Difference Discretization Scheme Research Articles

Numerical investigation of generalized tempered-type integrodifferential equations with respect to another function

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct a semi-discrete BE scheme. Second, the backward differentiation formula (BDF) and second-order interpolation quadrature rule are adopted to establish a semi-discrete second-order BDF (BDF2) scheme. Additionally, the stability and convergence of two semi-discrete methods are deduced in detail. To further demonstrate the effectiveness of proposed techniques, fully discrete BE and BDF2 finite difference schemes are formulated. Subsequently, the theoretical results of two fully discrete difference schemes are presented. Finally, the numerical results demonstrate the accuracy and competitiveness of the theoretical analysis.

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

THE HIGHER FINITE DIFFERENCE METHOD FOR SOLVING THE DYNAMICAL MODEL OF COVID-19

In the present paper, the SIR model tracks the numbers of susceptible, infected and recovered individuals during an epidemic with the help of ordinary differential equations (ODE). First, we give the model formulation of our phenomena. Secondly, a fully discrete difference scheme is derived for the SIR model.At the end of this aper, we give the simulation results of the model. A comparison of the obtained numerical results of both the models is performed in the absence of an exact solution.

Complex dynamics of a discrete‐time seasonally forced SIR epidemic model

In this paper, a discrete‐time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark–Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results.

An IDDES study of the near-wake flow topology of a simplified heavy vehicle

Abstract The complex wake flow of a GTS (ground transportation system) model contributes to large percentage of the aerodynamic drag force. Therefore, predicting accurate wake flow will help carry out the drag reduction strategies. In this paper, the near-wake flow topology of the GTS was studied at Re = 2.7×104 to assess the capability of a hybrid RANS/LES (Reynolds-averaged Navier–Stokes/large eddy simulation) approach, known as IDDES (improved delayed detached eddy simulation). The current study also aims to understand the effects of different computational parameters, e.g. the spatial resolution, time step, residual level, discretization scheme and turbulence model, on this asymmetrical wake flow configuration. A comparison of IDDES with previous water channel tests, well-resolved LES, partially averaged Navier–Stokes and URANS (unsteady RANS) was included to better understand the benefits of this hybrid RANS/LES approach. The results show that on the medium and fine grids, the IDDES produces an asymmetrical flow topology (known as flow state I) in the near-wake of the vertical midplane, as reported in previous studies. The recommended parameters for the time step (1×10–4 s) and residual level (1×10–4) provide sufficient accuracy of wake predictions to show good agreement with experiments. For the convective term of the momentum equation in IDDES, the bounded central difference discretization scheme is proposed to be adopted for discretization. Additionally, URANS cannot accurately capture this asymmetrical flow field. IDDES proves to be capable of predicting the wake flow field of this simplified heavy vehicle with high accuracy. All obtained conclusions can provide references for the aerodynamic drag reduction of the GTS.

A time-fractional dual-phase-lag framework to investigate transistors with TMTC channels (TiS3, In4Se3) and size-dependent properties

In this study, a time fractional dual-phase-lag model with temperature jump boundary condition as a choice for the Fourier's law replacement in thermal modeling of transistors, is utilized. In more details, the numerical simulation of heat transfer in newly proposed TMTC field effect transistors using fractional DPL equation has been investigated. Moreover, the Caputo fractional derivative is employed to formulate the finite difference scheme for discretization of the fractional DPL model. In order to obtain more precise results for the peak temperature rise, the temperature and heat flux profiles, the size-dependent thermal properties are taken into account. Also, the temperature jump boundary condition has been also applied by means of a mixed-type boundary condition. It is obtained that considering size-dependent thermal characteristics for transistors under study, results in an increase of the peak temperature rise. As an instance, considering the film dependent thermal properties for a 1-D silicone MOSFET with uniform heat generation at the upper part, makes the maximum temperature increase up to 250%. Furthermore, considering constant bulk thermal properties for the silicon MOSFET, certain oscillations are observed in the time-variation of the peak temperature rise for α = 0.7, 0.9 and 1. This presents the so-called negative bias temperature instability appearing in electronic nano-semiconductor devices. Finally, the hotspot temperature has been researched in transistors containing two-dimensional materials with quasi one-dimensional band structure channels. It is obtained that among the studied FETs, titanium trisulfide with maximum temperature increase of 19.63 K exhibits the least peak temperature rise. This presents that TiS3 may be an acceptable silicon channel replacement as far as the thermal issues are concerned.

