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The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin mixed finite element method based on Banach spaces for the stationary Navier–Stokes equation in pseudostress–velocity formulation. Specifically, a modified pseudostress tensor, which depends on the pressure as well as the diffusive and convective terms, is introduced as an auxiliary unknown, and the incompressibility condition is then used to eliminate the pressure, which is subsequently computed using a postprocessing formula. Consequently, to discretize the resulting mixed formulation, it is sufficient to provide a tensorial weak Galerkin space for the pseudostress and a space of piecewise polynomial vectors of total degree at most ’k’ for the velocity. Moreover, the weak gradient/divergence operator is utilized to propose the weak discrete bilinear forms, whose continuous version involves the classical gradient/divergence operators. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method’s good performance and confirming the theoretical rates of convergence are presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to implement the high-order local discontinuous Galerkin method (LDGM) for solving partial integro-differential equations (PIDEs) in two dimensions. Time marching method and the transform-based method known as the non-time marching method can be used to discretize temporal terms. The combination of a small time step size in time marching methods and a high-order scheme in space with many degrees of freedom requires a considerable amount of computational time. In order to address this limitation, we propose an algorithm based on a combination of the Laplace transform and LDGM for solving fourth-order time-fractional (TF) PIDEs with weakly singular kernels. Unlike time-marching approaches, the transform-based method has a much lower computational complexity and can take advantage of parallel computing. A numerical experiment validates the accuracy and applicability of the proposed temporal discretization approach and also shows that k-degree LDG solutions have a k + 1 convergence rate. [ABSTRACT FROM AUTHOR]

Coupled linear matrix equations arise in many fields of engineering, including control theory, system identification, and signal processing. In this manuscript, we propose an iterative algorithm based on the generalized shift-splittings and using the hierarchical identification principle (GSSHIP) approach for efficiently solving coupled Sylvester matrix equations. The efficacy of the iterative methods presented in this study has been demonstrated through the use of some numerical examples. Finally, we will explore the practical implementation of the proposed approach utilizing Newton's method to solve continuous-time algebraic Riccati matrix equations. [ABSTRACT FROM AUTHOR]

Optoelectronic oscillators are essential tools for generating pure microwave and millimeter-wavelength signals used in high-speed telecommunications, signal processing, radar, and related domains. This research investigates the numerical solution of the retarded optoelectronic oscillator model, described by delay Volterra integro-differential equations. We present a technique by employing the discrete collocation method and locally supported radial basis functions constructed on a set of scattered points. This scheme has the potential to achieve a accurate approximate solution by solving several smaller systems rather than one large system, resulting in a minimization of computational volume compared to its global counterpart. The discretization process is accomplished using the composite Gauss–Legendre quadrature formula. We estimate the error bound and convergence rate of our proposed scheme, supported by simulations that showcase the method's reliability and efficiency. Additionally, we conducted a comparison of the computational efficiency between the offered method and its global counterpart, confirming that the introduced method is better than its global equivalent. The numerical results obtained confirm its alignment with the theoretical error estimates. [ABSTRACT FROM AUTHOR]

In this paper, the cubic–quintic complex Ginzburg–Landau (CQCGL) equation is numerically studied in 1D, 2D and 3D spaces. First, by the Strang splitting technique, the CQCGL equation is decomposed into three subproblems. After spatial discretization, the first and third problems lead to nonlinear ODEs that are solved by the Runge–Kutta technique. For the second problem, involving the spatial derivative, a fourth-order RBF—generated Hermite finite difference (RBF–HFD) method is used. The RBF–HFD scheme improves the accuracy in space direction compared with RBF-FD method, but temporal convergence order remains two due to the use of Strang splitting and Crank–Nicolson schemes in time direction. To improve the order of convergence in time direction, a temporal Richardson extrapolation technique is applied. Numerical results show that the order of convergence is improved from O (τ 2 + h 4) to O (τ 4 + h 4) . Numerical tests are provided to confirm the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson–Nernst–Planck (PNP) and Navier–Stokes (NS) equations. These equations model microfluidic and electrochemical systems that include the diffuse transport of charged species within incompressible fluids coupled through electrostatic forces. A mixed VEM is employed to discretize the NS equations whereas classical VEM in primal form is used to discretize the PNP equations. The stability, existence and uniqueness of solution of the associated VEM are proved by fixed point theory. The global mass conservation and electric energy decay of the scheme are also established. Also, we rigorously derive unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the L 2 and H 1 -norms, as well as for the velocity and pressure of NS equations in the L 2 , H 1 - and L 2 -norms, respectively. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the satisfactory performance of the method in simulating the onset of electrokinetic instabilities in ionic fluids, and studying how they are influenced by different values of ion concentration and applied voltage. These tests are relevant in applications of water desalination. [ABSTRACT FROM AUTHOR]

