Second-order Difference Scheme Research Articles

Temporal second-order difference schemes for the nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay

This paper investigates a nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay. The problem is particularly challenging due to its nonlinear nature, the presence of a time delay, and the incorporation of both fractional diffusion and fractional wave terms, introducing computational complexities for numerical analysis. To address this, we transform the model into a new generalized form involving Riemann–Liouville fractional integrals and a Caputo derivative of order α∈(0,1). Subsequently, a temporal second-order linearized difference scheme is presented to approximate the model, and its unconditional stability is rigorously proven based on discrete Gronwall’s inequality. To reduce computational and storage costs, we extend the discussion to a fast variant of the proposed difference scheme. The sum of exponentials approach is employed to approximate the kernel function in fractional-order operators, leading to fast difference analogs for the Riemann–Liouville fractional integral and the Caputo derivative. A fast variant of the developed direct method is introduced based on these analogs. Numerical results are provided to validate our theoretical findings and assess the accuracy and efficiency of the difference schemes.

A second-order three-point difference scheme on a posteriori mesh for a singularly perturbed convection–diffusion problem

In this paper a three-point difference scheme based on a posteriori mesh is used to solve a second-order singularly perturbed convection–diffusion problem. A hybrid difference method is constructed on an arbitrary mesh. A posteriori error analysis is developed for the three-point difference scheme on an arbitrary mesh. A solution-adaptive algorithm based on a posteriori error estimation is devised by equidistributing a monitor function. Numerical experiments illustrate that the scheme is second-order uniformly convergent.

3-D forward modeling of shale gas fracturing dynamic monitoring using the borehole-to-ground transient electromagnetic method

The borehole-ground transient electromagnetic method enhances the detection range and resolution by placing the transmitter electrode within the borehole, allowing for close-proximity excitation of subsurface targets and compensating for the shortcomings of traditional ground-based methods. Utilizing an unstructured vector finite element method combined with a second-order backward Euler variable time-stepping difference scheme and the MUMPS solver, we have achieved three-dimensional forward modeling simulation of the transient electromagnetic field for a borehole-ground electric source. Based on the validation of its accuracy, we analyzed the effectiveness of this method for dynamic monitoring of shale gas reservoir fracturing. Through forward modeling, we examined the characteristics of the radial electric field response during shale gas reservoir fracturing monitoring and evaluated the effectiveness of the borehole-ground TEM method in dynamic monitoring of shale gas reservoirs. The research results indicate that this method can meet the requirements for reservoir fracturing dynamic monitoring and has a promising application prospect.

A second-order-accuracy finite difference scheme for numerical computation on two-dimensional three-neighboring-node unstructured grids

The computation of the finite difference method in curvilinear coordinates usually necessitates coordinate transformation. Eliminating geometric errors induced by coordinate transformation can be achieved with geometric conservation law. However, no research has been conducted to implement a difference scheme on two-dimensional three-neighboring-node unstructured meshes and achieve a second-order accuracy. In this paper, a new finite difference method is proposed for a numerical simulation on these meshes, and it can achieve a second-order accuracy with the use of the gradient interpolation and the discrete criterion, freezing the metrics and Jacobian at local nodes. The results of several numerical tests demonstrate that the method achieved reliable performance and robustness on unique and randomized unstructured grids for a complex flow with discontinuity, subsonic inflow, and diffraction. The method had comparable accuracy to the second-order difference scheme on structured grids.

A Temporal Second-Order Difference Scheme for Variable-Order-Time Fractional-Sub-Diffusion Equations of the Fourth Order

In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for the existence and uniqueness, as well as the stability of scheme, along with a priori error estimates. Finally, we validate our theoretical results through various numerical computations.

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

On the unique weak solvability of second-order unconditionally stable difference scheme for the system of sine-Gordon equations

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements.

Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

A second-order difference scheme for two-dimensional two-sided space distributed-order fractional diffusion equations with variable coefficients

A second-order difference scheme for two-dimensional two-sided space distributed-order fractional diffusion equations with variable coefficients

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

A modification of the numerical method of characteristics is developed to solve the initial-boundary value problem that arises when modeling the rectification process in the column. The process is described by a system of first-order hyperbolic equations. A specific peculiarity of the model is in the boundary conditions of a special type. At each of the boundaries, boundary conditions are determined from a system of ordinary differential equations, which also includes unknown values of functions on another boundary. A characteristic difference grid is constructed on the base of a linear transformation of a classical rectangular grid. Implicit second-order difference schemes are used, taking into account the features of the problem at the boundaries. The advantage of this approach is in consideration of the specifics of the propagation of perturbations in hyperbolic equations. Numerical implementation of the method was carried out. An illustrative example shows the effectiveness of the proposed modification of the characterization method. This method is a base for further solution of optimal control problems of flows in columns.

Theoretical and numerical study on the well-balanced regularized lattice Boltzmann model for two-phase flow.

In the multiphase flow simulations based on the lattice Boltzmann equation(LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation(WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.

Differential-Difference Equations with Optimal Parameters

The paper considers difference schemes with optimal parameters for solving Maxwell’s equations. Using Laguerre transforms, the numerical values of the optimal parameters are determined and differential-difference equations are constructed. Differential-difference equations are solved by the finite-difference method with iterations over small optimal parameters. Optimal second-order difference schemes for one-dimensional and two-dimensional Maxwell’s equations are considered. Optimal parameters of difference schemes are given. It is shown that the use of optimal difference schemes leads to an increase in the accuracy of solution.

Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O(nlog⁡n) per iteration and the memory requirement of O(n), where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques.

Efficient Finite Difference Approaches for Solving Initial Boundary Value Problems in Helmholtz Partial Differential Equations

This study presents numerical solutions for initial boundary value problems of homogeneous and nonhomogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes’ robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.

A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.

Bounded Solutions of Semi-Linear Parabolic Differential Equations with Unbounded Delay Terms

In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS) of this problem is established. The application of the main theorem to four different semi-linear delay parabolic differential equations is presented. The first- and second-order accuracy difference schemes (FSADSs) for the solution of a one-dimensional semi-linear time-delay parabolic equation are considered. The new desired numerical results of this paper and their discussion are presented.

NUMERICAL SIMULATION OF A FLUID FLOW IN THE CYLINDRICAL DUCT CONTAINING TWO DIAPHRAGMS WITH ORIFICES OF DIFFERENT DIAMETERS

The flow of a viscous incompressible fluid in a cylindrical duct with two serial diaphragms with orifices of different diameters was studied based on the numerical solution of the unsteady Navier-Stokes equations. The solution algorithm was based on the finite volume method using second-order accurate difference schemes in both space and time. The TVD (Total-Variation Diminishing) form of a central-difference scheme with a flux limiter was used for the interpolation of the convective terms. The combined evaluation of the velocity and pressure fields was carried out using the PISO (Pressure Implicit Split Operator) procedure. The problem was solved using OpenFOAM (Open-source Field Operation And Manipulation) open-source toolkit libraries using the computing power of the cluster supercomputer of the V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine. It was shown that within a certain range of the diameter ratio of the orifices of the diaphragms, a circulation movement is established in the cavity between the diaphragms. The boundary layer breaks off from the surface of the first diaphragm and forms an annular shear layer. When approaching the second diaphragm, a sequence of ring vortices is formed in it, which interact with the surface of the diaphragm and lead to the emergence of tonal sound. When the ratio of the orifice diameter of the second diaphragm to the first decreases, the share of the jet’s kinetic energy, which participates in the circulation in the middle of the cavity between the diaphragms, increases. As a result, the amplitude of velocity fluctuations in the orifice of the second diaphragm decreases. When the ratio of the diameters of the orifices increases, the share of energy involved in the circulation decreases, and when a certain value of the ratio is reached, the interaction between the vortices in the shear layer and the surface of the diaphragm does not occur. As a result, the excitation of the tonal sound stops. The Strouhal number of self-oscillations practically does not change when the diameter ratio of the orifices changes. During the calculations, two different modes of self-oscillation were obtained, which is consistent with previous works.

A Second-Order Difference Scheme for Solving a Class of Fractional Differential Equations

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.

A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions

Abstract In the current work, we build a difference analog of the Caputo fractional derivative with generalized memory kernel (𝜇L2-1𝜎 formula). The fundamental features of this difference operator are studied, and on its ground, some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid L 2 L_{2} -norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.