Second-order Difference Scheme Research Articles

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.

The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems

This is a discussion on the second-order accuracy difference schemes for approximate solution of the integral-type time-nonlocal parabolic problems. The theorems on the stability of r-modified Crank-Nicolson difference schemes and second-order accuracy implicit difference scheme for approximate solution of the integral-type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second-order accuracy in t difference schemes for the one and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given.

Improved fast iterative method for higher calculation accuracy of traveltimes

Fast Iterative Method (FIM), a grid-based ray-tracing algorithm, has a computational efficiency far exceeding that of conventional fast marching method (FMM) in serial computing, because it adopts an iterative algorithm and can be calculated in parallel. However, the finite difference scheme of the FIM only employs the first-order Godunov upwind finite difference scheme, resulting in low calculation accuracy of arrival traveltimes. This paper proposes a new algorithm of FIM, which utilizes second-order and mixed-order difference schemes under the parallel computing structure, effectively improves the calculation accuracy of traveltimes, and basically maintains the original computational efficiency. Especially, improved finite difference calculation in the diagonal direction of square grids and the double-grid technique near the source are applied to further reduce overall calculation error. Numerical experiments show that for the given model, the calculation error can be reduced from 2 ms of the original FIM calculation to 0.05 ms, and the calculation accuracy is increased by 40 times, while the time consumption increases only 27%.

Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation

In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers’ equation, from which L1-discretization formula is applied to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete L1 compact difference scheme on uniform meshes can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability of the proposed scheme are rigorously proved in the L∞-norm by the energy argument and mathematical induction. We also establish a temporal second-order compact difference scheme on graded time meshes for solving the problem with weak initial singularity. Finally, several numerical experiments are provided to test the accuracy of two numerical schemes and verify the theoretical analysis.

Physically plausible and conservative solutions to Navier–Stokes equations using physics-informed CNNs

The physics-informed neural network (PINN) is an emerging approach for efficiently solving partial differential equations (PDEs) using neural networks. The physics-informed convolutional neural network (PICNN), a variant of PINN enhanced by convolutional neural networks (CNNs), has achieved better results on a series of PDEs since the parameter-sharing property of CNNs is effective in learning spatial dependencies. However, applying existing PICNN-based methods to solve Navier–Stokes equations can generate oscillating predictions, which are inconsistent with the laws of physics and the conservation properties. To address this issue, we propose a novel method that combines PICNN with the finite volume method to obtain physically plausible and conservative solutions to Navier–Stokes equations. We derive the second-order upwind difference scheme of Navier–Stokes equations using the finite volume method. Then we use the derived scheme to calculate the partial derivatives and construct the physics-informed loss function. The proposed method is assessed by experiments on steady-state Navier–Stokes equations under different scenarios, including convective heat transfer and lid-driven cavity flow. The experimental results demonstrate that our method can effectively improve the plausibility and accuracy of the predicted solutions from PICNN.

Parameter uniform second-order numerical approximation for the integro-differential equations involving boundary layers

The work handles a Fredholm integro-differential equation involving boundary layers. A fitted second-order difference scheme has been created on a uniform mesh utilizing interpolating quadrature rules and exponential basis functions. The stability and convergence of the proposed discretization technique are analyzed and one example is solved to display the advantages of the presented technique.

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

Stable and efficient time second-order difference schemes for fractional Klein–Gordon–Zakharov system

In this paper, we develop and analyze two stable and efficient time second-order difference schemes for space fractional Klein–Gordon–Zakharov (KGZ) system. The former is based on the multi-point weighted time second-order scheme to construct energy conservative linearized difference scheme. The latter is on the basis of time second-order splitting technique to develop accurate, efficient, linear and decoupled difference scheme. A main idea of the latter is to split the original fractional KGZ system into linear and nonlinear parts, and then to advance the subproblems with three stages. Furthermore, by utilizing the cut-off technique, energy analysis method and mathematical induction method, we obtain the error estimates of the former scheme, without any restrictive conditions on the grid ratio, compared with the restrictive conditions demanded in the existing literature. The feature of proposed schemes are linear decoupled, and easy to be applied in parallel computing, especially in long time simulations. At last, some numerical results are provided to validate the accuracy and efficiency of the proposed schemes, and simulate wave dynamics and interaction of the fractional KGZ system in one and two dimensions.

