Finite Difference Method

The landfill construction has caused many negative impacts on the surrounding environment, particularly groundwater. Evaluation of the function of the leachate collection pipe at the landfill site is indispensable for managing the landfill operation. 3D groundwater flow simulation may be applicable but it requires much capacity of computer and time consumption comparing with 2D groundwater flow simulation due to the huge calculations. Therefore, the 2D horizontal groundwater flow simulation (2D h ) was carried out. However, the most difficulty is the assignment of the groundwater head at the leachate collection pipe buried for leachate drainage. Therefore, in the present paper the 2D vertical model (2D v ) was applied to calculate the change of groundwater table above the leachate collection pipe. This paper paid attention to examine the validation of the assignment of the leachate collection pipe boundary by applying the results of the 2D v . The 2D h was coupled with the rainwater recharge model to solve the partial differential equation of groundwater flow. Finite difference method and iterative successive over relaxation were applied. The drainage volume of leachate collection was summed up in the whole landfill site and compared with the average volume of treated waste water. The study demonstrated that the groundwater level at the vicinity of the drainage pipe in the 2D v analysis is reasonably assigned for the 2D h .

This paper presents a numerical simulation of steady two-dimensional channel ow with a partially compliant wall. Navier-Stokes equation is solved using an unstructured ÿnite volume method (FVM). The deformation of the compliant wall is determined by solving a membrane equation using ÿnite di erence method (FDM). The membrane equation and Navier-Stokes equation are coupled iteratively to determine the shape of the membrane and the ow ÿeld. A spring analogy smoothing technique is applied to the deformed mesh to ensure good mesh quality throughout the computing procedure. Numerical results obtained in the present simulation match well with that in the literature. Copyright

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We are interested in optimal design of 3D complex geometries, such as radial turbomachines, in large control space. The calculation of the gradient of the cost function is a key point when a gradient based method is used. Finite difference method has a complexity proportional to the size of the control space and the adjoint method requires important extra coding. We propose to consider the incomplete sensitivities method for optimal design of radial turbomachinery blades. The central point of the paper is how to adapt some formulations in radial turbomachinery to the validity domain of incomplete sensitivities. Also, we discuss on how to improve the accuracy of incomplete sensitivities using reduced order models based on physical assumptions. Fine/Turbo flow solver is coupled with gradient based optimization algorithms based on CAD-connected frameworks. Newton methods together with incomplete expressions of gradients are used. The approach is validated through optimization of centrifugal pumps. Finally the results are considered and discussed.

We developed a 1.5-m band TM-mode waveguide optical isolator that makes use of the nonreciprocalloss phenomenon. The device was designed to operate in a single mode and consists of an InGaAlAs͞InP ridge-waveguide optical amplifier covered with a ferromagnetic MnAs layer. The combination of the optical waveguide and the magnetized ferromagnetic metal layer produces a magneto-optic effect called the nonreciprocal-loss phenomenon-a phenomenon in which the propagation loss of light is larger in backward propagation than it is in forward propagation. We propose the guiding design principle for the structure of the device and determine the optimized structure with the aid of electromagnetic simulation using the finite-difference method. On the basis of the results, we fabricated a prototype device and evaluated its operation. The device showed an isolation ratio of 7.2 dB͞mm at a wavelength from 1.53 to 1.55 m. Our waveguide isolator can be monolithically integrated with other waveguide-based optical devices on an InP substrate.

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black-Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.

This research was conducted to solve one dimensional heat equation and groundwater flow equation using Finite Difference Method. Three Finite Difference methods were chosen to solve parabolic Partial Differential Equations which are Explicit, Implicit and Crank-Nicolson method. The algorithm for each method has been developed and the solution of the problem is simplified using MATLAB software. The result obtained by the explicit method is given the most accurate and the best results compared to the Crank-Nicolson method and implicit method.

In this work the mathematical model of a spatial pattern in chemical and biological systems is investigated numerically. The proposed model considered as a nonlinear reaction-diffusion equation. A computational approach based on finite difference and RBF-collocation methods is conducted to solve the equation with respect to the appropriate initial and boundary conditions. The ability and robustness of the numerical approach is investigated using two test problems.

