Implicit Difference Scheme Research Articles

A new fast predictor-corrector method for nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms

Time-fractional reaction-diffusion (TFRD) equation is an important fractional parabolic equation, and the research of its numerical solution has scientific significance and engineering application value. In this paper, a new fast predictor-corrector (FP-C) scheme is constructed based on fast L1 approximation of Caputo fractional derivative for solving nonlinear TFRD equation with nonhomogeneous terms. The linearized implicit difference scheme is used for the predictor step and Crank-Nicolson(C-N) scheme is used for the corrector step. Theoretical analysis proves that FP-C scheme is convergent and stable unconditionally for nonlinear TFRD equation. Numerical analysis and experiments show that the computational accuracy of FP-C scheme is $ O(\tau^{2-\alpha}+h^2) $ under the strong regularity condition and $ O(\tau^{\alpha}+h^2) $ under the weak regularity condition. Compared with the classical predictor-corrector (P-C) scheme based on standard L1 approximation, the FP-C scheme improves the computational efficiency without losing the computational accuracy. It is shown that the new FP-C scheme is an efficient method to solve nonlinear TFRD equation with nonhomogeneous terms.

ROBUST IMPLICIT DIFFERENCE APPROACH FOR THE TIME-FRACTIONAL KURAMOTO–SIVASHINSKY EQUATION WITH THE NON-SMOOTH SOLUTION

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy.

СРАВНИТЕЛЬНЫЙ АНАЛИЗ ЭФФЕКТИВНОСТИ ПАРАЛЛЕЛЬНЫХ ВЫЧИСЛЕНИЙ ПО ЯВНЫМ И НЕЯВНЫМ РАЗНОСТНЫМ СХЕМАМ ДЛЯ ЗАДАЧ ВЫЧИСЛИТЕЛЬНОЙ АЭРОДИНАМИКИ

One of the main areas of parallel computing application is solving of computational aerodynamicsproblems. The paper considers parallel modeling of gas dynamics with quasi-onedimensionalsystem of equations. The system describes the gas flow through a channel with variablecross section using implicit and explicit difference schemes. The purpose of this work is tostudy the methods efficiency for organizing parallel computations with implicit and explicit differenceschemes for solving internal aerodynamics problems. The article presents a comparativeanalysis of the proposed parallel models for quasi-one-dimensional equations system of gas dynamics.The system describes flows in a channel of variable cross section. Various parallel algorithmsare used to solve systems of such type numerically. The method of splitting by physical processesis used for an implicit difference scheme. To suppress the solution oscillations a smoothingoperator at the correction stage is introduced in calculations according to the scheme of the predictor-corrector type. In scheme fractional steps the parallel scalar sweep algorithm is used tosolve tridiagonal systems with the parametric unknown choice. Besides to compare scheme mentionedabove a parallel algorithm is constructed for McCormack's explicit scheme. Such algorithmis widely used in computational aerodynamics. Parallel computations are held by computingstructures with distributed memory and by another one – with linear switching dependence betweencomputing devices of the working field. The paper presents time estimations for each computingstage (both by implicit and explicit difference schemes). It helped to calculate the developedparallel algorithms efficiency. It is concluded that the acceleration factor in explicit scheme dependslinearly on the computing devices number.