Constructing Exact and Approximate Diffusion Wave Solutions for a Quasilinear Parabolic Equation with Power Nonlinearities

The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existence of these solutions being proved in our previous studies. A stepwise algorithm of the numerical solution with a time-difference scheme is proposed, the second-order difference scheme being used in such problems for the first time. At each step the problem is solved iteratively on the basis of a radial basis function (RBF) collocation method. In order to verify the numerical solution algorithm, two classes of exact generalized traveling wave solutions are proposed, whose construction is reduced to solving a Cauchy problem for second order ordinary differential equations (ODEs) with a singularity at the higher derivative. The theorem of the existence and uniqueness of the analytical solution in the form of a power series is proved for it, and the estimates of the radius of convergence are obtained. The Euler method is used to prove a similar statement concerning the existence of a continuous solution in the non-analytical case. The RBF collocation method is also applied for the approximate solution of the Cauchy problem. The solutions to the Cauchy problem are numerically analyzed, and this has enabled us to reveal and describe some of their properties, including those not previously observed, and to assess the accuracy of the method.

A second-order difference scheme for the singularly perturbed Sobolev problems with third type boundary conditions on Bakhvalov mesh

We consider a singularly perturbed initial-third boundary value Sobolev problem. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete energy norm. Finally, three numerical examples are solved to validate the theory.

A high-order numerical scheme for right Caputo fractional differential equations with uniform accuracy

<abstract><p>In this paper, we mainly study the high-order numerical scheme of right Caputo time fractional differential equations with uniform accuracy. Firstly, we construct the high-order finite difference method for the right Caputo fractional ordinary differential equations (FODEs) based on piecewise quadratic interpolation. The local truncation error of right Caputo FODEs is given, and the stability analysis of the right Caputo FODEs is proved in detail. Secondly, the time fractional partial differential equations (FPDEs) with right Caputo fractional derivative is studied by coupling the time-dependent high-order finite difference method and the spatial central second-order difference scheme. Finally, three numerical examples are used to verify that the convergence order of high-order numerical scheme is $ 3-\lambda $ in time with uniform accuracy.</p></abstract>

A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem

This work constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space. In the temporal direction, the Crank-Nicolson (CN) method and product-integration (PI) rule are employed, from which the non-uniform meshes are used to eliminate the singular behaviour of the exact solution at t=0. Then, a fully discrete scheme is obtained by the spatial discretization based on the finite difference method. Simultaneously, an alternating direction implicit (ADI) algorithm is designed to reduce the computational cost. The stability in L2 norm and convergence are derived via the energy method, in which the convergence orders are O(k2+h2) with certain suitable assumptions, where k and h are corresponding space-time step sizes, respectively. Numerical results confirm the theoretical analysis.

Numerical simulation of flow field in a Laval nozzle based on one-dimensional Euler equation

Numerical simulation refers to a computer research method which runs calculation based on a specific mathematical model to simulate actual physical processes. It is a powerful tool to analyse complex engineering problems. In this paper, a numerical method for solving the steady one-dimensional Euler equations using a two-step second-order difference scheme is developed. The method is implemented by a Python code. The method is applied to numerical simulation of flows in a Laval nozzle and used to investigate the influence of shapes of Laval nozzle. It is found that the physical quantities of the nozzle flow show a positive correlation trend with different throat positions. In terms of the temperature, density, and pressure, they increase during the initial evolutions, then reach the maximum point and produce significant fluctuations, indicating the flow flows into the transonic stage. Subsequently, those physical properties gradually tend to a stable value during the supersonic stage. Moreover, it is observed that the closer the throat is to the exit, the lower the Mach number at which it eventually stabilizes. Finally, a suggestion about the best shape of the nozzle which can realize the max efficiency is concluded. Despite using an inviscid flow model, the steady pressures are quite satisfactorily predicted over the range of frequencies studied.

A FAST SOLVER BASED ON DEEP NEURAL NETWORK FOR DIFFERENCE EQUATION

In recent years, artificial neural networks (ANNs), especially deep neural networks (DNNs), have become a promising new approach in the field of numerical computation due to their high computational efficiency on heterogeneous platforms and their ability to fit high-dimensional complex systems. In the process of numerically solving the partial differential equations, the large-scale linear equations are usually the most time-consuming problems; therefore, utilizing the neural network methods to solve linear equations has become a promising new idea. However, the direct prediction of deep neural networks still has obvious shortcomings in numerical accuracy, which becomes one of the bottlenecks for its application in the field of numerical computation. To break this limitation, a solving algorithm combining Residual network architecture and correction iteration method is proposed in this paper. In this paper, a deep neural network-based method for solving linear equations is proposed to accelerate the solving process of partial differential equations on heterogeneous platforms. Specifically, Residual network resolves the problems of network degradation and gradient vanishing of deep network models, reducing the loss of the network to 1/5000 of the classical network model; the correction iteration method iteratively reduce the error of the prediction solution based on the same network model, and the residual of the prediction solution has been decreased to 10−5 times of that before the iteration. To verify the effectiveness and universality of the proposed method, we combined the method with the finite difference method to solve the heat conduction equation and the Burger’s equation. Numerical results demonstrate that the algorithm has more than 10 times the acceleration effect for equations of size larger than 1000, and the numerical error is lower than the discrete error of the second-order difference scheme.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.

