires above and under the phase wires. EMTP simulation results agree qualitatively well with the measurement and FDTD results. [ABSTRACT FROM AUTHOR]

This paper presents a finite difference time domain formulation incorporating conductor losses due to skin and proximity effects for a uniform lossy two-conductor transmission line. At high frequency the per-unit-length effective internal impedance model is based on a boundary element method formulation enforcing high-order surface impedance boundary conditions. A smooth transition from the low to the high frequency model is obtained using first-order low- and high-pass filters. The effective internal impedance model is implemented into the finite difference time domain method via the discretization of the convolution. Finally, a computationally efficient finite difference time domain discretization of the two-conductor transmission line equation is obtained applying a recursive convolution technique. The proposed formulation is validated by comparison with the conventional inverse fast Fourier transform method and shows an improvement with respect to the well-known Paul's method, which does not consider the proximity effect. [ABSTRACT FROM AUTHOR]