Alternating Direction Implicit Finite Difference Research Articles

Stability and Super Convergence Analysis of ADI-FDTD for the 2D Maxwell Equations in a Lossy Medium

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved. These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations (2D-ADI-FDTD). It is proved that 2D-ADI-FDTD is approximately energy conserved, unconditionally stable and second order convergent in the discrete L2 and H1 norms, which implies that 2D-ADI-FDTD is super convergent. By this super convergence, it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate. It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H1 norm. Experimental results to confirm the theoretical analysis on stability, convergence and energy conservation are presented.

Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation

Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

AbstractIn this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Modeling of simultaneous heat and mass transfer during convective oven ring cake baking

The convective oven ring cake baking process was investigated experimentally and numerically as a simultaneous heat and mass transfer process. The mathematical model described previously by the authors for cup cake baking was modified to simulate ring cake baking. The heat and mass transfer mechanisms were defined by Fourier’s and Fick’s second laws, respectively. The implicit alternating direction finite difference technique was used for the numerical solution of the representative model. Prior to the utilization of the developed model in predicting the temperature and moisture profiles for ring cake baking, the results of the numerical model were compared with analytical results involving only heat or mass transfer with constant thermo-physical properties. Excellent agreement was observed. The numerical temperature and moisture contents predicted by the model were compared with the experimental profiles. They agreed generally reasonably well with the experimental temperature and moisture profiles.

A new fourth‐order numerical algorithm for a class of three‐dimensional nonlinear evolution equations

AbstractIn this article, a new compact alternating direction implicit finite difference scheme is derived for solving a class of 3‐D nonlinear evolution equations. By the discrete energy method, it is shown that the new difference scheme has good stability and can attain second‐order accuracy in time and fourth‐order accuracy in space with respect to the discrete H1 ‐norm. A Richardson extrapolation algorithm is applied to achieve fourth‐order accuracy in temporal dimension. Numerical experiments illustrate the accuracy and efficiency of the extrapolation algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

General ADI-FDTD Formulations for Multi-Term Dispersive Electromagnetic Applications

General formulations of the alternating direction implicit finite difference time domain method are presented for modeling multi-term dispersive electromagnetic applications. The proposed formulations, which are based on the exponential evolution operator scheme, allow modeling different types of frequency dependent materials in the same manner and can be easily incorporated with the nearly perfectly matched layer absorbing boundary conditions to model open region problems. 2-D numerical example is included to show the validity of the proposed formulations.

An O( N log 2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O( N 3) per time step and memory of O( N 2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O( N log 2 N) per time step and memory of O( N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5 h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

Novel fast photonic crystal multiplexer-demultiplexer switches

In this paper, a 3 × 1 Multiplexer/Demultiplexer (MUX/DEMUX) Photonic Crystal (PhC) based structure is presented. This is achieved by carefully considering the coupling length of the propagating wave and accurately engineering the geometrical design of the microcavities. The design is highly selective, such that, a microcavity embedded between two waveguides selects a particular wavelength to couple from one waveguide into an adjacent waveguide. The numerical technique used for the designs throughout this paper is the Complex Envelope Alternating Direction Implicit Finite Difference Time Domain (CE-ADI-FDTD).

Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions

New shallow-water equations (SWEs), for sloshing in three dimensions (two horizontal and one vertical) in a vessel which is undergoing rigid-body motion in 3-space, are derived. The rigid-body motion of the vessel (roll–pitch–yaw and/or surge–sway–heave) is modelled exactly and the only approximations are in the fluid motion. The flow is assumed to be inviscid but vortical, with approximations on the vertical velocity and acceleration at the surface. These equations improve previous shallow-water models. The model also extends to three dimensions the essence of the Penney–Price–Taylor theory for the highest standing wave. The surface SWEs are simulated using a split-step alternating direction implicit finite-difference scheme. Numerical experiments are reported, including comparisons with existing results in the literature, and simulations with vessels undergoing full 3-D rotations.

