Showing papers on "Finite difference method published in 2020"

Conservative Finite-Difference Methods on General Grids

Abstract: INTRODUCTION Governing Equations Elliptic Equations Heat Equation Equation of Gas Dynamic in Lagrangian Form The Main Ideas of Finite-Difference Algorithms 1-D Case 2-D Case Methods of Solution of Systems of Linear Algebraic Equation Methods of Solution of Systems of Nonlinear Equations METHOD OF SUPPORT-OPERATORS Main Stages The Elliptic Equations Gas Dynamic Equations System of Consistent Difference Operators in 1-D Inner Product in Spaces of Difference Functions and Properties of Difference Operators System of Consistent Difference Operators in 2-D THE ELLIPTIC EQUATIONS Introduction Continuum Elliptic Problems with Dirichlet Boundary Conditions Continuum Elliptic Problems with Robin Boundary Conditions One-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Centered Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Two-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Valued Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Conclusion Two-Dimensional Support Operator Algorithms Discretization Spaces of Discrete Functions The Prime Operator The Derived Operator Multiplication by a Matrix and the Operator D The Difference Scheme for the Elliptic Operator The Matrix Problem Approximation and Convergence Properties HEAT EQUATION Introduction Finite-Difference Schemes for Heat Equation in 1-D Finite-Difference Schemes for Heat Equation in 2-D LAGRANGIAN GAS DYNAMICS Kinematics of Fluid Motions Integral Form of Gas Dynamics Equations Integral Equations for One Dimensional Case Differential Equations of Gas Dynamics in Lagrangian Form The Differential Equations in 1D. Lagrange Mass Variables The Statements of Gas Dynamics Problems in Lagrange Variables Different Forms of Energy Equation Acoustic Equations Reference Information Characteristic Form of Gas Dynamics Equations Riemann's Invariants Discontinuous Solutions Conservation Laws and Properties of First Order Invariant Operators Finite-Difference Algorithm in 1D Discretization in 1D Discrete Operators in 1D Semi-Discrete Finite-Difference Scheme in 1D Fully Discrete, Explicit, Computational Algorithm Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Stability Conditions Homogeneous Finite-Difference Schemes. Artificial Viscosity Artificial Viscosity in 1D Numerical Example Finite Difference Algorithm in 2D Discretization in 2D Discrete Operators in 2D Semi-Discrete Finite-Difference Scheme in 2D Stability Conditions Finite-Difference Algorithm in 2D Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Artificial Viscosity in 2D Numerical Example APPENDIX: FORTRAN CODE DIRECTORY General Description of Structure of Directories on the Disk Programs for Elliptic Equations Programs for 1D Equations Programs for 2D Equations Programs for Heat Equations Programs for 1D Equations Programs for 2D Equations Programs for Gas Dynamics Equations Programs for 1D Equations Programs for 2D Equations Bibliography

Dynamical Modeling and Boundary Vibration Control of a Rigid-Flexible Wing System

Abstract: A boundary control approach is used to control a two-link rigid-flexible wing in this article. Its design is based on the principle of bionics to improve the mobility and the flexibility of aircraft. First, a series of partial differential equations (PDEs) and ordinary differential equations (ODEs) are derived through the Hamilton's principle. These PDEs and ODEs describe the governing equations and the boundary conditions of the system, respectively. Then, a control strategy is developed to achieve the objectives including restraining the vibrations in bending and twisting deflections of the flexible link of the wing and achieving the desired angular position of the wing. By using Lyapunov's direct method, the wing system is proven to be stable. The numerical simulations are carried out with the finite difference method to prove the effectiveness of designed boundary controllers.

Analytical and numerical study of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model

Abstract: In this work, we introduce a numerical and analytical study of the Peyrard-Bishop DNA dynamic model equation. This model is studied analytically by hyperbolic and exponential ansatz methods and numerically by finite difference method. A comparison between the results obtained by the analytical methods and the numerical method is investigated. Furthermore, some figures are introduced to show how accurate the solutions will be obtained from the analytical and numerical methods.