Two-dimensional model of grain boundary diffusion and oxidation

The grinding of the structure of materials is accompanied by a change in the physical and mechanical properties. This occurs largely due to the accumulation of energy and defects in the structure, which activates the diffusion of impurities contained in the material. The increase in the number of grain boundaries and joints can cause the inelastic behavior of the material, its additional chemical activation. For some metals and alloys this leads to strengthening, while for others it leads to rapid degradation of mechanical properties. Grain boundary diffusion in such materials is the main mechanism of transport of alloying components or harmful impurities, so its study is important. This paper presents a two-dimensional model of grain boundary diffusion in a material with an explicit structure assignment. The model takes into account the presence of chemical transformations that can determine the corrosion mechanisms under operating conditions. For simplicity of calculation the material structure is taken symmetrical and contains two phases: grains and a boundary phase. Diffusion and kinetic parameters of the phases may differ. The model is represented in dimensionless form so that the distances between neighboring grains or the widths of the boundary phase are the same and the grain sizes can vary. Depending on the ratio of phase sizes we can speak about micro- and nanocrystalline structure. The problem was solved numerically using implicit difference scheme and coordinate splitting. Diffusion and kinetic parameters, which are close to the parameters of oxygen grain boundary diffusion in titanium and titanium oxidation, respectively, were taken for the calculations. Integral concentrations reflect the dynamics (kinetics) of oxygen and oxides accumulation over the calculation area. Results showing the role of changes in the oxidation kinetics due to changes in the reaction constants in the phases and the phase size ratio are presented. The model can be useful for assessing the degree of influence of grain boundary diffusion on the oxidation process and the accompanying change in properties, as well as for setting up appropriate experiments.

ВИЗНАЧЕННЯ КОЕФІЦІЄНТУ ШОРСТКОСТІ ДЛЯ РОЗРАХУНКУ ПОТОКУ НЕУСТАЛЕНОГО РУХУ РІДИНИ

The analysis of the literature devoted to calculation of hydrotechnical structures showed that there are recommendations that take into account the influence roughness of the channel on magnitude transformation transit flow of liquid along the length of the channel. But these recommendations are difficult to apply to reclamation systems, which include complexes of hydraulic engineering and transport facilities, because they do not take into account the characteristic features of unsteady fluid movement in such systems. Unfortunately, as of today, this issue has not been adequately addressed in modern scientific literature. The article considers the issue of determining the roughness coefficient based on the materials of field experiments on irrigation systems of Ukraine for its further use in the Saint-Venant equations for unsteady water movement in open prismatic channels. At one time, professor V.O. Bolshakov applied the method of running according to the implicit difference scheme, provided that the roughness coefficients are known, to solve the Saint-Venant equations. The authors conducted an analysis errors of field measurements of maximum depths, calculation area of live sections, etc. The absolute error of finding the drop in water levels in individual sections of the channel was determined. The results. Among the issues that were covered in the discussed publication, the main attention was paid to the issue of studying the influence of the roughness coefficient on flow elements during unsteady motion in meliorational (irrigation) systems. Also, the characteristics conducted field measurements are given, a detailed overview field data is provided, and the measured parameters at different sections research channel are listed. Conclusions. An indicative evaluation and analysis errors of the performed measurements was carried out, and recommendations were given for determining the average values of roughness coefficients for irrigation channels (channels) with a well-planned bottom and slopes, as well as for banks not covered with vegetation. The results of data processing of field observations are summarized in a table, which presents a comparison calculated values of the roughness coefficients.

Influence of material functional heterogeneity on non-stationar oscillations of piezoceramic bodies

The influence of material functional heterogeneity on mechanical oscillations of piezoelement under non-stationary electrical loading is investigated. Within the assumption of functional distribution of material characteristics by thickness of the piezoelectric element, which corresponds to the physical properties of the body, a unified system of solving equations was obtained to describe the thickness fluctuations of piezoelectric plates, cylinders, and balls. For controllingof accuracy, the calculation is carried out using an explicit and implicit difference scheme.
 Unsteady oscillations of a flat piezoceramic layer, cylinder, and sphere are investigated with a parabolic distribution of all material characteristics along the thickness of the element. It is assumed that the average value of the function along the thickness is equal to the tabular value of the material characteristic, and the value on electrodes is proportional to the area of electrodes. At such conditions, we obtained a decrease in the speed of disturbances propagation and a slight change in the amplitude associated with the curvature of the element. The increase in amplitude reaches 3% for balls. It should be noted that at given load oscillations occur in the compressed zone without entering the undeformed state. The considered cylinder and ball have a rather large curvature, for bodies with a smaller curvature the influence of the described effect will be smaller. The additional analysis indicates that the shape of the distribution curve under described above conditions also has little effect on the results.
 It was established that the effect of functional heterogeneity within the same material has little effect on the oscillations of the piezoelement, that is, it is really possible to average the material characteristics by thickness at calculating, since the deviation between the results is within acceptable limits (up to 2.5%). Also, an important result is the confirmation of the assumption that for curved bodies such as cylinder and sphere, the material characteristics can be considered constant on thickness, regardless of the curvature of the body.
 The proposed technique can be applied for studyingof the vibrations of different geometries bodies with significantly heterogeneous functional material or what are combined from several materials with a gradient transition between them.