There are various methods for calculating rarefied gas flows, in particular, statistical methods and deterministic methods based on the finite-difference solutions of the Boltzmann nonlinear kinetic equation and on the solutions of model kinetic equations. There is no universal method; each has its disadvantages in terms of efficiency or accuracy. The choice of the method depends on the problem to be solved and on parameters of calculated flows. Qualitative theoretical arguments help to determine the range of parameters of effectively solved problems for each method; however, it is advisable to perform comparative tests of calculations of the classical problems performed by different methods and with different parameters to have quantitative confirmation of this reasoning. The paper provides the results of the calculations performed by the authors with the help of the Direct Simulation Monte Carlo method and finite-difference methods of solving the Boltzmann equation and model kinetic equations. Based on this comparison, conclusions are made on selecting a particular method for flow simulations in various ranges of flow parameters.

In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for obtaining the approximate solutions of an initial boundary value problem. We develop Forward Time Centered Space (FTCS) and Crank-Nicolson (CN) finite difference schemes for one-dimensional heat equation using the Taylor series. Later, we use these schemes to solve our governing equation. The stability criterion is discussed, and the stability conditions for both schemes are verified. We exhibit the results and then compare the results between the exact and approximate solutions. Finally, we estimate error between the exact and approximate solutions for a specific numerical problem to present the convergence of the numerical schemes, and demonstrate the resulting error in graphical representation.

This paper presents a comprehensive numerical study of the two-dimensional time-dependent heat conduction equation using the Forward Time Centered Space (FTCS) finite difference scheme. The heat equation is a fundamental parabolic partial differential equation, models the diffusion of thermal energy in a medium and is applicable in areas such as thermal insulation design, microchip cooling, and biological heat transfer. Due to the limitations of analytical methods in handling complex geometries and boundary conditions, we employ the FTCS scheme. The problem is formulated with Dirichlet boundary conditions and a sinusoidal initial condition for which an exact analytical solution is known. We derive the FTCS discretization using Taylor series-based approximations and perform a detailed von Neumann stability analysis to establish the Courant-Friedrichs-Lewy (CFL) condition. The scheme's performance is evaluated through numerical simulations on a uniform grid, with results compared against the exact solution. Simulation results show that the FTCS scheme achieves L 2 and max-norm errors on the order of 10-11 and 10-10 , respectively, under stable conditions. Graphical comparisons further demonstrate excellent agreement between numerical and analytical solutions. Overall, the FTCS method proves to be a robust and reliable tool for solving heat conduction problems, provided the stability criterion is satisfied.

Optical modulation of the effective refractive properties of a “fishnet” metamaterial with a Ag∕Si∕Ag heterostructure is demonstrated in the near-IR range and the associated fast dynamics of negative refractive index is studied by pump-probe method. Photoexcitation of the amorphous Si layer at visible wavelength and corresponding modification of its optical parameters is found to be responsible for the observed modulation of negative refractive index in near-IR.

This book illustrates the use of computational methods in engineering analyses by focussing on thermofluid analyses and by using a general-purpose spreadsheet application which is Microsoft Excel. The Excel-based modelling platform described in the book has four elements; (i) Excel with its user-interface and built-in functions, (ii) the Solver add-in that comes with Excel, (iii), the integrated programming language Visual
Basic for Applications (VBA) and (iv) an Excel add-in for fluid properties called Thermax. While the two main components, Excel and Solver, are adequate for most fluid mechanics and heat-transfer analyses, Thermax helps the students to perform thermodynamic analyses with Excel. VBA is needed for the development of custom functions when the analytical model cannot be executed by only using Excel’s built-in functions and Thermax functions. Properly used, the Excel-based modelling platform minimises the effort of developing the analytical computer model so that more attention can be paid to the application of the relevant thermofluid principles. Apart from the wide availability of Excel on personal computers, the modelling platform allows the students to be more involved in the process of developing their models compared to other applications that are dedicated to thermofluid analyses.

The book has been compiled from the first two books of a set of three books that illustrate the use of the Excel-based modelling platform for various types of computer-aided thermofluid analyses. The selected topics it includes are intended to suit a 2 credit-hour course on computer-applications for engineers that is preferably taken after completing the basic three thermofluid courses. The book builds on the students’ theoretical background at an intermediate level and does not discuss advanced related topics such as CFD (Computational Fluid Dynamics), the finite-element method, exergo-economics, or multi-objective optimisation. The last two topics are discussed in the third book of this
set and can be added if required. Most of the examples given in the book are based on relevant cases or examples given in standard thermofluid textbooks or obtained from the published literature so that the students can look for any additional information needed to validate their Excel models and verify their results. Exercises are given at the end of each chapter to help students sharpen their skills related to the particular topic.