A direct RBF-PU method for simulating the infiltration of cytotoxic T-lymphocytes into the tumor microenvironment

Predicting the antitumor responses of cytotoxic T-lymphocytes (i.e., CD8+T cells) can be accomplished by numerical simulations of a mathematical model of tumor-immune cell interactions to control tumor growth in cancer therapy. Cytotoxic T lymphocytes are considering as the main effector cells to target tumor cells that are in a dormant avascular state through the chemotactic mechanism and also during the cytotoxic activity of immune cells. We study a dynamical model that is formulated based on the nonlinear partial differential equations describing the interactions between tumor-infiltrating cytotoxic lymphocytes, tumor cells, TICL-tumor cell complexes, inactivated tumor-infiltrating cytotoxic lymphocytes, dead tumor cells, and chemokine. Here, a new meshless method, i.e., the direct radial basis function partition of unity (RBF-PU) technique for the spatial discretization and a semi-implicit backward differentiation formula of order 1 for the temporal discretization are employed. The proposed meshless technique is a generalized form of the original RBF-PU method such that the computational cost of computing the derivatives could be reduced in comparison with its standard form. Besides, the method depends only on the location of nodes located on the domain of the problem for the approximation, where both types of radial functions, i.e., a conditionally positive definite function, i.e., the polyharmonic splines and a positive definite function, namely Matérn function have been used The obtained fully discrete difference scheme based on the new spatial approximation is solved using an iterative method, i.e., the generalized minimal residual method with a zero-fill incomplete lower–upper preconditioner. Finally, we present some simulation results, in which we conclude that the tumor’s growth entirely depends on CTLs activation, and this behavior can be predicted by simulating the proposed mathematical model, which may have a significant role in improving cancer immunotherapy. Besides, the developed numerical method is more accurate than the classical finite element method for solving the studied model as shown in one example.

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.

Dynamic Measurements with the Bicone Interfacial Shear Rheometer: The Effects of the Numerical Implementation of the Interfacial Boundary Condition

The increasing popularity of flow field-based data analysis (FFBDA) techniques has a paradigmatic example in the routines already developed for the rotational oscillating bicone bob interfacial shear rheometer. Such routines use a second order centered finite difference (SOCFD) discretization scheme, in both the vertical and radial coordinates, for the velocity field in the bulk fluid subphase and a first order forward finite difference (FOFFD) scheme in the vertical coordinate for the velocity field at the air/water interface. Such a mixture of schemes causes non-smooth flow fields at the interface that can be tackled by appropriately devising a SOCFD scheme for the vertical coordinate at the interface using a line of “phantom” nodes that merely serve to adequately merge the Navier–Stokes equations and the Boussinesq–Scriven boundary condition at the interface. Here we report on a detailed analysis of the quantitative improvements of such a scheme over the previous one by comparing the structure of the flow fields at and close to the interface, the differences in the interfacial and bulk drag torques on the bicone bob, and the differences in the torque/displacement complex amplitude ratio.

Nonlinear Supratransmission in Quartic Hamiltonian Lattices With Globally Interacting Particles and On-Site Potentials

Abstract We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. To simulate a semi-infinite chain, we will consider a relatively long chain with string coupling. Using reliable finite difference discretization schemes, we establish the existence of nonlinear supratransmission for both short-range and long-range interactions, and demonstrate that the presence of quadratic interactions is not necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on-site potentials is also not crucial for the system to present supratransmission.