Numerical study of buoyancy‐driven transitional flow in a square cavity with partially heated and cooled vertical walls

PurposeThe purpose of this paper is to numerically study transient natural convective flow in a square cavity with partially heated and cooled vertical walls, thermally insulated top wall and linearly heated bottom wall.Design/methodology/approachThe governing equations of motion are non‐dimensionalized and reformulated using stream function‐vorticity approach. Alternating direction implicit finite difference scheme is used to solve the coupled equations.FindingsThe transient results obtained for different values of Grashof number (Gr) and fixed Prandtl number Pr = 0.733 are presented in the form of isotherms, streamlines, bifurcation diagram and time series. The transition from steady to oscillatory motions is analyzed in detail with respect to Gr. The flow is observed to be steady up to Gr ≈ 2 × 104. A time‐periodic unsteady solution first appears at Gr = 20,900 and the amplitude of the fluctuation grows as Gr is increased.Research limitations/implicationsThe study is limited to laminar flow in a square cavity. Further extension of this work could include the influence of various choices of Prandtl number and the effect of aspect ratio. Buoyancy‐driven convection in a sealed cavity with differentially heated walls is a prototype of many industrial applications such as energy‐efficient design of buildings and rooms, convective heat transfer associated with boilers, etc.Originality/valueThe paper presents an original computer program written in FORTRAN to solve the partial differential equations.

Improved design of photonic crystal-based multiplexer/demultiplexer devices

In this paper, an improved design of photonic crystal (PhC)-based multiplexer/demultiplexer (MUX-DEMUX) devices is presented. This is achieved using careful geometrical design of microcavities with respect to the coupling length of the propagating wave. The nature of the design means that a microcavity embedded between two waveguides selects a particular wavelength to couple from one waveguide into the adjacent waveguide showing high selectivity. The numerical technique used for the designs throughout this article is the complex envelope alternating direction implicit finite difference time domain (CE-ADI-FDTD). Also, improvements have been made on the performance of the perfectly matched layers (PML) by presenting an accurate physical model of discretisation error.

A Parallel Implementation on GPUs of ADI Finite Difference Methods for Parabolic PDEs with Applications in Finance

We study a parallel implementation on a Graphics Processing Unit (GPU) of Alternating Direction Implicit (ADI) time-discretization methods for solving time-dependent parabolic Partial Differential Equations (PDEs) in three spatial dimensions with mixed spatial derivatives in a variety of applications in computational finance. Finite differences on uniform grids are used for the spatial discretization of the PDEs. As examples, we apply the GPU-based parallel methods to price European rainbow and European basket options, each written on three assets. Numerical results showing the efficiency of the parallel methods are provided.

Reduction of numerical errors in frequency dependent ADI-FDTD

A new method modifying the frequency dependent alternating direction implicit finite difference time domain (FD-ADI-FDTD) is presented. The method can improve the accuracy of FD-ADI-FDTD without significant increase of computational costs. The proposed method is validated by numerical tests.

Numerical Investigation of Conformal ADI-FDTD Schemes with Second-Order Convergence

This paper presents unconditionally stable and conformal FDTD schemes which are based on the alternating-direction implicit finite difference time domain (ADI-FDTD) method for accurate modeling of perfectly electric conducting (PEC) objects. The proposed schemes are formulated within the framework of the matrix-vector notation of the finite integration technique (FIT), which allows a systematic and consistent extension of finite difference solution of Maxwell's equations on dual grids. As possible choices of second-order convergent conformal method, we apply the partially filled cell (PFC) and the uniformly stable conformal (USC) schemes for the ADI-FDTD method. The unconditional stability and the rates of convergence of the proposed conformal ADI-FDTD (CADI-FDTD) schemes are verified by means of numerical examples of waveguide problems.

Three-Dimensionally Nonorthogonal Alternating-Direction Implicit Finite-Difference Time-Domain Algorithm for the Full-Wave Analysis of Microwave Monolithic Circuit Devices

An alternating-direction implicit finite-difference time-domain (FDTD) algorithm is applied to the full wave analysis of microwave integrated circuit devices. A 3-D multidomain method is developed in nonorthogonal coordinates. Nonorthogonal grids are only used for the anomalistic regions of a complex structure, whereas the standard FDTD lattice is used for the other regions. By using the Jacobian coordinate transformation, curvilinear coordinates can be converted into conventional FDTD format expediently. The perfectly matched layer is used to truncate the boundary. Accurate griddings using the new scheme are obtained, and the complexity of the algorithm is minimal. To illustrate the theory, a sinusoidal plane wave and a Gaussian pulse that propagate through a localized nonorthogonal grid space is used, and the stability of our code is examined. A newly developed compact microstrip bandpass filter is analyzed using the proposed method. The simulated results agree very well with measurements. As compared to other nonorthogonal FDTD (NFDTD) method, the proposed algorithm is much more efficient than other NFDTD counterpart when complex structures are analyzed.