We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.

During many earthquakes, soil liquefaction causes dramatic damage. The use of a granular column is a ground-improvement technique used to mitigate soil liquefaction. The present paper studies the performance of reinforcement with granular columns as a method for mitigation of potential risk of liquefaction during cyclic loading. The generation and dissipation of excess pore pressure are analyzed by considering the granular-column installation effect. This effect is described by a decrease in horizontal permeability within a disturbed zone around the column, which is described by constant, linear, and parabolic variations. The evolution of excess pore pressure is studied as a function of the disturbed zone in terms of reduced horizontal permeability. Obtained results show that a granular-column installation has a significant influence on the mitigation of liquefaction.

An experimental realization of the dispersion compensating optical fiber (DCF) for the dense wavelength-division-multiplexed (DWDM) fiber-optic link is reported. The negative dispersion is obtained within ±1.5 ps/kmnm from central wavelength in the wavelength region of 1.45 μιη to 1.63 μιη. The characterization of the fiber in terms of spectral attenuation, bend loss, single mode cutoff wavelength, mode field diameter and dispersion has been carried out. This work is useful for the optical fiber communication link that employs DWDM at minimum dispersion.

Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete favorably with the variational regularization method for stable calculating the derivatives of noisy functions.

Many times a scientist is choosing a programming language or a software for a specific purpose. For the field of scientific computing, the methods for solving differential equations are one of the important areas. What I would like to do is take the time to compare and contrast between the most popular offerings. This is a good way to reflect upon what's available and find out where there is room for improvement. I hope that by giving you the details for how each suite was put together (and the "why", as gathered from software publications) you can come to your own conclusion as to which suites are right for you. (Full disclosure, I am the lead developer of DifferentialEquations.jl. You will see at the end that DifferentialEquations.jl does offer pretty much everything from the other suite combined, but that's no accident: our software organization came last and we used these suites as a guiding hand for how to design ours.

The hydromagnetic convective boundary layer flow past a stretching porous wall embedded in a porous medium with heat and mass transfer in the presence of a heat source and under the influence of a uniform magnetic field is studied. Exact solutions of the basic equations of motion, heat and mass transfer are obtained after reducing them to nonlinear ordinary differential equations. The reduced equations of heat and mass transfer are solved using a confluent hypergeometric function. The effects of the flow parameters such as a suction parameter (N ), magnetic parameter (M ), permeability parameter (K p ), wall temperature parameter (r), wall concentration parameter (n), and heat source/sink parameter (Q) on the dynamics are discussed. It is observed that the suction parameter appears in the boundary condition ensuring the variable suction at the surface. Transverse component of the velocity increases only when magnetic field strength exceeds certain value, but the thermal boundary layer thickness and concentration distribution increase for all values. Results presented in this paper are in good agreement with the work of the previous author and also in conformity with the established theory.

In this paper, the results of simulations of natural circulation loop performance, obtained by Cathare and Relap codes, are reported. Both series of results are analyzed and compared with experimental data gathered in the MTT-1 loop, a rectangular natural circulation loop realized by DITEC at the University of Genova. Both Cathare and Relap codes, in absolute terms, show poor agreement with experimental data. At low power, the Cathare code shows a good capability to predict the steady state quantities, after the initial transient. On the other hand, no unstable behavior is predicted at each analyzed power level. The Relap code is able to show oscillating quantities, but not at the same power levels as in the experiments.

During recent years a computational strategy has been developed at the Technical University of Denmark for numerical simulation of water wave problems based on the high-order finite-difference method, [2],[4]. These methods exhibit a linear scaling of the computational effort as the number of grid points increases. This understanding is being applied to develop a tool for predicting the added resistance (drift force) of ships in ocean waves. We expect that the optimal scaling properties of this solver will allow us to make a convincing demonstration of convergence of the added resistance calculations based on both near-field and far-field methods. The solver has been written inside a C++ library known as Overture [3], which can be used to solve partial differential equations on overlapping grids based on the high-order finite-difference method. The resulting code is able to solve, in the time domain, the linearised potential flow forward-speed hydrodynamic problems; namely the stead...