An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel

An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel

Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity

This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank–Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann–Liouville fractional integral term. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at t=0 so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, the stability and convergence are proved by means of the energy argument. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.

An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by using the Galerkin method based on piecewise linear test functions. The Riemann–Liouvile fractional integral terms are treated by convolution quadrature. In order to obtain a fully discrete method, the standard central difference approximation is used to discretize the spatial derivative. The stability and convergence are rigorously proved by the discrete energy method. In addition, the existence and uniqueness of numerical solutions for nonlinear systems are proved strictly. Additionally, we introduce and compare the Besse relaxation algorithm, the Newton iterative method, and the linearized iterative algorithm for solving the nonlinear systems. Numerical results confirm the theoretical analysis and show the effectiveness of the method.

NUMERICAL MODELING OF AIR POLLUTION NEAR THE HIGHWAY WITH PROTECTIVE BARRIERS

Problem statement. The problem of predicting chemical pollution zones near the highway, where protective barriers are located, is considered. The protective barriers influence the aerodynamics of air flow and reduce the level of working areas chemical pollution near the highway. It is necessary to have mathematical models that allow such evaluation at the design stage in order to evaluate the effectiveness of protective barriers promptly. The purpose of the article. Development of a numerical multi-parameter model and computer program for evaluating the effectiveness of the protective barriers using near the highway to reduce the chemical contamination level of work areas. Methodology. A two-dimensional mass transfer equation is used for mathematical modeling of chemical hazardous substances spreading that are emitted from a car and spread near the highway. This simulation equation takes into account the wind direction, atmospheric diffusion, the intensity of chemically hazardous substance emission from the car, the emission source location, the car body and the location of protective barriers near the highway. The Navier − Stokes equation and the potential motion model both are used to solve the aerodynamics problem of determining the uneven air velocity field near the highway. An implicit splitting difference scheme is used to numerically integrate the vortex transfer equation. Two numerical schemes are used to numerically integrate the equation for the current function. An explicit difference scheme is used to numerically integrate the Laplace equation for the velocity potential. Based on the constructed numerical model the package of programs is developed. To solve numerically equation of pollutant transfer the finite difference schemes of splitting are used. Scientific novelty. To calculate the formation of chemical pollution areas for atmospheric air near the highway during emissions from vehicles numerical model is proposed. This model allow to estimate the protective barriers effectiveness for reducing air pollution in the working areas. Practical significance. Based on the developed model the code for quickly calculation of air pollution process both near the highway and in work areas protecting with barriers is created. Conclusions. The developed numerical model and the computer code implementing it allow to investigate the process of chemical hazardous substance spreading in the atmospheric air near the highway. The model allows to quickly determine the effectiveness of protective barriers using near the highway. The developed computer program can be implemented on low and medium power computers. The results of a computational experiment are presented.

