Standard Finite Difference Method Research Articles

Numerical solutions for nonlinear partial differential equations arising from modelling dye-sensitized solar cells

Dye-sensitized solar cells have generated diverse research directions, which include a mathematical model based on the diffusion of electrons in the conduction band of a nano-porous semiconductor (traditionally TiO\(_2\)). We solve the nonlinear diffusion equation under its boundary conditions, as stated by Anta et al. [J. Phys. Chem. B 110 (2006) pp 5372--5378]. We employ a standard finite difference method, a fourth order finite difference method scheme and a Runge--Kutta scheme. We calculate errors and evaluate the utility of each scheme as it applies to this boundary value problem. 
 
 References J. A. Anta, F. Casanueva, and G. Oskam. A numerical model for charge transport and recombination in dye-sensitized solar cells. J. Phys. Chem. B, 110(11):5372–5378, 2006. doi:10.1021/jp056493h. F. Cao, G. Oskam, G. J. Meyer, and P. C. Searson. Electron transport in porous nanocrystalline TiO\(_2\) photoelectrochemical cells. J. Phys. Chem., 100(42):17021–17027, 1996. doi:10.1021/jp9616573. A. J. Frank, N. Kopidakis, and J. van de Lagemaat. Electrons in nanostructured TiO\(_2\) solar cells: transport, recombination and photovoltaic properties. Coordin. Chem. Rev., 248:1165–1179, 2004. doi:10.1016/j.ccr.2004.03.015. Y. Gacemi, A. Cheknane, and H. S. Hilal. Simulation and modelling of charge transport in dye-sensitized solar cells based on carbon nano-tube electrodes. Phys. Scripta, 87(3):035703–035714, 2013. doi:10.1088/0031-8949/87/03/035703. B. O'Regan and M. Gratzel. A low-cost, high-efficiency solar cell based on dye-sensitized colloidal TiO\(_2\) films. Nature, 353:737–740, 1991. doi:10.1038/353737a0. S. Sodergren, A. Hagfeldt, J. Olsson, and S. Lindquist. Theoretical models for the action spectrum and the current-voltage characteristics of microporous semiconductor films in photoelectrochemical cells. J. Phys. Chem., 98:5552–5556, 1994. doi:10.1021/j100072a023.

High-order compact finite difference scheme for option pricing in stochastic volatility jump models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its performance to standard finite difference and finite element methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial $LU$-factorisation of a sparse matrix to perform the option price valuation. Compared to finite element approaches, it is very parsimonious in terms of memory requirements and computational effort, since it achieves high-order convergence without requiring additional unknowns, unlike finite element methods with higher polynomial order basis functions. The new high-order compact scheme can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.

Modeling Hydraulically Fractured Shale Wells Using the Fast-Marching Method With Local Grid Refinements and an Embedded Discrete Fracture Model

Summary Recently, fast–marching–method (FMM) –based flow simulation has shown great promise for rapid modeling of unconventional oil and gas reservoirs. Currently, the application of FMM–based simulation has been limited to using tartan grids to model the hydraulic fractures (HFs). The use of tartan grids adversely impacts the computational efficiency, particularly for field–scale applications with hundreds of HFs. Our purpose in this paper is to extend FMM–based simulation to incorporate local grid refinements (LGRs) and an embedded discrete fracture model (EDFM) to simulate HFs with natural fractures, and to validate the accuracy and efficiency of the methodologies. The FMM–based simulation is extended to LGRs and EDFM. This requires novel gridding through the introduction of triangles (2D) and tetrahedrons (2.5D) to link the local and global domain and solution of the Eikonal equation in unstructured grids to compute the diffusive time of flight (DTOF). The FMM–based flow simulation reduces a 3D simulation to an equivalent 1D simulation using the DTOF as a spatial coordinate. The 1D simulation can be carried out using a standard finite–difference method, thus leading to orders of magnitude of savings in computation time compared with full 3D simulation for high–resolution models. First, we validate the accuracy and computational efficiency of the FMM–based simulation with LGRs by comparing them with tartan grids. The results show good agreement and the FMM–based simulation with LGRs shows significant improvement in computational efficiency. Then, we apply the FMM–based simulation with LGRs to the case of a multistage–hydraulic–fractured horizontal well with multiphase flow, to demonstrate the practical feasibility of our proposed approach. After that, we investigate various discretization schemes for the transition between the local and global domain in the FMM–based flow simulation. The results are used to identify optimal gridding schemes to maintain accuracy while improving computational efficiency. Finally, we demonstrate the workflow of the FMM–based simulation with EDFM, including grid generation; comparison with FMM with unstructured grid; and validation of the results. The FMM with EDFM can simulate arbitrary fracture patterns without simplification and shows good accuracy and efficiency. This is the first study to apply the FMM–based flow simulation with LGRs and EDFM. The three main contributions of the proposed methodology are (i) unique mesh–generation schemes to link fracture and matrix flow domains, (ii) DTOF calculations in locally refined grids, and (iii) sensitivity studies to identify optimal discretization schemes for the FMM–based simulation.

Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence

We extend the nonstandard finite difference (NSFD) method of solutions to the study of an SIS epidemic model with standard incidence. We show that the proposed NSFD schemes preserve two essential properties of the continuous model: positivity and global asymptotic stability properties. The reproduction number of the model is calculated by the next generation matrix approach. It is worth noting that the global asymptotic stability of the disease-free equilibrium point of the proposed numerical schemes is proved theoretically by the use of an extension of the Lyapunov stability theorem. Besides, the global asymptotic stability of the endemic equilibrium point is investigated by the use of the Lyapunov indirect method and numerical simulations. Consequently, NSFD schemes which are dynamically consistent with the continuous model are obtained. Some numerical simulations are presented to validate the theoretical results and to show that the NSFD schemes are effective and appropriate for solving the continuous model. We employ the standard finite difference (SFD) method as a means of comparison to NSFD schemes. The numerical simulations indicate that the use of SFD method is not suitable as it produced solutions that do not correspond exactly to solutions of the continuous model.

Learning data-driven discretizations for partial differential equations

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.

A High-Order Kernel-Free Boundary Integral Method for the Biharmonic Equation on Irregular Domains

This work proposes second-order and fourth-order versions of a Cartesian grid based kernel-free boundary integral (KFBI) method for the biharmonic equation on both bounded irregular domains and singly periodic irregular domains. It is further development of the previous KFBI method for second-order elliptic PDEs. It reformulates boundary value problems of the fourth-order PDE as boundary integral equations of the first kind but the solution never needs to know the fundamental solution or Green’s function of the elliptic operator. Evaluation of boundary or volume integrals in the solution of boundary integral equations is made by solving equivalent interface problems on Cartesian grids with standard finite difference methods and fast Fourier transform based solvers. The work decomposes the biharmonic equation into two Poisson equations. It assumes the solution to one Poisson equation, which has no boundary conditions, as the sum of a volume integral with a double layer boundary integral, and applies Green’s third identity to derive a scalar boundary integral equation from the other Poisson equation that are subject to two boundary conditions. In the solution of the scalar boundary integral equation, each volume or boundary integral is evaluated with the KFBI method. Numerical examples are presented to demonstrate the solution accuracy and algorithm efficiency. A remarkable point of the work is that the nine-point compact difference scheme in dealing with each split second-order elliptic interface problem on irregular domains yields fourth-order accurate solution for the biharmonic equation.

Fractional optimal control in transmission dynamics of West Nile virus model with state and control time delay: a numerical approach

In this paper, an optimal control for a novel fractional West Nile virus model with time delay is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the Grünwald–Letnikov sense. Stability analysis of fixed points is studied. Corresponding fractional optimal control problem, with time delays in both state and control variables, is formulated and studied. Two simple numerical methods are used to study the nonlinear fractional delay optimal control problem. The methods are standard finite difference method and nonstandard finite difference method. Comparative studies are implemented, it is found that the nonstandard finite difference method is better than the standard finite difference method.