High-precision quantum algorithms for partial differential equations

Abstract: Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $\mathrm{poly}(1/\epsilon)$, where $\epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $\mathrm{poly}(d, \log(1/\epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

An efficient numerical technique for solving time fractional Burgers equation

Abstract: A finite difference scheme which depends on a new approximation based on an extended cubic B-spline for the second order derivative is used to calculate the numerical outcomes of time fractional Burgers equation. The presented scheme uses Caputo’s formulation for the time derivative. Finite difference method will be used to discretize the Caputo’s fractional derivative. The proposed scheme will be shown to be unconditionally stable by Von-Neumann method. The convergence analysis of the numerical scheme will be presented of order O ( h 2 + τ 2 - α ) . The presented scheme is tested on four numerical examples. The numerical results are compared favorably with other computational schemes.

Mesoscopic simulation of three-dimensional pool boiling based on a phase-change cascaded lattice Boltzmann method

Abstract: In this paper, a three-dimensional (3D) cascaded lattice Boltzmann method (CLBM) is implemented to simulate the liquid–vapor phase-change process. The multiphase flow field is solved by incorporating the pseudopotential multiphase model into an improved CLBM, the temperature field is solved by the finite difference method, and the two fields are coupled via a non-ideal equation of state. Through numerical simulations of several canonical problems, it is verified that the proposed phase-change CLBM is applicable for both the isothermal multiphase flow and the liquid–vapor phase-change process. Using the developed method, a complete 3D pool boiling process with up to hundreds of spontaneously generated bubbles is simulated, faithfully reproducing the nucleate boiling, transition boiling, and film boiling regimes. It is shown that the critical heat flux predicted by the 3D simulations agrees better with the established theories and correlation equations than that obtained by two-dimensional simulations. Furthermore, it is found that with the increase in the wall superheats, the bubble footprint area distribution changes from an exponential distribution to a power-law distribution, in agreement with experimental observations. In addition, insights into the instantaneous and time-averaged characteristics of the first two largest bubble footprints are obtained.

Mean Field Games and Applications: Numerical Aspects

Abstract: The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker–Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

Certain new models of the multi space-fractional Gardner equation

Abstract: In this paper, we study an accurate numerical method for solving the multi space-fractional Gardner equation (MSFGE) with the Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) fractional derivatives where the space-fractional terms are under the sense of Caputo. To the best knowledge of the reader, we are first to use the spectral collocation method based on the third Chebyshev approximations to reduced the multi space-fractional Gardner equation to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them via the finite difference method (FDM). By computing the absolute errors we present the effectiveness and accuracy of the proposed methods. The present paper investigates the dynamics of Gardner Equation by considering two fractional operators that is the Caputo–Fabrizio and Atangana–Baleanu, this is entirely new idea by using the two operators on a Gardner equation. Our results prove that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.

Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations

Abstract: The present article develops a semi-discrete numerical scheme to solve the time-fractional stochastic advection–diffusion equations. This method, which is based on finite difference scheme and radial basis functions (RBFs) interpolation, is applied to convert the solution of time-fractional stochastic advection–diffusion equations to the solution of a linear system of algebraic equations. The mechanism of this method is such that time-fractional stochastic advection–diffusion equation is first transformed into elliptic stochastic differential equations by using finite difference scheme. Then meshfree method based on RBFs has been used to approximate the resulting equation. In other words, the approximate solution of time-fractional stochastic advection–diffusion equation is achieved with discrete the domain in the t-direction by finite difference method and approximating the unknown function in the x-direction by generalized inverse multiquadrics RBFs. In this method, the noise terms are directly simulated at the collocation points in each time step and it is the most important advantage of the suggested approach. Stability and convergence of the scheme are established. Finally, some test problems are included to confirm the accuracy and efficiency of the new approach.

On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

Abstract: The exact solution is calculated for fractional telegraph partial differential equation depend on initial boundary value problem. Stability estimates are obtained for this equation. Crank-Nicholson difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied to deal with fractional telegraph differential equation defined by Caputo fractional der

A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model

Abstract: In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of L1 discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations; second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence in L2-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system.