A Crank–Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity

A Crank–Nicolson-type finite-difference scheme is proposed and analyzed for a nonlinear partial integro-differential equation arising from viscoelasticity. The time derivative is approximated by the Crank–Nicolson scheme and the Riemann–Liouville fractional integral term is treated by means of the trapezoidal convolution quadrature rule. To construct a fully discrete difference scheme, the standard centered difference formula is utilized to approximate the second-order spatial derivative and the Galerkin method based on piecewise linear test functions is used to discrete the nonlinear convection term. Theoretical results of stability and convergence are derived using the non-negative character of the real quadratic form associated with the convolution quadrature, and combining with the discrete Gronwall inequality. Besides, a fixed point iterative numerical algorithm, which fills the gap that the existed numerical schemes have only theoretical analysis but no numerical results, is presented. Numerical results show the efficiency and feasibility of our scheme, and the orders of convergence are in good agreement with the theoretical results.

Simulation of non-equilibrium gas kinetic processes in the multitube Knudsen pump on the basis of the Boltzmann equation

A periodic system of multitube micropumps with pumping down based on the Knudsen effect is simulated in this work. The simulation is performed on an unstructured mesh. The Boltzmann equation is solved, and the advection and relaxation processes are simulated separately. The differential equation for advection is approximated with the first-order discrete difference scheme, and the collision integral is calculated with the projection method. The dependences of pump-down on the Knudsen number and temperature gradient were obtained from supercomputer computations. Data on the pumping rate were also obtained.

Three semi-implicit compact finite difference schemes for the nonlinear partial integro-differential equation arising from viscoelasticity

ABSTRACT Three semi-implicit compact finite difference schemes are described for the nonlinear partial integro-differential equation arising from viscoelasticity. In our methods, the time derivative is approximated by the backward-Euler, Crank–Nicolson-type and formally second-order backward differentiation formula scheme, respectively, and the convolution quadrature formula is employed to process the Riemann–Liouville fractional integral term. Fully discrete difference schemes are constructed with the spatial discretization by the compact finite difference formula. Meanwhile, the semi-implicit technique is used to deal with nonlinear convection term uux . In the numerical experiment, the comparisons between our methods and existing methods demonstrate the efficiency of our methods.

Numerical Solutions of Unsteady Advection-Diffusion Equations by Using EG Iteration with Wave Variable Transformation

The primary goal of this paper is to investigate the effectiveness of the 4-point Explicit Group (4-point EG) iterative method for solving one-dimensional unsteady advection-diffusion problems via similarity transform. By using this transformation approach, the proposed problem can be reduced into the corresponding two-point boundary volume problem. By imposing the second-order central finite difference discretization scheme, then the corresponding approximation equation can be derived to construct a system of linear equations. Having a large linear system, the 4-point EG iterative method has been used to solve the generated system of linear equations. The formulation of the 4-point EG method is also derived. Some numerical experiments are conducted that to verify the 4-point EG method is more effective than the Gauss-Seidel (GS) method.

Four-Point EGSOR Iteration for the Grünwald Implicit Finite Difference Solution of One-Dimensional Time-Fractional Parabolic Equations

In this paper, our main concerned is on the application of the formulation of a four-point explicit group successive over-relaxation (4EGSOR) iterative method in solving one-dimensional time-fractional parabolic equations based on the second-order Grünwald implicit approximation equation. The formulation of the 4EGSOR method is constructed by using the implicit approximation equation which is derived by the Grunwald derivative operator and the implicit finite difference discretization scheme. In order to access the performance results of the 4EGSOR iterative method, another block and point iterative methods which are the four-point EGGS (4EGGS) and the Gauss-Seidel (GS) were also presented as control methods. The results of three numerical experiments show substantial improvement in terms of the number of iterations and execution time.