LibreGrowth: A tumor growth code based on reaction–diffusion equations using shared memory

In recent years, in-silico experimentation within the field of oncological medicine has been intensively investigated with the aim of better understanding tumor dynamics and dose–response relationships in cancer treatments. In a series of previous works, Luján et al. (2018, 2017, 2016)we described the micro-environmental influence on micro-tumor infiltration patterns through in-silico/in-vitro experimentation. Here we present the latest version of the software utilized for, but not limited to, those studies: LibreGrowth, a libre tumor growth code able to simulate the core growth and peripheral tumor cell infiltration, considering a benign and a malignant stages. We implemented a reaction–diffusion based model, with spatially variable diffusion coefficient, into a three-dimensional domain, using C++ and OpenMP over a GNU/Linux system. LibreGrowth aims to provide a flexible implementation for depicting heterogeneous tissues and infiltration processes, and to shed light in current therapy optimization strategies. Program summaryProgram Title: LibreGrowthProgram Files doi:http://dx.doi.org/10.17632/zp2my52xpv.1Licensing provisions: GPLv3Programming language: C++, OpenMPSupplementary material:Nature of problem: In the field of computational oncology, mathematical models based on reaction–diffusion equations describing tumor proliferation and invasion into peripheral host tissue have proved to be of clinical relevance. In-silico/in-vitro experimentation could help in the design of new strategies able to predict as much as possible the invasive behavior of a tumor, based on its particular properties and the bio-physicochemical characteristics of its microenvironment.Solution method: We introduced LibreGrowth, the latest version of the codes used in our previous studies to model the growth and infiltration of a tumor. It implements mentioned reaction–diffusion model through the standard finite difference method, using C++ and OpenMP parallelization technology.

Energy-conserving Galerkin approximations for quasigeostrophic dynamics

A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) [1] to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recombined Legendre basis from Shen (1994) [2] for the streamfunction. The method is tested in the context of linear baroclinic instability calculations, where it is compared to the standard second-order finite-difference method and to a Chebyshev collocation method. The Galerkin scheme is quite accurate even for a small number of degrees of freedom N, and growth rates converge much more quickly with increasing N for the Galerkin scheme than for the finite-difference scheme. The Galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom. The energy-conserving Galerkin scheme is of comparable accuracy to the Chebyshev collocation scheme in most linear stability calculations, but not in the Eady problem where the Chebyshev scheme is significantly more accurate. Finally the three methods are compared in the context of a simplified version of the nonlinear equations: the two-surface model with zero potential vorticity. The Chebyshev scheme is the most accurate, followed by the Galerkin scheme and then the finite difference scheme. All three methods conserve energy with similar accuracy, despite not having any a priori guarantee of energy conservation for the Chebyshev scheme. Further nonlinear tests with non-zero potential vorticity to assess the merits of the methods will be performed in a future work.

On controllability methods for the Helmholtz equation

When the Helmholtz equation is discretized by standard finite difference or finite element methods, the resulting linear system is highly indefinite and thus notoriously difficult to solve, in fact increasingly so at higher frequency. The exact controllability approach (Bristeau et al., 1994) instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period, the controllability method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. The original energy functional used to penalize the departure from periodicity is strictly convex only for sound-soft scattering problems. To extend the controllability approach to general boundary-value problems governed by the Helmholtz equation, new penalty functionals are proposed, which are numerically efficient. Numerical experiments for wave scattering from sound-soft and sound-hard obstacles, inclusions, but also for wave propagation in closed wave guides illustrate the usefulness of the resulting controllability methods.

A partial-low-rank method for solving acoustic wave equation

Numerical solutions of the acoustic wave equation, especially in anisotropic media, is crucial to seismic modeling, imaging and inversion as it provides efficient, practical, and stable approximate representation of the medium. However, a clean implementation (free of shear wave artifacts and dispersion) of wave propagation, especially in anisotropic media, requires an integral operator, the direct evaluation of which is extremely expensive. Recently, the low-rank method was proposed to provide a good approximation to the integral operator utilizing Fourier transforms. Thus, we propose to split the integral operator into two terms. The first term provides a differential operator that approximates that can be approximated with a standard finite-difference method. We, then, apply the low-rank approximation on the residual term of the finite-difference operator. We implement the two terms in two complementing steps, in which the spectral step corrects for any errors admitted by the finite difference step. Even though we utilize finite-difference approximations, the resulting algorithm admits spectral accuracy. Also, through the finite difference step, the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner. Numerical examples show that the method is of high accuracy and efficiency.