The modified mild slope equation of is solved using a combination of the boundary element method (BEM) and the finite difference method (FDM). The exterior domain of constant depth and infinite horizontal extent is solved by a BEM using linear or quadratic elements. The interior domain with variable depth is solved by a flexible order of accuracy FDM in boundary-fitted curvilinear coordinates. The two solutions are matched along the common boundary of two methods (the BEM boundary) to ensure continuity of value and normal flux. Convergence of the individual methods is shown and the combined solution is tested against several test cases. Results for refraction and diffraction of waves from submerged bottom mounted obstacles compare well with experimental measurements and other computed results from the literature.

Several theoretical and numerical aspects concerning the highly accurate Boussinesq-type equations of are discussed. A re-derivation of the model recently presented by Bingham et al. (2009) is outlined. This provides a more general framework for the model establishing the correct relationship between a velocity formulation and a velocity potential formulation and correcting previous errors in the potential formulation. A new shoaling enhancement operator is introduced which enables the derivation of new models which differ from the existing ones at O(∇h). The performance of the new formulation is validated using computations of linear and nonlinear shoaling problems. The behaviour on a rapidly varying bathymetry is also checked using linear wave reflection from a shelf and Bragg scattering from an undulating bottom. A new stable discretization scheme around structural corners within the fluid domain is also presented.

This paper describes the extension of a finite difference model based on a recently derived highly accurate Boussinesq formulation to include domains having arbitrary piecewise-rectangular bottom-mounted structures. The resulting linearized system is analyzed for stability on a structurally divided domain, and it is shown that exterior corner points pose potential stability problems, as well as other numerical difficulties. These are mainly due to the discretization of high-order mixed-derivative terms near these points, where the flow is theoretically singular. Fortunately, the system is receptive to dissipation, and these problems can be overcome in practice using high-order filtering techniques. The resulting model is verified through numerical simulations involving classical linear wave diffraction around a semi-infinite breakwater, linear and nonlinear gap diffraction, and highly nonlinear deep water wave run-up on a vertical plate. These cases demonstrate the applicability of the model over a wide range of water depth and nonlinearity.

The problem of mixed convection along an inclined wall, acting as a source or sink with lateral mass flux, embedded in a porous medium is solved by the local non similarity method. The wall surface temperature, the flow free stream velocity and lateral surface velocity vary in a power law along the wall. The two components of buoyancy force are retained. The inclination of the plate ranges from horizontal to vertical. Numerical solutions are carried out up to the third level of truncation and results are presented for a wide range of mass flux parameter at different wall temperature distributions and inclination angles.

In this paper a parabolic-parabolic chemotaxis system of PDEs that describes the evolution of a population with non-local terms is studied. We derive the discretization of the system using the meshless method called Generalized Finite Difference Method. We prove the conditional convergence of the solution obtained from the numerical method to the analytical solution in the twodimensional case. Several examples of the application are given to illustrate the accuracy and efficiency of the numerical method. We also present two examples of a parabolic-elliptic model, as generalized by the parabolic-parabolic system addressed in this paper, to show the validity of the discretization of the non-local terms.

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This paper focuses on the numerical analysis of a discrete version of a nonlinear reaction-diffusion system consisting of an ordinary equation coupled to a quasilinear parabolic PDE with a chemotactic term. The parabolic equation of the system describes the behavior of a biological species, while the ordinary equation defines the concentration of a chemical substance. The system also includes a logistic-like source, which limits the growth of the biological species and presents a time-periodic asymptotic behavior. We study the convergence of the explicit discrete scheme obtained by means of the generalized finite difference method and prove that the nonnegative numerical solutions in two-dimensional space preserve the asymptotic behavior of the continuous ones. Using different functions and long-time simulations, we illustrate the efficiency of the developed numerical algorithms in the sense of the convergence in space and in time.

In the present paper we propose the Generalized Finite Difference Method (GFDM) for numerical solution of a cross-diffusion system with chemotactic terms. We derive the discretization of the system using a GFD scheme in order to prove and illustrate that the uniform stability behavior/ convergence of the continuous model is also preserved for the discrete model. We prove the convergence of the explicit method and give the conditions of convergence. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the GFDM.