INFLUENCE OF THE DEGREE FLOODING AND COMPRESSION CROSS-SECTION OF A HYDRAULIC STRUCTURE WITH UNSTEADY LIQUID MOVEMENT

The article is devoted to topical issues of the influence of hydraulic structures on the behavior of channel flow. Issues related to the operation of hydraulic structures located on irrigation canals were considered: - the influence of the degree of flooding on the elements of unsteady movement in open channels; - the influence of flow compression on the elements of unsteady motion in open flows. Quantitative assessment of the impact degree of flooding and compression on the elements of the flow during steady motion was carried out taking into account the recommendations of prof. Bolshakova VO, which are based on the use of the method of prof. Vasilieva OF The question influence of hydraulic structures on the behavior of the channel flow was solved using the equations of Saint-Venan by the numerical method, namely the method of run by the implicit-difference scheme. To close the system when using this method, the following conditions were taken into account: a - initial; b - left and right boundary conditions. The initial conditions are the presence of uniform movement in the channel. The left boundary condition is determined by the schedule of water supply to the channel, which has the form of a triangular hydrograph. The right boundary condition is determined by the known formula of a spillway with a wide threshold. The initial data were obtained from field observations. Quantitative assessment of the impact of flooding and flow compression on the final flow, velocity and depth results was performed. The issue of distribution of the support along the channel bed, which was formed due to the compression of the flow, was solved using the recommendations of E.V. Eremenko. The equation of flow continuity was considered - under the condition of changing the volume of water in the elementary section of the channel. The time of increase in the volume of water due to compression was determined from the formula obtained in the calculation process. Based on the condition that the time factor is a known value, it is possible to obtain a mathematical expression that determines the length of the propagation of the compression effect. Thanks to the obtained formulas, the calculated graphs of the relative maximum depth depending on the degree of flooding were constructed. With the help of these graphs it is possible to solve the problem of water supply in irrigation canals in the presence of flooding and compression of the flow (in case of unsteady movement).

The chemical interaction charged particles with target material under surface treatment of metal with particle beam

AbstractThe paper presents a nonlinear coupled nonisothermal model of the process of material implementation into the target surface under conditions of surface treatment by a particle beam in the form of one or few consecutive pulses. The model takes into account the interaction of impurity diffusion, heat propagation, and mechanical disturbances, as well as the chemical reaction between the introduced impurity and the target material. The problem is solved numerically using an implicit symmetric difference scheme of the second‐order approximation in time and spatial coordinates. The results of solution are obtained for different time intervals. It was found that the interaction of different processes leads to the appearance of distortions in the front of waves. The distributions of temperature and impurity concentration have a wave character. The increment of the reaction product in the dynamics for different numbers of pulses is presented. The interaction mechanisms of the studied processes do not change and do not depend on the number of pulses. However, a more complicated picture is observed under treatment by several successive pulses due to the appearance of new extremes and distortions.

INVERSE PROBLEM FOR RECONSTRUCTING THE RIGHT SIDE OF A ONE-DIMENSIONAL WAVE EQUATION

An inverse problem of recovering the unknown parameter of an external source for a one-dimensional wave equation with a nonlocal additional condition is considered. It is assumed that the unknown source parameter depends only on time. By integration, the problem is transformed to an inverse boundary value problem with local conditions. A difference analogue of the differential problem in the form of an implicit difference scheme is constructed and a non-iterative computational algorithm for solving the resulting system of difference equations is proposed. As a result, an explicit formula is obtained for determining the approximate value of the sought parameter for each discrete value of the time variable

Time-Domain Boundary Element Method with Broadband Impedance Boundary Condition

Acoustic liners are an effective tool for noise reduction and are characterized by a frequency-dependent impedance value. In this paper, a time-domain boundary element method for acoustic scattering coupled with a broadband impedance boundary condition is studied for the case of no mean flow. A Burton-Miller reformulation of the time domain boundary element method for acoustic scattering is carried out with both the pressure and its surface normal derivative terms retained. A multipole impedance model is converted into the time domain and used for enforcing an impedance boundary condition over a wide range of frequencies. Discretization of the coupled system is described. In particular, the time-domain impedance boundary condition is discretized by the third-order implicit backward difference scheme. Stability of the coupling is assessed by an eigenvalue analysis. Scattering solutions that demonstrate the validity and stability of the numerical method are presented.

An implicit difference scheme for the fourth-order nonlinear partial integro-differential equations

In this paper, an implicit difference scheme is constructed and analyzed for the fourth-order nonlinear partial integro-differential equations. The backward Euler scheme and convolution quadrature are combined for time derivative and Riemann–Liouville (R–L) fractional integral terms. A fully discrete difference scheme is established with the space discretization by the standard central difference approximation. For nonlinear convection term , by the Galerkin method based on piecewise linear test functions, we can handle nonlinear term implicitly and attain a system of nonlinear algebraic equations. The stability and convergence are derived by the energy method. Especially, the existence and uniqueness of the numerical solutions are strictly proved for the nonlinear system. Also, we introduce three methods for solving the nonlinear systems, where two existing methods and a new linearized iterative algorithm are given to show the efficiency of our scheme. Numerical results are consistent with the theoretical analysis.