Vortex-Induced Vibrations of A Long Flexible Cylinder in Linear and Exponential Shear Flows

A numerical study based on a wake oscillator model was conducted to determine the response performance of vortex-induced vibration (VIV) on a long flexible cylinder with pinned-pinned boundary conditions subjected to linear and exponential shear flows. The coupling equations of a structural vibration model and wake oscillator model were solved using a standard central finite difference method of the second order. The VIV response characteristics including the structural displacement, structural frequency, structural wavenumber, standing wave behavior, travelling wave behavior, structural velocity, lift force coefficient and transferred energy from the fluid to the structure with different flow profiles were compared. The numerical results show that the VIV displacement is a combination of standing waves and travelling waves. For linear shear flow, standing waves and travelling waves dominate the VIV response within the low-velocity and high-velocity zones, respectively. The negative values of the transferred energy only occur within the low-velocity zone. However, for exponential shear flow, travelling waves dominate the VIV response and the negative energy occurs along the entire length of the cylinder.

MULTISCALE SEAMLESS-DOMAIN METHOD FOR NONPERIODIC FIELDS: NONLINEAR HEAT CONDUCTION ANALYSIS

A multiscale numerical solver called the seamless-domain method (SDM) is used in linear heat conduction analysis of nonperiodic simulated fields. The practical feasibility of the SDM has been verified for use with periodic fields but has not previously been verified for use with nonperiodic fields. In this paper, we illustrate the mathematical framework of the SDM and the associated error factors in detail. We then analyze a homogeneous temperature field using the SDM, the standard finite difference method, and the conventional domain decomposition method (DDM) to compare the convergence properties of these methods. In addition, to compare their computational accuracies and time requirements, we also simulated a nonperiodic temperature field with a nonuniform thermal conductivity distribution using the three methods. The accuracy of the SDM is very high and is approximately equivalent to that of the DDM. The mean temperature error is less than 0.02% of the maximum temperature in the simulated field. The total central processing unit (CPU) times required for the analyses using the most efficient SDM model and the DDM model represent 13% and 17% of that of the finite difference model, respectively.

Numerical prediction of vortex-induced vibrations of a long flexible cylinder in uniform and linear shear flows using a wake oscillator model

A numerical study based on a wake oscillator model was conducted to determine the response performance of vortex-induced vibration (VIV) on a long flexible cylinder with pinned-pinned boundary conditions subjected to uniform and linear shear flows. The coupling equations of a structural oscillator model and wake oscillator model were solved using a standard central finite difference method of the second order. The structural displacement, structural frequency, standing waves, travelling waves, lift force coefficient and transferred energy for uniform and linear shear flows were compared. The numerical results show that the VIV response is a combination of standing waves and travelling waves. In a uniform flow, the VIV response is dominated by the standing waves along the total cylinder span. In linear shear flows, standing waves and travelling waves dominate the VIV response near the cylinder ends and midpoint, respectively, and the VIV displacements display multi-frequency characteristics when monitoring their characteristics during a long time. However, for a given time, the displacements instantaneously exhibit mono-frequency characteristics. The variation of the transferred energy from a positive value to a negative value is due to the decrease of the phase lag between the structural displacement and the lift force coefficient.

Numerical Modeling of the Static Electric Field Effect on the Director of the Nematic Liquid Crystal Director

A two-dimensional model of the Frederiks effect is used to investigate the static electric field effect on the orientation of the nematic liquid crystal (LC) director in a side-electrode cell. The solutions are obtained by the standard finite-difference methods. The programs for the numerical solution of a two-dimensional parabolic partial differential equation are developed both in FORTRAN and C/C++. The Frederiks transition threshold for the central part of the cell and the dependences of the director’s orientation distribution on a high electric field are obtained. The results of the calculation are compared with the experimental data.

Theory of the Drift-Wave Instability at Arbitrary Collisionality

A numerically efficient framework that takes into account the effect of the Coulomb collision operator at arbitrary collisionalities is introduced. Such a model is based on the expansion of the distribution function on a Hermite-Laguerre polynomial basis to study the effects of collisions on magnetized plasma instabilities at arbitrary mean-free path. Focusing on the drift-wave instability, we show that our framework allows retrieving established collisional and collisionless limits. At the intermediate collisionalities relevant for present and future magnetic nuclear fusion devices, deviations with respect to collision operators used in state-of-the-art turbulence simulation codes show the need for retaining the full Coulomb operator in order to obtain both the correct instability growth rate and eigenmode spectrum, which, for example, may significantly impact quantitative predictions of transport. The exponential convergence of the spectral representation that we propose makes the representation of the velocity space dependence, including the full collision operator, more efficient than standard finite difference methods.