Delay integral equations can be used to model a large variety of phenomena more realistically by intervening in the history of processes. Indeed, the past exerts its influences on the present and, hence, on the future of these models. This paper presents a numerical method for solving nonlinear Volterra integral equations of the second kind with delay arguments. The method uses the discrete collocation approach together with thin plate splines as a type of free-shape parameter radial basis functions. Therefore, the offered scheme establishes an effective and stable algorithm to estimate the solution, which can be easily implemented on a personal computer with normal specifications. We employ the composite Gauss–Legendre integration rule to estimate all integrals that appeared in the method. The error analysis of the presented method is provided. The convergence validity of the new technique is examined over several nonlinear delay integral equations, and obtained results confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

In this research, a radial basis functions weighted essentially non-oscillatory (RBF-WENO) scheme in the framework of finite difference is developed for solving non-linear degenerate parabolic (NDP) equations which may contain discontinuous solutions. The traditional WENO schemes for solving NDP equations are based on the polynomial interpolation. In this paper, a non-polynomial WENO finite difference scheme in order to enhance the local accuracy is proposed. For a non-polynomial interpolation basis, the infinitely smooth RBFs such as the multi-quadratic are adopted. By optimizing the free parameter in RBFs, the local error generated by the RBFs interpolation can be easily controlled. The formulation of scheme will be described in detail and numerical tests have been prepared to demonstrate the efficiency of the scheme in one and two-dimension. The numerical results demonstrate that the developed non-polynomial WENO schemes enhance the local accuracy and give sharper solution profile than the original WENO based on the polynomial interpolation for solving NDP equations. [ABSTRACT FROM AUTHOR]

Awareness of the possible velocity and load of dust particles carried by the wind could be beneficial for preventive measures and preparation for health, transportation, military, and other damages of this atmospheric phenomenon. Aerosol optical depth (AOD) is closely related to visibility (V), and both of these parameters are so important and useful to studies connected to aerosols, tropospheric dust, air pollution, and atmospheric radiation budget. This research analyzed the mathematical relation between the aforementioned parameters and wind speed (WS) as a key index in storms by using evolutional genetic algorithm (GA). The study area is Yazd Province as representative of central Iran for 20 years (2000–2019). In fact, the purpose of this analysis has been to reach a relation that can estimate unavailable quantities of mentioned data parameters based on the available data of other existing parameters with the least error. To attain this mathematic equation, the linear regression and several other well-known functions have been compared. Meanwhile, the binomial function was selected as the best fitness function. The result was a mathematical equation between AOD parameters, visibility, and wind speed during the occurrence of 30–35 synoptic codes based on a binomial function with the confidence level of 95% was as follows: WS = 19.115 − 0/0033 × V + 1.2 × AOD. The equation presented in this research is an experimental equation that is presented for the first time and has the ability to be used and to calculate the mentioned parameters throughout central Iran and areas similar to the environmental conditions of central Iran. [ABSTRACT FROM AUTHOR]

The current work concerns to develop a new local meshless procedure to simulate the one-, two- and three-dimensional space Galilei invariant fractional advection-diffusion (GI-FAD) equations. The fractional derivative is discretized by a second-order finite difference formula. The unconditional stability and rate of convergence of the time-discrete scheme are analytically studied. Then, we employ the shape functions of moving Kriging interpolation for the differential quadrature (DQ) method. According to this combination, we can derive a new local meshless technique. At the end, some examples are solved to confirm the theoretical results and efficiency of the proposed meshless method. [ABSTRACT FROM AUTHOR]

This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. Then we prove that the optimal parameters of the new algorithm satisfy a constrained optimization problem. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study and compared the results by some existing algorithms. [ABSTRACT FROM AUTHOR]