In this paper it is shown the application of the generalized finite difference method (GFDM) for solving numerically the Telegraph equation in two and three-dimensional spaces. The explicit time discretization is used and for approximating the spatial variables a GFDM is applied. The GFDM is a truly meshless method. The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM). The stability condition using von Neumann method is done, which is provided in Theorem 1. Some numerical results have been reported and compared with the exact solutions for showing the accuracy and efficiency of the proposed numerical scheme.

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (PDEs): wave propagation, advection-diffusion, plates, beams, etc. The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear parabolic PDEs. This paper illustres that the GFD explicit formulae developed to obtain the different derivatives of the pde's are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Criteria for convergence of fully explicit method using GFDM for different non linear parabolic pdes are given. This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.

This paper shows the displacements and velocity-stress formulations for the wave propagation problem with the aim of comparing its effectivity when they are implemented with the Generalized Finite Difference Method (GFDM). Schemes in GFD with approximations of the second and fourth order have been used and absorbing boundaries have been simulated with perfectly matched layers. The results obtained using both formulations are compared by using several examples in 2-D. The influence of the type of discretization of the clouds of nodes (regular and irregular) is studied.

The Generalized finite difference method (GFDM) is a meshfree method that can be applied for solving problems defined over irregular clouds of points. The GFDM uses the Taylor series development and the moving least squares approximation to obtain explicit formulae for the partial derivatives. In this paper, this meshfree method is used for solving elliptic and parabolic partial differential equations in 3-D. The influence of the main parameters involved in the approximation and the treatment of the Neumann boundary condition are shown. Parabolic equations have been solved using an explicit method and the criterion for stability has been improved taking into account the irregularity of the cloud of points. The numerical results show the high accuracy obtained.

Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points. In this paper the extension of the GFD to the explicit solution of parabolic and hyperbolic equations has been developed for partial differential equations with constant coefficients in the cases of considering one, two or three space dimensions. The convergence of the method has been studied and the truncation errors over irregular grids are given. Different examples have been solved using the explicit finite difference formulae and the criterion of stability. This has been expressed in function of the coefficients of the star equation for irregular clouds of nodes in one, two or three space dimensions. The numerical results show the accuracy obtained over irregular grids. This paper also includes the study of the maximum local error and the global error for different examples of parabolic and hyperbolic time-dependent equations.

In this paper it is possible to appreciate the great eciency of the generalized ®nite dierence method (GFD), that is to say with an irregular arrangements of nodes, to solve second-order partial dierential equations which represent the behaviour of many physical processes. The method solves any type of second-order dierential equation, in any type of domain and boundary condition (Dirichlet, Neumann and mixed), and immediately obtains the values of derivatives of the nodes through the application of the formulae in dierences obtained. This paper analyses the in¯uences of key parameters of the method, such as the number of nodes of the star, the arrangement of the same, the weight function and the stability parameter in time-dependent problems. This analysis includes solutions obtained for dierent types of problems, represented by dierent dierential equations, including time-dependent equations and under dierent boundary conditions.

One of the most universal and effective methods, in wide use today, for approximately solving equations of mathematical physics is the finite difference (FD) method. An evolution of the FD method has been the development of the generalized finite difference (GFD) method, which can be applied over general or irregular clouds of points. The main drawback of the GFD method is the possibility of obtaining illconditioned stars of nodes. In this paper a procedure is given that can easily assure the quality of numerical results by obtaining the residual at each point. The possibility of employing the GFD method over adaptive clouds of points increasing progressively the number of nodes is explored, giving in this paper a condition to be accomplished to employ the GFD method with more efficiency. Also, in this paper, the GFD method is compared with another meshless method the, so-called, element free Galerkin method (EFG). The EFG method with linear approximation and penalty functions to treat the essential boundary condition is used in this paper. Both methods are compared for solving Laplace equation.

An efficient and thorough strategy to introduce undergraduate students to a numerical approach of calculating flow is outlined. First, the basic steps, especially discretization, involved when solving Navier-Stokes equations using a finite-volume method for incompressible steady-state flow are developed with the main aim being for the students to follow through from the mathematical description of a given problem to the final solution of the governing equations in a transparent way. The well-known 'driven-cavity' problem is used as the problem for testing coding written by the students, and the Navier-Stokes equations are initially cast in the vorticity-streamfunction form. This is followed by moving on to a solution method using the primitive variables and discussion of details such as, closure of the Navier-Stokes equations using turbulence modelling, appropriate meshing within the computation domain, various boundary conditions, properties of fluids, and the important methods for determining that a convergence solution has been reached. Such a course is found to be an efficient and transparent approach for introducing students to computational fluid dynamics.