Improved singlet oxygen generation and antimicrobial activity of sulphur-doped graphene quantum dots coupled with methylene blue for photodynamic therapy applications.

Due to their many unique properties, graphene quantum dots (GQDs) have attracted much attention and are a promising material with potential applications in many fields. One application of GQDs is as a photodynamic therapy agent that generates singlet oxygen. In this work, GQDs were grown by focusing nanosecond laser pulses into benzene and then were later combined with methylene blue (MB) and used to eradicate the Gram-negative bacteria, Escherichia coli, and Gram-positive bacteria, Micrococcus luteus. Theoretical calculation of pressure evolution was calculated using the standard finite difference method. Detailed characterization was performed with transmission electron microscopy (TEM), scanning electron microscopy (SEM), Fourier-transform infrared (FTIR), UV-vis (UV-vis), and photoluminescence (PL) spectra. Furthermore, MB-GQD singlet oxygen generation was investigated by measuring the rate of 9,10-anthracenediyl-bis(methylene) dimalonic acid photobleaching. Combining MB with GQDs caused enhanced singlet oxygen generation. Our results show that the MB-GQD combination efficiently destroys bacteria. The (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide) (MTT) assay was used to determine if GQDs in dark conditions caused human cellular side-effects and affected cancer and noncancer cellular viability. We found that even high concentrations of GQDs do not alter viability under dark conditions. These results suggest that the MB-GQD combination is a promising form of photodynamic therapy.

Nonstandard finite difference approach for solving 3-compartment pharmacokinetic models.

Complex nature of the analytical solutions to 3-compartment pharmacokinetic models leads to the discrete approximation of the continuous differential equation been mostly used. In this paper, we applied nonstandard finite difference method to 3-compartment pharmacokinetic models. This method was introduced to compensate for the weaknesses of methods such as the standard finite difference methods, numerical instabilities being a prime example of such. Three-compartment pharmacokinetic model with 2 different routes of administration (IV bolus injection and IV bolus infusion) is considered for simulations. For the case when the system is homogeneous (models arising from IV bolus injection mode of administration), "exact" finite difference scheme is obtained for any step size while in the case of nonhomogeneous (models arising from IV bolus infusion route of administration), scheme that has the same qualitative behavior as the analytical solution for all step sizes is obtained. It was shown computationally that the nonstandard finite difference scheme for 3-compartment pharmacokinetic model with IV bolus mode of administration is exact while 3-compartment pharmacokinetic model with IV infusion is dynamically consistent with the continuous model for all step sizes.

Generalized finite difference approach verification on circular plates

The best-known method, finite differences (abbreviated in text as - FDM), consists of replacing each derivative by a difference algebraic quotient in the classic formulation. It is that discretization method which simple to code and economic to compute. In a sense, a finite difference formulation offers a more direct approach to the numerical solution of partial differential equations (abbreviated in text as - PDE) than does a method based on other formulations. The main drawback of the finite difference methods is the flexibility. Standard finite difference methods requires more regularity of the solution and triangulation (e.g. uniform meshing). Difficulties also arise in imposing boundary conditions.Nowadays, FDM predominates the numerical solutions of PDE not less in pair with the method of finite elements (abbreviated in text as - FEM).In This paper, an advanced static structural-deflection analysis procedure of circular plates through generalized (modernized) finite difference method is presented. The generalized approach considers a lot of parameters been regarded-less by the classical one. The thing which negatively affects the accuracy (convergence to the exact solution values) and consequently; tendency of results it generates. Besides, the newly presented can be easily applied to many cases that the classical even fails to solve. A theoretical mathematical statement been reviewed containing all the necessary explanatories, graphs and indications followed by verification testing problems simultaneously with a comparison been held among variety of results generated via different techniques. The verification comparison indicates the powerfulness of the innovated approach relative to the classical one and the easiness and precision it offers in relevant to the others.

A study of mimetic and finite difference methods for the static diffusion equation

Two second-order finite difference methods in a staggered mesh to solve the static diffusion equation are proposed in this article. These methods were compared with a standard finite difference method and with two numerical schemes naturally established in staggered grids: mimetic method and conservative method. Also, mimetic discretization is presented in a formal manner. The methods were tested using different configurations, including boundary layers and heterogeneous media. The study shows that the two proposed finite difference methods produce numerical solutions that are comparable to those given by mimetic methods, in terms of rates of convergence and magnitude of the approximation error.