Modeling credit spreads with the Cheyette model and its application to credit default swaptions

In this paper we apply Cheyette's Markov representation of the Heath-Jarrow-Morton framework to the modeling of stochastic credit spreads. As an application of this framework, the volatility of the credit spread process is modeled by considering the constant elasticity of variance approach of Ritchken and Sankarasubramanian and the Andersen-Andreasen displaced approach. To examine the practicability of this approach, we calibrate the model to market prices of credit default swaptions. Thereby we use Monte Carlo simulation and the alternating direction implicit finite-difference method.

High‐order accurate FDTD method based on split‐step scheme for solving Maxwell's equations

AbstractA new split‐step finite difference time domain (SS‐FDTD) method with high‐order accuracy is presented, which is proven to be unconditionally stable and has four substeps. The numerical dispersion error and the numerical anisotropic error of the proposed method are reduced than the alternating direction implicit finite difference time domain method, the conventional SS‐FDTD method and the SS‐FDTD method based on the Strang‐splitting scheme. The proposed method has new splitting forms, which is different from the SS‐FDTD method based on the exponential evolution operator. At each time step, an important aspect is that the proposed method produces 33% reduction of the total number of arithmetic operators than the SS‐FDTD method based on the exponential evolution operator. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 562–565, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24100

Mathematical simulation of dispersed phase dynamics for the nonisothermal free convection of a heterogeneous medium in a vertical cylindrical reactor

The motion of monodisperse spherical particles in a heterogeneous medium for the nonisothermal free convection of the carrying viscous incompressible liquid in a cylindrical reactor with nonuniform wall temperature profiles is mathematically simulated. The simulation is based on the simultaneous solution of the Navier-Stokes equations written in terms of Eulerian variables for the carrying liquid in the Boussinesq approximation and the equations written in terms of Lagrange variables for the dispersed particle. The system of Navier-Stokes equations is solved in terms of curl-stream function variables using the alternating direction implicit finite difference scheme. The monotone Samarskii’s approximation is used for the convective terms in the curl equation. The calculated results demonstrate the evolution of the curl, stream function, temperature, and particle trajectory.

Heat transfer of horizontal parallel pipe ground heat exchanger and experimental verification

The ground heat exchangers (GHE) consist of pipes buried in the soil and is used for transferring heat between the soil and the heat exchanger pipes of the ground source heat pump (GSHP). Because of the complexity of the boundary conditions, the heat conduction equation has been solved numerically using alternating direction implicit finite difference formulation. A software was developed in MATLAB environment and the effects of solution parameters on the results were investigated. An experimental study was carried out to test the validity of the model. An experimental GSHP system is installed at Yıldız Technical University Davupasa Campus on 800 m 2 surface area with no special surface cover. Temperature data were collected using thermocouples buried in soil horizontally and vertically at various distances from the pipe center and at the inlet and the outlet of the ground heat exchanger. Experimental and numerical simulation results calculated using experimental water inlet temperatures were compared. The maximum difference between the numerical results and the experimental data is 10.03%. The temperature distribution in the soil was calculated and compared with experimental data also. Both horizontal and vertical temperature profiles matched the experimental data well. Simulation results were compared with the other studies.

An accurate analysis of numerical dispersion for 3‐D ADI‐FDTD with artificial anisotropy

AbstractIn this article, an accurate numerical dispersion relationship is developed for 3‐D alternating direction implicit finite difference time domain (ADI FDTD) with artificial anisotropy. The numerical dispersion relation with accurate mathematical model helps to calculate the anisotropic parameters, which are used to control the error of the numerical phase velocity. Numerical example shows that the proposed method helps to reduce the numerical dispersion significantly. The effects of anisotropic parameters on the numerical dispersion with different mesh resolution, courant number, and cell dimension ratio, are shown for three‐dimensional anisotropic ADI‐FDTD. A comparison has been made for anisotropic ADI FTDD and conventional ADI FDTD in terms of accuracy and CPU time for the calculation of resonant frequencies of rectangular cavity. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 3109–3112, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22961