In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vector potential satisfies is solved using a version of FD-WENO developed for Hamilton-Jacobi equations. The resulting numerical method is endowed with several important properties: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as point values on the same mesh (i.e., there is no mesh staggering); (2) both the spatial and temporal orders of accuracy are fourth-order; (3) no spatial integration or multidimensional reconstructions are needed in any step; and (4) special limiters in the magnetic vector potential update are used to control unphysical oscillations in the magnetic field. Several 2D and 3D numerical examples are presented to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks.

The American Physical Society Discontinuous Galerkin Methods for Magnetohydrodynamics JAMES ROSSMANITH, University of Wisconsin -Madison -Standard shockcapturing numerical methods fail to give accurate solutions to the equations of magnetohydrodynamics (MHD). The essential reason for this failure is that by ignoring the divergence-free constraint on the magnetic field, these methods can be shown to be entropy unstable. In this talk we will briefly review the entropy stability theorem for discontinuous Galerkin (DG) methods. We will then present a class of constrained transport (CT) methods that we will show give both stable and accurate results on several test cases. The proposed CT approach can be viewed as a predictor-corrector method, where an approximate magnetic field is first predicted by a standard DG method, and then corrected through the use of a magnetic potential. Finally, we will briefly describe efforts to extend this approach to Hall MHD and genuinely two-fluid plasma models.

In this study, we obtain approximate solution for singularly perturbed problem of differential equation having two integral boundary conditions. With this purpose, we propose a new finite difference scheme. First, we construct this exponentially difference scheme on a uniform mesh using the finite difference method. We use the quasilinearization method and the interpolating quadrature formulas to establish the numerical scheme. Then, as a result of the error analysis, we show that the method under study is convergent in the first order. Consequently, theoretical findings are supported by numerical results obtained with an example. Approximate solutions curves are compared on the chart to provide concrete indication. The maximum errors and convergence rates obtained are given on the table for different ε and N values.

An exact closed form solution is derived for the mechanical behaviour of a linear viscoelastic Burgers rock around an axisymmetric tunnel, supported by a linear elastic ring. Analytical formulae are provided for the displacement of the rock/lining interface and for the pressure exerted by the rock on the lining, taking into account the stiffness and its installation time. Results calculated from these formulae do validate the corresponding numerical results of a 2D finite differences code. Further, comparison to previous existing solutions for the same viscoelastic model indicates similarities and differences. A parametric study is performed to investigate the effect of the viscoelastic constants, the stiffness and installation time of the support. The derived closed form solution is used to construct the time-dependent Supported Ground Reaction Curves of the viscoelastic rock, i.e. the time contour plots on the convergence confinement diagram. The importance of the effect of the support on the restrained rock creep and the exerted pressure on the lining, during the design life of a structure, is examined.

In this paper, coupling characteristics of dualcore photonic crystal fiber (PCF) are studied extensively using vector finite element method, which has the potential to realize wavelength selective MUX-DEMUX for wavelength division multiplexing (WDM) application. Dispersion characteristic is also reported and demonstrates the wavelength region where it can support short duration soliton like pulses.

This work presents a numerical technique for simulating incompressible, isothermal, viscoelastic flows of fluids governed by the upper-convected Maxwell (UCM) and K-BKZ (Kaye-Bernstein, Kearsley and Zapas) integral models. The numerical technique described herein is an extension of the GENSMAC method to the solution of the momentum and mass conservation equations to include integral constitutive equations. The governing equations are solved by the finite difference method on a staggered grid using a Marker-and-Cell approach. The Finger tensor B t 0 ðtÞ is computed in an Eulerian framework using the ideas of the deformation fields method. However, improvements to the deformation fields method are introduced: the Finger tensor B t 0 ðx; tÞ is obtained by a second-order accurate method and the stress tensor sðx; tÞ is computed by a second-order quadrature formula. The numerical method presented in this work is validated by comparing the predictions of velocity and stress fields in two-dimensional fully-developed channel flow of a Maxwell fluid with the corresponding analytic solutions. Furthermore, the flow through a planar 4:1 contraction is investigated and the numerical results were compared with the corresponding experimental data. Finally, the UCM and the K-BKZ models were used to simulate the planar 4:1 contraction flow over a wide range of Reynolds and Weissenberg numbers and the numerical results obtained are in agreement with published data.

