Alternating Direction Explicit Method Research Articles

Fast operator splitting methods for obstacle problems

The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator–splitting methods to solve the linear and nonlinear obstacle problems. The proposed methods have three ingredients: (i) Utilize an indicator function to formularize the constrained problem as an unconstrained problem, and associate it to an initial value problem. The obstacle problem is then converted to solving for the steady state solution of an initial value problem. (ii) An operator–splitting strategy to time discretize the initial value problem. After splitting, a heat equation with obstacles is solved and other subproblems either have explicit solutions or can be solved efficiently. (iii) A new constrained alternating direction explicit method, a fully explicit method, to solve heat equations with obstacles. The proposed methods are easy to implement, do not require to solve any linear systems and are more efficient than existing numerical methods while keeping similar accuracy. Extensions of the proposed methods to related free boundary problems are also discussed.

An explicit stable finite difference method for the Allen–Cahn equation

We propose an explicit stable finite difference method (FDM) for the Allen–Cahn (AC) equation. The AC equation has been widely used for modeling various phenomena such as mean curvature flow, image processing, crystal growth, interfacial dynamics in material science, and so on. For practical use, an explicit method can be applied for the numerical approximation of the AC equation. However, there is a strict restriction on the time step size. To mitigate the disadvantage, we adopt the alternating direction explicit method for the diffusion term of the AC equation. As a result, we can use a relatively larger time step size than when the explicit method is used. Numerical experiments are performed to demonstrate that the proposed scheme preserves the intrinsic properties of the AC equation and it is stable compared to the explicit method.

Fast Operator Splitting Methods for Obstacle Problems

The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator-splitting methods to solve the linear and nonlinear obstacle problems. The proposed methods have three ingredients: (i) Utilize an indicator function to formularize the constrained problem as an unconstrained problem, and associate it to an initial value problem. The obstacle problem is then converted to solving for the steady state solution of an initial value problem. (ii) An operator-splitting strategy to time discretize the initial value problem. After splitting, a heat equation with obstacles is solved and other subproblems either have explicit solutions or can be solved efficiently. (iii) A new constrained alternating direction explicit method, a fully explicit method, to solve heat equations with obstacles. The proposed methods are easy to implement, do not require to solve any linear systems and are more efficient than existing numerical methods while keeping similar accuracy. Extensions of the proposed methods to related free boundary problems are also discussed.

Alternating Direction Explicit Method for a Nonlinear Model in Finance

In this article, at first standard linear Black-Scholes model and then some nonlinear Black-Scholes models will be considered and thereupon alternating direction explicit (ADE) method is applied firstly for solving the standard Black-Scholes model and then for Barles and Soner model which is one of the most complete and comprehensive nonlinear Black-Scholes models. Furthermore, the stability of this method has been considered and its accuracy will be compared with other numerical methods such as finite difference methods. Since in solving nonlinear Black-Scholes models by the ADE methods, we need to solve only some scalar nonlinear equations instead of a full nonlinear system of equations that we should solve in implicit methods, so this method can be a suitable choice for solving such models.

Reconstructing the Time-Dependent Thermal Coefficient in 2D Free Boundary Problems

The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefficients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and additional measurements is, for the first time, numerically investigated. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For instance, steel annealing, vacuum-arc welding, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, we apply the alternating direction explicit method along with the Tikhonov regularization to find a stable and accurate numerical solution of finite differences. The root mean square error (rmse) values for various noise levels p for both smooth and non-smooth continuous time-dependent coefficients Examples are compared. The resulting nonlinear minimization problem is solved numerically using the MATLAB subroutine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.

Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions

ABSTRACT The aim of this paper is to identify numerically the timewise thermal conductivity coefficients in the two-dimensional heat equation in a rectangular domain using initial and Dirichlet boundary conditions and the local heat flux as over-specification conditions. The measurement data represented by the local heat flux is shown to ensure the unique solvability of the inverse problem solution. The two-dimensional inverse problem is discretized using an alternating direction explicit method. The resulting constrained optimization problem is minimized iteratively by employing the MATLAB subroutine. Exact and noisy input data are inverted numerically. The root mean square error values for various noise levels p for both smooth and non-smooth continuous timewise thermal conductivity coefficients Examples are compared. Numerical results are presented and discussed in order to illustrate the performance of the inversion for thermal conductivity components identification. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficient identification problems with applications in heat transfer and porous media.