Visible light communication (VLC) network is an appropriate solution to meet the 5G and 6G. It can overcome the spectrum scarcity and capacity in the radio frequency (RF) networks. This network faces some challenges. Obstacles can break links and disconnect the network. Mobile nodes cannot receive data continuously because the transmitters have few fields of view. Using common MAC solutions in wireless networks does not solve VLC network problems. VLC MAC solutions should solve deafness, hidden node, full-duplex capability, channel utilization, and connectivity. This paper provided a survey in MAC layer solutions in VLC networks based on the systematic literature review (SLR). The methods have been recently published from 2011 to 2021. We analyzed analytically and statistically and the technical taxonomy presented with the SLR process according to the proposed solutions. Solutions were categorized into half-duplex, full-duplex, cooperative, and VLC/RF categories. Also, other proposed methods were described briefly. We explained the features, advantages, and disadvantages of each method. MAC layer design challenges in VLC networks were discussed to fill the gap in the previous literature. By classification and problem analysis, fewer attention topics that need more research were extracted. We found that fairness and reliability metrics received less attention and dynamic resource allocation was the main context in the VLC MAC solutions. Finally, future research challenges and open issues in VLC MAC solutions were proposed. [ABSTRACT FROM AUTHOR]

Todays by equipping vehicles with wireless technologies, Vehicular Ad Hoc Network (VANET) has been emerged. This type of network can be utilized in many fields such as emergency, safety or entertainment. It is also considered as a main component of intelligent transportation system. However, due to the nodes velocity (vehicles velocity), varying density, obstacles and lack of fixed infrastructure, finding and maintaining a route between nodes are always challenging in VANET. Any routing protocol can be effective only if the nodes can learn and adapt themselves with such a dynamic environment. One way to achieve this adaptation is using machine learning techniques. In this paper we try to reach this goal by applying Multi-Agent Reinforcement Learning (MARL) that enables agents to solve routing optimization problems in a distributed way. Although model-free Reinforcement Learning (RL) schemes are introduced for this purpose, such techniques learn using a trial and error scheme in a real environment so they cannot reach an optimal policy in a short time. To deal with such a problem, we have proposed a mode-based RL based routing scheme. We have also developed a Fuzzy Logic (FL) system to evaluate the quality of links between neighbor nodes based on parameters such as velocity and connection quality. Outputs of this fuzzy system have been used to form the state transition model, needed in MARL. Results of evaluations have shown that our approach can improve some routing metrics like delivery ratio, end-to-end delay and traffic overhead. [ABSTRACT FROM AUTHOR]

Generalized preconditioned Hermitian and skew-Hermitian splitting (GPHSS) method is a numerical method for obtaining the solution of linear systems with complex symmetric semi-positive definite coefficient matrix. This method relates to the two relaxed parameters that should be chosen properly and is a hard task mathematically. In this study, based on the work in [J. Comput. Appl. Math. 255, 142–149 (2014)] that investigates the optimum parameter for the HSS method, we expand the results of this paper for computing optimum parameters for GPHSS method. We will show that these parameters are obtained by minimizing a function of two variables. Finally in the numerical section, we study some test problems to support the theoretical discussion. [ABSTRACT FROM AUTHOR]

With the expansion in the use of IoT, increasing the efficiency of these networks has become even more significant. Objects need reliable communications at suitable speed to be able to reach the expected performance. In a heterogeneous network of IoT, the objects can include users and related devices. Despite the hybrid Li-Fi and Wi-Fi networks, IoT needs were somewhat met. However, the efficiency of these networks depends on the choice of access points at the network level. In this study, a new algorithm was proposed based on the access point selection model, considering the strategic behavior of the objects and users. In this algorithm, inspired by access point selection model, a new adaptive algorithm was selected by choosing the access point by Markov game to enhance the load balancing and efficiency in IoT networks based on Wi-Fi and Li-Fi combination. According to the simulation results, it is seen that the proposed method could greatly increase the efficiency of IoT network and better distributes network load between access points with different technology. Overall network throughput was estimated at an average of at least 10% compared to the load-balancing method with fuzzy logic approach proposed in before research. [ABSTRACT FROM AUTHOR]

The virtual element method (VEM) is a recent technology that can make use of very general polygonal/polyhedral meshes without the need to integrate complex nonpolynomial functions on the elements and preserving an optimal order of convergence. In this article, the VEM is formulated and analyzed to solve the Brusselator model on polygonal meshes. Also an optimal a priori error estimate (under a small data assumption) is derived. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed scheme and to plot the Turing patterns of the Brusselator equation on a set of different computational meshes. [ABSTRACT FROM AUTHOR]