A diffusion equation to describe the isothermal absorption of liquid water in a spherical solid that undergoes uniform swelling was derived. The resulting partial differential equation was solved using a finite difference method, taking into consideration water content dependence of the diffusion coefficient. The developed model was applied to simulate the water uptake of brown rice for the soaking temperatures of 25, 45, 55 and 65 °C. The estimated ''differential" diffusion coefficients were a strongly increasing function of moisture content for all temperatures tested, approaching to the self-diffusion coefficients of water for brown rice moisture contents near to the saturation values. The ''integral" diffusion coefficient corresponding to range of moisture content resulting from soaking conditions were calculated and correlated according to Arrhenius equation with an activation energy of 32.5 kJ/mol.

This word discusses approximate solutions of linear parabolic equations with initial-boundary conditions. The primary focus is on methods that effectively find such solutions by employing a moving finite difference analog of the differential equation. This approach allows us to formulate an approximate analytical solution, significantly simplifying the computation process. By transitioning from the differential equation to an algebraic equation, we obtain a single equation, the solution of which represents an approximate analytical solution to the original problem. However, to achieve higher accuracy in this solution, we apply additional moving nodes, which enhances the results. By using multipoint moving nodes, we can form a system of algebraic equations, the solution of which provides us with an improved analytical solution. The article also presents numerical experiments that confirm the effectiveness of the proposed method and its advantages over traditional approaches.

Put option is a contract to sell some underlying assets in the future with a certain price. On European put options, selling only can be exercised at maturity date. Behavior of European put options price can be modeled by using the Black-Scholes model which provide an analytical solution. Numerical approximation such as binomial tree, explicit and implicit finite difference methods also can be used to solve Black-Scholes model. Some numerical methods are applied and compared with the analytical solution to determine the best numerical method. The results show that numerical approximations using the binomial tree is more accurate than explicit and implicit finite difference method in pricing European put options. Moreover when the value of T is higher then the error obtained is also higher, while the error obtained is lower when the value of N is higher. The value of T and N cause the increase of the computation time. When the value of T is higher the computation time is lower, while computation time is higher if the value of N is higher. Overall, the lowest computation time is obtained by using an explicit finite difference method with an exceptional as the value of T is big and the value of N is small. The lowest computation time is obtained by using a binomial tree method.

A shallow water model with linear time-dependent dispersive waves in an unbounded domain is considered. The domain is truncated with artificial boundaries B where a sequence of high-order non-reflecting boundary conditions (NRBCs) proposed by Higdon are applied. Methods devised by Givoli and Neta that afford easy implementation of Higdon NRBCs are refined in order to reduce computational expenses. The new refinement makes the computational effort associated with the boundary treatment quadratic rather than exponential (as in the original scheme) with the order. This allows for implementation of NRBCs of higher orders than previously. A numerical example for a semi-infinite channel truncated on one side is presented. Finite difference schemes are used throughout.

This paper proposes a new multichannel time reversal focusing (MTRF) method for circumferential Lamb waves which is based on modified time reversal algorithm and applies this method for detecting different kinds of defects in thick-walled pipe with large-diameter. The principle of time reversal of circumferential Lamb waves in pipe is presented along with the influence from multiple guided wave modes and propagation paths. Experimental study is carried out in a thick-walled and large-diameter pipe with three artificial defects, namely two axial notches on its inner and outer surface respectively, and a corrosionlike defect on its outer surface. By using the proposed MTRF method, the multichannel signals focus at the defects, leading to the amplitude improvement of the defect scattered signal. Besides, another energy focus arises in the direct signal due to the partial compensation of dispersion and multimode of circumferential Lamb waves, alongside the multichannel focusing, during MTRF process. By taking the direct focus as a time base, accurate defect localization is implemented. Secondly, a new phenomenon is exhibited in this paper that defect scattered wave packet appears just before the right boundary of truncation window after time reversal, and to which two feasible explanations are given. Moreover, this phenomenon can be used as the theoretical basis in the determination of defect scattered waves in time reversal response signal. At last, in order to detect defects without prior knowing their exact position, a largerange truncation window is used in the proposed method. As a result, the experimental operation of MTRF method is simplified and defect detection and localization are well accomplished.