An accurate, robust, and efficient finite difference scheme with graded meshes for the time-fractional Burgers’ equation

This work presents the L1 implicit difference scheme based on non-uniform meshes for the time-fractional Burgers equation. We find that the difficulty arising from the singularity of the exact solution at t=0 can be overcome by non-uniform meshes. We yield the unconditional stability and optimal convergence rate of h2+τmin{rα,2−α} by means of the energy method, when selecting r=(2−α)/α. Convergence rates are higher than some recently studied schemes. Numerical experiments confirm the predicted theoretical estimate.

Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model.

The paper is concerned with development of an accurate and effective positivity-preserving high-order compact difference method for solving the Keller-Segel chemotaxis model, which is a kind of nonlinear parabolic-parabolic system in mathematical biology. Firstly, a stiffly-stable five-step fourth-order fully implicit compact difference scheme is proposed. The new scheme not only has fourth-order accuracy in the spatial direction, but also has fourth-order accuracy in the temporal direction, and the computational strategy for the nonlinear chemotaxis term is provided. Then, a positivity-preserving numerical algorithm is presented, which ensures the non-negativity of cell density at all time without accuracy loss. And a time advancement algorithm is established. Finally, the proposed method is applied to the numerical simulation for chemotaxis phenomena, and the accuracy, stability and positivity-preserving of the new scheme are validated with several numerical examples.

MODELING OF PRODUCING PRESSURE IN HETEROGENEOUS OIL-BEARING RESERVOIRS

Numerical modeling of the distribution of the reservoir pressure drop in the vicinity of an operating well was carried out taking into account the inhomogeneous distribution of filtration characteristics (permeability and oil viscosity) in the near and distant zones of the well operation in order to study the practical aspects of filtration in heterogeneous oil-bearing formations based on a combined finite-element-difference method for non-stationary problem of piezoconductivity. The use of the combined finite-element-difference method enables to combine the advantages of the finite-element method and the finite difference method: to model geometrically complex areas, to find the value at any point of the object under study, while the implicit difference scheme.
 It is shown that the intensity of filtration processes in the vicinity of the operating well depends mainly on the permeability, and, to a lesser extent, on the viscosity of the oil. Moreover, the influence of the permeability of the oil phase in the remote zone (Rd < 5 m) is greater than the effect in the close zone (Rd > 5 m) of the operating well. In the case of low permeability of the oil phase in the vicinity of the existing well, to maintain stable oil production, it is necessary to place an injection well near the production well. Using the method suggested, it is possible to predict the effect of the injection well on the formation pressure distribution in the formation.
 The scientific novelty of the work lies in the study of the influence of the heterogeneous permeability and oil viscosity distribution on the reservoir pressures distribution around the wells by modeling filtration processes based on a combined finite-element-difference method.
 The practical significance of the research results comes down to confirming the close relationship between the heterogeneity of the porous medium and the reservoir pressures distribution around an operating producing well. The combined finite-element-difference method used in this work can be used to solve other filtration problems (for example, to calculate the gas saturation of a reservoir, create a method for calculating well flow rates, assess the effect of injection wells on filtration processes).

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.

Identification of stationary source in the anomalous diffusion equation

In this paper, we consider the initial-boundary value problem of determining the stationary right-hand side function in the anomalous diffusion equation with a Caputo fractional derivative with respect to time. The value of the solution of the problem at the final time moment is set as the overdetermination condition. In order to carry out the numerical solution, the iterative conjugate gradient method is used, while at each iteration a direct problem is solved by the finite-difference method using a purely implicit difference scheme. The computational experiment results for the model problem are presented to confirm the efficiency of this new method.