The shallow water equation based on the generalized equal width (GEW) equation is a PDE that can be classified in the category of hyperbolic PDEs. The current paper concerns developing an efficient and robust numerical technique to solve the shallow water equation based on the generalized equal width model. For this aim, first, the space derivative is approximated by the local collocation method via a compact integrated radial basis function. Second, the time derivative is approximated by the method of lines that yields a system of nonlinear ODEs. Furthermore, the constructed system of ODEs is solved by a fourth-order algorithm to get high-numerical results. Also, an adaptive collocation method based upon IRBF is presented for the solution of GEW. The effectiveness of our adaptive technique is shown in four examples such as interpolation, the motion of single solitary, two solitary waves and Maxwellian initial condition. It is explained in detail how the adaptive method refines the resolution. [ABSTRACT FROM AUTHOR]

Many works on solving matrix equations are related to the complex nonlinear Yang–Baxter matrix equation A X A = X A X , where A ∈ C n × n is a given matrix and X is an unknown matrix. The Yang–Baxter matrix equation has been widely studied by its application in various fields of mathematics and physics. In this paper, we introduce an iterative method based on the Hermitian and skew-Hermitian splitting of coefficient matrix A for solving complex nonlinear Yang–Baxter matrix equation. Then, we prove the convergence of the new scheme subject to some conditions. Finally, an example is solved to discover the applicability of the new method via comparing it with some related previous methods. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The temporal variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O (τ 3 2) . Also, the space variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique. [ABSTRACT FROM AUTHOR]

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to present new and simple numerical methods for solving the time-dependent transport equation on the sphere in spherical coordinates. We use two techniques, namely generalized moving least squares and moving kriging least squares to find the new formulations for approximating the advection operator in spherical coordinates such that any singularities have been omitted. Another feature of the methods considered here is that they do not depend on the background mesh or triangulation for approximation, which they could be applied on the transport equation in spherical coordinates easily with different distribution points. Furthermore, due to the eigenvalue stability of the dicretized advection operator via two proposed approximations, an implicit-explicit linear multistep method has been applied to discretize the time variable. The fully discrete scheme obtained here yields a linear system of algebraic equations at each time step, which is solved via the biconjugate gradient stabilized algorithm with zero-fill incomplete lower-upper (ILU) preconditioner. Three well-known test problems, namely "solid body rotation", "vortex roll-up" and "deformational flow" are solved to demonstrate our developments. Also for the first test problem, we apply a simple positivity-preserving filter at the end of each time step, which keeps the transported variable positive. [ABSTRACT FROM AUTHOR]

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order O (τ 2 - α) as 0 < α < 1 . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to analysis of a novel meshless numerical procedure for solving the inverse tempered fractional diffusion equation. The employed numerical technique is based on a modification of element-free Galerkin (EFG) method, and the shape functions of interpolating moving least squares approximation are utilized for ingredients of the test and trial functions. At the first stage, the time derivative is discretized by a Crank–Nicolson idea to derive a semi-discrete scheme. In the next stage, the space variable is approximated by the EFG procedure. The convergence rate and stability of the time-discrete formulation are analyzed. Furthermore, the error estimate of the full-discrete plan is discussed in detail. In the end, some numerical experiments are investigated to check the theoretical results and the efficiency of the developed technique. [ABSTRACT FROM AUTHOR]

The current paper proposes an efficient numerical procedure for solving the two-dimensional Brusselator reaction–diffusion system. First, the time derivative is discretized by a semi-implicit finite difference scheme such that the nonlinear terms are approximated by using one term of the Taylor expansion. It is shown if the nonlinear terms satisfy the Lipshitz condition, then the proposed difference scheme is stable. In the next, the spatial derivative is discretized by the differential quadrature technique based upon the shape functions of radial point interpolation method. We study some practical problems to show the efficiency and acceptable results of the developed technique. [ABSTRACT FROM AUTHOR]

Accelerated double-step scale splitting (ADSS) approach is an efficient and fast method for solving a class of large complex symmetric linear equations which has been recently presented by parameterized DSS method. In this paper, we will apply ADSS scheme for solving complex Sylvester matrix equation. It will be proved analytically that the ADSS iteration method is faster than the DSS iteration method. Moreover, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and results will be reported to support the theoretical claims. [ABSTRACT FROM AUTHOR]