Numerical Treatment of a Two-Dimensional Vertically-Averaged Groundwater Flow Model Using Alternating Direction Explicit Methods

Leachate from a landfill can flow down and contaminate groundwater. Mathematical models are often used to describe groundwater flow, which can help designers to identify an appropriate location for a landfill area. We focused on simulation of hydraulic head of groundwater under landfill construction in a rural area. We used a partial differential equation model called two-dimensional vertically-averaged groundwater flow model to explain groundwater volume, in particular, to explain the hydraulic head of the groundwater which strongly affected the groundwater quality. We applied two alternating direction explicit methods to the groundwater flow model under general boundary conditions and rates of change of boundary conditions to obtain an approximate hydraulic head of groundwater. An alternating direction explicit method for all types of boundaries is used to find approximate solutions. The traditional alternative direction explicit method was modified to form our modified alternative direction explicit method. The results of all method are closed. The traditional alternative direction explicit method and the modified alternative direction explicit method led to more accurate solutions than forward time centered space in heterogeneous aquifer, two methods ensured that a stable solution of all interior points can be obtained by using a fine grid spacing.

Groundwater-quality Assessment Models with Total Nitrogen Transformation Effects

Nitrogen is emitted extensively by industrial companies, increasing nitrogen compounds such as ammonia, nitrate, and nitrite in soil and water as a result of nitrogen cycle reactions. Groundwater contamination with nitrates and nitrites impacts human health. Mathematical models can explain groundwater contamination with nitrates and nitrites. Hydraulic head model provides the hydraulic head of groundwater. Groundwater velocity model provided x- and y- direction vector in groundwater. Groundwater contamination distribution model provides nitrogen, nitrate and nitrite concentration. Finite difference techniques are approximate the models solution. Alternating direction explicit method was used to clarify hydraulic head model. Centered space explained groundwater velocity model. Forward time central space was used to predict groundwater transportation of contamination models. We simulate different circumstances to explain the pollution in leachate water underground, paying attention to the toxic nitrogen, ammonia, nitrate, nitrite blended in the water.

Reconstruction of the timewise conductivity using a linear combination of heat flux measurements

The reconstruction of the timewise conductivity in the heat equation from an observation consisting of a linear combination of heat flux measurement data is considered. This inverse formulation results in a local uniquely solvable problem. The two-dimensional inverse problem is discretized using an alternating direction explicit method. The resulting constrained optimization problem is minimized iteratively by employing a MATLAB toolbox subroutine.

An alternating direction explicit method for time evolution equations with applications to fractional differential equations

We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.

Non symmetric alternating direction explicit method applied to the calculation of ADS transients

ADS systems – Accelerator Driven Systems, along with Generation IV reactors, have been receiving great attention, especially due to their capability of transmutation and eventually incineration of nuclear waste. To study ADS safety aspects, kinetic models are used to analyze transients. Currently, there are quite a few studies involving transient in ADS systems. Furthermore, most existing studies make use of the Point Kinetics equations. In this context, this work presents a proposal to implement a space dependent kinetics formalism, the Non-Symmetric Alternating Direction Explicit Method, to analyze ADS systems under transient conditions. The computer code developed in this study has satisfactorily reproduced results in the literature of the transients tested. The results are in accordance to what should be expected from the physical point of view.

Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.