The discrete maximum principle and energy stability of a new second-order difference scheme for Allen-Cahn equations

This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated.

An efficient numerical method for pricing a Russian option with a finite time horizon

In this paper, we present a finite difference scheme for a linear complementarity problem with a mixed boundary condition arising from pricing a Russian option with a finite time horizon. An implicit Euler method for the temporal discretization and second-order difference schemes on a piecewise uniform mesh for the spatial discretization are used to solve the linear complementarity problem with a mixed boundary condition. It is shown that the transformed discrete operator satisfies a maximum principle, which is used to derive the error estimate. It is proved that the scheme is first- and second-order convergent with respect to the temporal and spatial variables, respectively. Numerical experiments verify the validity of the theoretical results.

Mixed Finite Element-Second Order Upwind Fractional Step Difference Scheme of Darcy–Forchheimer Miscible Displacement and Its Numerical Analysis

The incompressible miscible displacement of three-dimensional Darcy–Forchheimer flow is discussed in this paper, and the mathematical model is formulated by two partial differential equations, a Darcy–Forchheimer flow equation for the pressure and a convection-diffusion equation for the concentration. The pressure plays an important role and determines the Darcy–Forchheimer velocity. A conservative mixed finite element is used to approximate the pressure and the velocity, and the computational accuracy of Darcy–Forchheimer velocity is improved by one order. A second-order upwind fractional step difference scheme is adopted to obtain the concentration, and numerical oscillation and dispersion are eliminated. Computational work is reduced greatly by decomposing the whole computation into successive one-dimensional subcomputations, where the speedup solvers are used. Applying energy-norm method, operator decomposition, speedup algorithm and induction hypotheses, we derive an optimal second-order estimates in $$L^2$$ norm. Numerical experiments are given to show the theoretical accuracy and the applicability for solving actual problems.

Mathematical modeling and research of heat and moisture transfer processes in porous media

A mathematical model, numerical algorithm, solution, and results of the computational experiment on a computer are developed to predict the process of heat and moisture transfer in porous media, taking into account such factors as the internal heat and moisture release of a porous natural product on the example of raw cotton and its products of hulling, seeds of various crops. It also considers the effect of temperature and moisture content changes in the environment on the storage and drying of porous materials. In this study, the developed mathematical support of the object under research makes it possible to predict the change in temperature and moisture content at arbitrary points of the porous body and serves to prevent the loss of quality and spontaneous combustion of materials under solar radiation and to analyze and make managerial decisions. Based on the method of coordinate-wise splitting, a numerical algorithm for calculating three-dimensional heat and moisture transfer problems in areas of the parallelepiped type is presented. An implicit second-order difference scheme for calculating the required functions is presented. Based on the numerical calculations performed, it was established that moisture and heat transfer and their exchange with the environment occurs in the upper layers of the cotton pile; in the inner layers, there is an increase in temperature and moisture due to the respiration of raw cotton and its seeds, which depends on the degree of moisture of the raw cotton and their products of hulling.

Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation

In the Galerkin weak form technique based on various kernels that they do not have δ-Kronecker property, in order to apply the essential boundary condition, there are two straight strategies that one of them is the Lagrange multiplier method and another one is the penalty method. In the penalty method the main boundary value problem (BVP) is converted to a new BVP with Robin boundary condition. So, we obtain a new BVP that it must be solved. The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. For this aim, we select fractional modified distributed-order anomalous sub-diffusion equation. At the first stage, we propose a second-order difference scheme for the temporal variable. The convergence and stability analysis for the time-discrete scheme are proposed. At the second stage, we derive the full-discrete scheme based on the Galerkin weak form and shape functions of reproducing kernel particle method (RKPM) as the mentioned shape functions do not have the δ-Kronecker property. Furthermore, it is shown that when the penalty parameter goes to infinity then the solutions of BVP with Robin boundary condition are convergent to the solutions of BVP based on the essential boundary condition. The proposed examples verify that the present error estimate is true.