An Efficient Nonmonotone Method for State-Constrained Elliptic Optimal Control Problems

This paper presents a novel numerical strategy based on combination of an adaptive semismooth Newton (ASN) method and the Lavrentiev regularization technique for the solution of elliptic optimal control problems with state constraints. Using the global convergence proof for a nonmonotone semismooth Newton method, we will exploit an adaptive nonmonotone line search method such that the nonmonotonicity degree of this method can be increased when the results are far from the optimum solution and it can be reduced when they are close to the optimizer. In this strategy, the role of the Lavrentiev regularization technique is converting the original optimal control problem to a regularized optimal control problem. Using the finite difference discretization scheme and a Newton–Cotes rule, the regularized optimal control problem is converted to a bound constrained optimization problem (BCOP). Then the ASN method is implemented to solve the resulting BCOP. Numerical results show the efficiency of the proposed procedure.

A formally second-order BDF finite difference scheme for the integro-differential equations with the multi-term kernels

In this article, a formally second-order backward differentiation formula (BDF) finite difference scheme is presented for the integro-differential equations with the multi-term kernels. In the time direction, the time derivative is approximated by a second-order BDF scheme and the Riemann-Liouville (R-L) fractional integral terms are discretized by the second-order convolution quadrature rule. We construct a fully discrete difference scheme with the space discretization by the standard central difference formula. The and -norms stability, and convergence in -norm are derived by the discrete energy method. In the numerical experiments, the results are consistent with the theoretical analysis.

Finite Volume Monte Carlo (FVMC) method for the analysis of conduction heat transfer

The numerical solution of the heat equation is a particularly challenging subject in complex, practical applications such as functionally graded materials for which analytical solution is not available or hardly attainable. The Monte Carlo method is a powerful technique with some advantages compared to the conventional methods and is often used when all else fail. In this paper, we introduce the Finite Volume Monte Carlo (FVMC) method for solving 3D steady-state heat equation where, instead of using the usual finite difference scheme for discretization of the heat equation, the finite volume scheme is used. The FVMC method is tested for three problems to assess the robustness of the method, first one in a simple geometry for validation and evaluation of the predictive performance, the second one in a complex geometry with unstructured mesh and the last one in a problem with a variable heat source and different kinds of boundary conditions. Comparisons were made to the analytical solution in the first test case, whereas for the remaining test cases, the CFD methods were utilized in the absence of the analytical solutions. It was observed that the FVMC temperature distribution agrees perfectly with analytical and CFD solutions in all problems. Despite expecting computational accuracy to improve by increasing total number of particles in the FVMC method, a very good accuracy was obtained for all considered problems after a small number of walks, and the calculated relative root-mean-square errors were below 1%.

A backward Euler difference scheme for the integro-differential equations with the multi-term kernels

ABSTRACTA backward Euler difference scheme is formulated and analyzed for the integro-differential equations with the multi-term kernels. The convolution quadrature is used to deal with the Riemann-Liouville fractional integral terms. A fully discrete difference scheme is constructed with time derivative by backward Euler scheme and space discretization by standard central difference approximation. We prove the stability and convergence based on the nonnegative character of the real quadratic form. The and -norms stability and convergence are given. Finally, numerical experiments validate the theoretical results.

Mathematical Modelling and Simulation of a Trickle-Bed Reactor for Hydrotreating of Petroleum Feedstock

Abstract In this work, a model for a trickle-bed reactor for catalytic hydrotreating (HDT) of oil fractions is developed and simulations are performed to investigate its behavior. The model considers dynamic, one-dimensional plug-flow to describe a heterogeneous, adiabatic trickle-bed reactor. It takes into consideration the main reactions present in the HDT process: hydrodesulfurization (HDS), hydrodenitrogenation (HDN), and hydrodearomatization (HDA) with a reconstructed petroleum feedstock using a practical approach of generation of pseudo-components by dividing the boiling point curves of the feedstock. The model is solved using the method of lines with a finite difference scheme for discretization in the axial direction and simulations are performed for an industrial hydrotreating unit to evaluate the behavior of the system under different conditions and assumptions e. g. related to the linear gas velocity. A study of the dynamics is carried out to investigate the behavior of the system with a change in the sulfur compound concentration of the feed. In addition, a sensitivity analysis of the most relevant model parameters is performed.