Parallelization of 2-D IADE-DY Scheme on Geranium Cadcam Cluster for Heat Equation

A parallel implementation of the Iterative Alternating Direction Explicit method by D’Yakonov (IADE-DY) for solving 2-D heat equation on a distributed system of Geranium Cadcam cluster (GCC) using the Message Passing Interface (MPI) is presented. The implementation of the scheduling of n tri-diagonal system of equations with the above method was used to show improvement on speedup, effectiveness, and efficiency. The Master/Worker paradigm and Single Program Multiple Data (SPMD) model is employed to manage the whole computation based on the use of domain decomposition. The completion of the execution can need task recovery and favorable configuration. The above mentioned details consist of a main report about the numerical validation of the parallelization through simulation to demonstrate the proposed method effectiveness on the cluster system. It was found that the rate of convergence decreases as the number of processors increases. The result of this paper suggests that the 2-D IADE-DY scheme is a good approach to solving problems, particularly when it is simulation with more processors.

Armadillo generation distributed system with geranium Cadcam cluster for solving 2D telegraphic problem

In this paper, the parallel implementation of the iterative alternating direction explicit method by D'Yakonov (IADE-DY) and the Mitchell and Fairweather double sweep method (DS-MF) compared with the alternative direction implicit (ADI) method for solving the 2D telegraphic problem on a distributed system of Armadillo cluster and geranium Cadcam cluster using the parallel virtual machine and message passing interface is presented. We implement the scheduling of n tridiagonal system of equations with the above methods to show improvement on speedup and efficiency with parallel strategies across the two platforms. The single program multiple data model was employed for the implementation. The IADE-DY, DS-MF and ADI are developed by the splitting of the implicit equation using the finite difference discretization on the 2D telegraphic problem. The implementation is discussed in relation to means of the performance parallel strategies and analysis. The model enhances overlap communication and computation to avoid unnecessary synchronization, hence, the method yields significant speedup by the use of the non-blocking communication. We present some analyses that are helpful for speedup and efficiency.

Parallel Implementation of 2-D Telegraphic Equation on MPI/PVM Cluster

In this paper, a parallel implementation of the Iterative Alternating Direction Explicit method by D’Yakonov (IADE-DY) to solve 2-D telegraphic problem on a distributed system using Message Passing Interface (MPI) and Parallel Virtue Machine (PVM) are presented. The parallelization of the program is implemented by a domain decomposition strategy. A Single Program Multiple Data (SPMD) model is employed for the implementation. The implementation is discussed in relation to means of the parallel performance strategies and analysis. The model enhances overlap communication and computation to avoid unnecessary synchronization, hence, the method yields significant speedup. The level of speedup observed from tables as the mesh increases are in the range of 5–10%. Improvement has been achieved by numbers of tables and figures in our experiment. We present some analyses that are helpful for speedup and efficiency. It is concluded that the efficiency is strongly dependent on the grid size, block numbers and the number of processors for both MPI and PVM. Different strategies to improve the computational efficiency are proposed.

Fast simulation code for heating, phase changes and dopant diffusion in silicon laser processing using the alternating direction explicit (ADE) method

Enhancements to our existing finite-differences code for the simulation of laser heating, melting and evaporation of silicon are presented. The Knudsen evaporation model has been added to the previously used enthalpy-based model in order to simulate laser pulses with pulse lengths down to a few nanoseconds. Fick’s diffusion law has also been incorporated allowing laser doping by dopant diffusion in silicon melt to be described. Finally, the basic equations for the alternating direction explicit method (ADE) have been adapted to consider nonlinear temperature-enthalpy relations, thus including affects due to phase changes. This improved the simulation speed by up to factor of 100 compared to standard explicit and implicit time integration methods. Details of the ADE algorithm and numerical stability issues are presented in this paper. Validation of the code is presented by comparing to different integration methods and to experimental results. The final code successfully simulates melting, evaporation and dopant diffusion by multiple laser pulses in three dimensions in an acceptable computing time.