In this paper, a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. The used numerical method is based on the integrated radial basis functions (IRBFs). First, the time derivative has been approximated using a finite difference scheme. Then, the IRBF method is developed to approximate the spatial derivatives. The two-dimensional version of these equations is solved using the presented method on different computational geometries such as the rectangular, triangular, circular and butterfly domains and also other irregular regions. The aim of this paper is to show that the integrated radial basis function method is also suitable for solving nonlinear partial differential equations. Numerical examples confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The numerical simulation of the tumor-induced angiogenesis process is an useful tool for the prediction of this mechanism and drug targeting using anti-angiogenesis strategy. In the current paper, we study numerically on the continuous mathematical model of tumor-induced angiogenesis in two-dimensional spaces. The studied model is a system of nonlinear time-dependent partial differential equations, which describes the interactions between endothelial cell, tumor angiogenesis factor and fibronectin. We first derive the global weak form of the model and discretize the time variable via a semi-implicit backward Euler method. To approximate the spatial variables of the studied model, we use a meshless technique, namely element-free Galerkin. Also, the shape functions of moving least square and moving Kriging approximations are used in this method. The main difference between two meshless methods proposed here is that the shape functions of moving least squares approximation do not satisfy Kroncker's delta property, while moving Kriging technique satisfies this property. Also, both techniques do not require the generation of a mesh for approximation, but a background mesh is needed to compute the numerical integrations, which are appeared in the derived global weak form. The full-discrete scheme obtained here gives the linear system of algebraic equations that is solved via an iterative method, namely biconjugate gradient stabilized with zero-fill incomplete lower upper (ILU) preconditioner. Some numerical simulations are provided to illustrate the ability of the presented numerical methods, which show the endothelial cell migration in response to the tumor angiogenesis factors during angiogenesis process as well. [ABSTRACT FROM AUTHOR]

With the recent advances in robotic technologies, field coverage using mobile sensors is now possible, so that a small set of sensors can be mounted on mobile robots and move to desired areas. Compared to static settings, area coverage is more complicated in a mobile sensor network due to the dynamics arising from the continuous movement of the sensors. This complication is even higher in the more realistic case where little or no prior metric information is available about the sensor field. In this paper, we consider the problem of self-deployment of a set of mobile sensors which have no knowledge of the area, the number of nodes, their location, and even the distances to each other. In this restricted setting, we formulate the problem as a multi-player game in which each sensor tries to maximize its coverage while considering the overlapping sensing areas by its neighbors. We propose a distributed learning algorithm for coordinating the movement of the sensors in the field, and prove its convergence to the equilibria of the formulated game. Simulation results demonstrate that for moderate density deployments, the proposed algorithm competes with the existing location-dependent mobility strategies, while outperforming location-free algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study an accelerated double-step scale splitting (ADSS) iteration method for solving complex linear systems. The convergence of the ADSS iteration method is determined under suitable conditions. Also each iteration of ADSS method requires the solution of two linear systems that their coefficient matrices are real symmetric positive definite. We analytically prove that the ADSS iteration method is faster than the DSS iteration method. Moreover, to increase the convergence rate of this method, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and simulation results will be reported to support the theoretical results. [ABSTRACT FROM AUTHOR]

We are proposing graphene (G)-based multilayered plasmonic spatial switch, operating at 10 THz. It is composed of hBN/Ag/hBN/G/hBN/G/hBN/SiO2/p+-Si multilayers. When a 10-THz transverse magnetic (TM)-polarized signal is normally incident upon the structure top surface, the nanoaperture devised in the Ag nanolayer, acting as a grating, excites surface plasmons at the top graphene micro-ribbons/hBN interface. These surface plasmons depending on the graphenes chemical potentials can be coupled to the lower-right or left graphene micro-ribbons and continue to propagate laterally towards the corresponding output port. Numerical simulations show that a change of ∆VG ≈ ± 2.7 V in the voltage, applied to the gated micro-ribbons, can modulate their chemical potentials sufficiently to switch the right (left) output port from ON (OFF) to OFF(ON) and vice versa. Besides its low power consumption, the switch ultra-small dimensions make it a potential spatial router suitable for THz-integrated circuit applications. [ABSTRACT FROM AUTHOR]

In this paper, we want to present a fast, efficient, and robust numerical procedure for solving a system of PDEs with regard to the fractional Laplacian equation. In the developed approximate scheme, the spatial direction is discretized by a second-order finite difference formula and the temporal direction is discretized by a first-order finite difference approximation. In order to improve the accuracy and to decrease the used CPU time an alternative direction implicit (ADI) method is employed. Furthermore, we use a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique to reduce the elapsed computational time. The developed numerical scheme is well known as the reduced order finite difference scheme. To emphasize the fast and efficiency of the proposed algorithm, we apply it for the two-dimensional case. [ABSTRACT FROM AUTHOR]