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The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.

A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.

The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.

A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

The purpose of this study is to develop a new method of lines for one-dimensional (1D) advection-reaction-diffusion (ADR) equations that is conservative and provides piecewise analytical solutions in space, compare it with other finite-difference discretizations and assess the effects of advection and reaction on both 1D and two-dimensional (2D) problems.

A conservative method of lines based on the piecewise analytical integration of the two-point boundary value problems that result from the local solution of the advection-diffusion operator subject to the continuity of the dependent variables and their fluxes at the control volume boundaries is presented. The method results in nonlinear first-order, ordinary differential equations in time for the nodal values of the dependent variables at three adjacent grid points and triangular mass and source matrices, reduces to the well-known exponentially fitted techniques for constant coefficients and equally spaced grids and provides continuous solutions in space.

The conservative method of lines presented here results in three-point finite difference equations for the nodal values, implicitly treats the advection and diffusion terms and is unconditionally stable if the reaction terms are implicitly treated. The method is shown to be more accurate than other three-point, exponentially fitted methods for nonlinear problems with interior and/or boundary layers and/or source/reaction terms. The effects of linear advection in 1D reacting flow problems indicates that the wave front steepens as it approaches the downstream boundary, whereas its back corresponds to a translation of the initial conditions; for nonlinear advection, the wave front exhibits steepening but the wave back shows a linear dependence on space. For a system of two nonlinearly coupled, 2D ADR equations, it is shown that a counter-clockwise rotating vortical field stretches the spiral whose tip drifts about the center of the domain, whereas a clock-wise rotating one compresses the wave and thickens its arms.

A new, conservative method of lines that implicitly treats the advection and diffusion terms and provides piecewise-exponential solutions in space is presented and applied to some 1D and 2D advection reactions.

The paper aims to compare and clarify the differences and between the two well-known decomposition spectral techniques; the Winer–Chaos expansion (WCE) and the Winer–Hermite expansion (WHE). The details of the two decompositions are outlined. The difficulties arise when using the two techniques are also mentioned along with the convergence orders. The reader can also find a collection of references to understand the two decompositions with their origins. The geometrical Brownian motion is considered as an example for an important process with exact solution for the sake of comparison. The two decompositions are found practical in analysing the SDEs. The WCE is, in general, simpler, while WHE is more efficient as it is the limit of WCE when using infinite number of random variables. The Burgers turbulence is considered as a nonlinear example and WHE is shown to be more efficient in detecting the turbulence. In general, WHE is more efficient especially in case of nonlinear and/or non-Gaussian processes.

The paper outlined the technical and literature review of the WCE and WHE techniques. Linear and nonlinear processes are compared to outline the comparison along with the convergence of both techniques.

The paper shows that both decompositions are practical in solving the stochastic differential equations. The WCE is found simpler and WHE is the limit when using infinite number of random variables in WCE. The WHE is more efficient especially in case of nonlinear problems.

Applicable for SDEs with square integrable processes and coefficients satisfying Lipschitz conditions.

This paper fulfils a comparison required by the researchers in the stochastic analysis area. It also introduces a simple efficient technique to model the flow turbulence in the physical domain.

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

The purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation.

A conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported.

The finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature.

A new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.

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 main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.

This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.

Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.

In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

This paper aims to establish a piecewise Maxwell stress analytical model of bearingless switched reluctance motor (BSRM) for the full rotor angular positions. The proposed model varies from the existing models, which are only applicable to the partial-overlapping positions of stator and rotor poles. By extending the applicable rotor angular positions, this model provides a basic analytical model for the multi-phase excitation control of BSRM.

The full rotor angular positions are classified into the partial-overlapping positions and the non-overlapping positions. At first, two different air gap subdividing methods are proposed, respectively, for the two-position ranges. Then, different integration paths are selected accordingly. Furthermore, two approximate methods are presented to calculate the average flux density of each air gap subdivision. Finally, considering the mutual coupling between the two perpendicular radial suspension forces, a piecewise Maxwell stress analytical model is derived for the full rotor angular positions of BSRM.

A piecewise Maxwell stress analytical model of BSRM is built for the full rotor angular positions, and applicable to the multi-phase excitation mode of BSRM. For the partial-overlapping positions and the non-overlapping positions, two sets of air gap subdividing methods, integration paths and approximate calculation methods of air gap flux densities are proposed, respectively. The accuracy and reliability of the proposed model are verified by the finite element method.

The piecewise Maxwell stress analytical model of BSRM for the full rotor angular positions is proposed for the first time. The novel air gap subdividing methods, integration paths, approximate calculation methods of air gap flux densities and the coupling between the two radial suspension forces are adopted to improve the modeling accuracy. As the applicable range of rotor angular position is extended, this model overcomes the limitation of the existing models only for single-phase excitation mode and contributes to the accurate control of BSRM multi-phase excitation mode.

We introduce a second‐order accurate time‐marching pressure‐correction algorithm to accommodate weakly‐compressible highly‐viscous liquid flows at low Mach number. As the incompressible limit is approached (Ma ≈ 0), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In the viscous‐dominated regime of low Reynolds number (zone of interest), the algorithm treats the viscous part of the equations in a semi‐implicit form. Two discrete representations are proposed to interpolate density: a piecewise‐constant form with gradient recovery and a linear interpolation form, akin to that on pressure. Numerical performance is considered on a number of classical benchmark problems for highly viscous liquid flows to highlight consistency, accuracy and stability properties. Validation bears out the high quality of performance of both compressible flow implementations, at low to vanishing Mach number. Neither linear nor constant density interpolations schemes degrade the second‐order accuracy of the original incompressible fractional‐staged pressure‐correction scheme. The piecewise‐constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.