Numerical methods for conjunctive two-dimensional surface and three-dimensional sub-surface flows

SUMMARY Sophisticated catchment runoff problems necessitate conjunctive modeling of overland flow and subsurface flow. In this paper, finite difference numerical methods are studied for simulation of catchment runoff of two-dimensional surface flow interacting with three-dimensional unsaturated and saturated sub-surface flows. The equations representing the flows are mathematically classified as a type of heat diffusion equation. Therefore, two- and three-dimensional numerical methods for heat diffusion equations were investigated for applications to the surface and sub-surface flow sub-models in terms of accuracy, stability, and calculation time. The methods are the purely explicit method, Saul’yev’s methods, the alternating direction explicit (ADE) methods, and the alternating direction implicit (ADI) methods. The methods are first examined on surface and sub-surface flows separately; subsequently, 12 selected combinations of methods were investigated for modeling the conjunctive flows. Saul’yev’s downstream (S-d) method was found to be the preferred method for two-dimensional surface flow modeling, whereas the ADE method of Barakat and Clark is a less accurate, stable alternative. For the three-dimensional sub-surface flow model, the ADE method of Larkin (ADE-L) and Brian’s ADI method are unconditionally stable and more accurate than the other methods. The calculations of the conjunctive models utilizing the S-d surface flow sub-model give excellent results and confirm the expectation that the errors of the surface and sub-surface sub-models interact. The surface sub-model dominates the accuracy and stability of the conjunctive model, whereas the sub-surface sub-model dominates the calculation time, suggesting the desirability of using a smaller time increment for the surface sub-model. Copyright © 2000 John Wiley & Sons, Ltd.

Solving the Navier–Stokes Systems with Weak Viscosity and Strong Heat Conduction Using the Flux-Corrected Transport Technique and the Alternating-Directional Explicit Method

The flux-corrected transport (FCT) technique and the alternating-direction explicit (ADE) method are coupled through a time-splitting technique. This new combination of both methods has been used successfully to solve the fully coupled Navier–Stokes system applied to ionospheric thermal plasma flows with a viscosity and strong heat conduction. The combined scheme gives convergent solutions within the time step set for nonlinear stability of the corresponding nondissipative flow fields, and the time-dependent solutions are consistent with other model results using different methods. To have a quantitative view of the flux-limiter of Boris' FCT version, a concept of local variation is defined to identify local extrema. The total variation diminishing scheme finds unique entropy solutions for vanishing dissipation. The ADE scheme, however, enables us to handle dissipation when the FCT technique alone can be inappropriate.

Performance Comparison of Overland Flow Algorithms

Two regional models, the South Florida Water Management Model (SFWMM) and the Natural System Model (NSM), are applied to analyze and predict water conditions in the Everglades and South Florida. Both of these models use an alternating direction explicit (ADE) type method to solve diffusion flow. In this paper, three finite-difference algorithms based on explicit, alternating direction implicit (ADI) and successive over relaxation (SOR) methods are examined as possible replacements for the ADE method. Various model solutions are verified using an axisymmetric test problem that is solved using an axisymmetric test model. A comparison of run time versus error plots proved that the ADI method has the best overall performance. The study includes a description of the relationship between the accuracies and run times of different algorithms and their spatial and temporal discretizations in dimensionless form. Linear and spectral analyses are used to derive theoretical expressions for numerical error, run time, and stability. Comparisons indicate that theoretical estimates of numerical error and run times closely approximate the experimental values. Results of this study are valuable as methods to determine optimum space and time discretizations of future modeling applications when the maximum allowable numerical error and the dimensions of the flow features to be simulated are known. Results can also be used to understand the magnitudes of numerical errors in existing modeling applications.

Curing simulation of thick thermosetting composites

A one-dimensional through-the-thickness simulation is developed in which heat conduction, kinetic, viscosity and flow (squeezed sponge model) equations are solved as a coupled system of equations. The control-volume method combined with the alternating direction explicit method is employed in the solution. Heat capacity, density, thermal conductivity and permeability of the composite are considered as functions of fibre volume fraction. It is shown that either temperature overshoot at the middle of the composite or incomplete through-the-thickness consolidation can put a restriction on the application of each cure cycle up to a certain thickness. Although the recommended cure cycles for a thick composite can satisfy the temperature conditions, complete through-the-thickness consolidation mostly cannot be achieved. Pressure effect, bleeding from the top and bottom and prebleeding are studied as possible solutions for incomplete consolidation. It seems that the prebleeding technique is the most promising method of fabrication for thick composites.