A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations

Abstract: In this paper, we propose a fast algorithm for the variable-order (VO) Caputo fractional derivative based on a shifted binary block partition and uniform polynomial approximations of degree r . Compared with the general direct method, the proposed algorithm can reduce the memory requirement from O ( n ) to O ( r log n ) storage and the complexity from O ( n 2 ) to O ( r n log n ) operations, where n is the number of time steps. As an application, we develop a fast finite difference method for solving a class of VO time-fractional diffusion equations. The computational workload is of O ( r m n log n ) and the active memory requirement is of O ( r m log n ) , where m denotes the size of spatial grids. Theoretically, the unconditional stability and error analysis for the proposed fast finite difference method are given. Numerical results of one and two dimensional problems are presented to demonstrate the well performance of the proposed method.

Flow analysis by Cattaneo–Christov heat flux in the presence of Thomson and Troian slip condition

Abstract: This communication considers entropy minimization in stagnation point flow of a hybrid nanofluid past a nonlinear permeable stretching sheet with Thomson and Troian boundary condition. Due to the porous medium, the Darcy–Forchheimer relation is added. The nonlinear thermal radiation, heat generation, and viscous dissipation by using Cattaneo–Christov heat flux model are explained. Further the influence of variable viscosity, activation energy, and variable mass diffusivity is taken into account. For first time, hybrid nanofluid consisting of carbon nanotubes with Thomson and Troian boundary conditions and induced MHD has been implemented and has not yet been studied. The finite difference method, i.e., bvp4c from Matlab, is utilized to solve the transformed ordinary differential equations (ODEs). This method has good certainty to solve this problem, compared to previous works. Comparison of simple nanofluid and hybrid nanofluid is graphically illustrated. It is noticed that the solid volume fraction decreases the velocity profile and enhances the temperature distribution. Further, compared to simple nanofluid, hybrid nanofluid has greater thermal conductivity and better heat transfer performance.

Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

Abstract: Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space A linearized technique is employed by the extrapolation We prove the unique solvability and stability for both numerical methods The convergence of both numerical methods is analyzed at length utilizing the energy argument, and the convergence orders under the optimal step size ratio are O ( τ 2 + h 2 ) and O ( τ 2 + h 4 ) in the sense of the discrete L 2 -norm, where τ is the time step size, h = max { h x , h y } , and h x , h y are spatial grid sizes in the x -direction and y -direction, respectively In addition, we construct a multistep alternating direction implicit (ADI) scheme and a multistep compact ADI scheme based on BDF2 for the efficiently numerical implementation Finally, numerical examples are carried out to verify our theoretical results

Comparison of Exact and Numerical Solutions for the Sharma–Tasso–Olver Equation

Abstract: In this work, we have considered a Sharma–Tasso–Olver (STO) equation in order to obtain some exact and numerical solutions by using the auto-Backlund transformation method (aBTM) and the finite forward difference method. We successfully obtain some kink-type solutions with exponential prototype structure to this equation and also we obtain some numerical solution by using the finite difference method (FDM). We illustrate the comparison between exact and numerical approximations and support the comparison with a graphic plot of these illustrations as well. Moreover, the Fourier–von Neumann stability analysis is used in checking the stability of the numerical scheme. The L2 and L∞ error norms of the solutions to this equation are also illustrated here.

Variational Quantum algorithm for Poisson equation

Abstract: The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose a Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we first adopt the finite difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only $2\log n+1$ items, of its coefficient matrix under a specific set of simple operators, where $n$ is the dimension of the coefficient matrix. This implies that the proposed VQA only needs $O(\log n)$ measurements, which dramatically reduce quantum resources. Additionally, we perform quantum Bell measurements to efficiently evaluate the expectation values of simple operators. Numerical experiments demonstrate that our algorithm can effectively solve the Poisson equation.

Generalized finite difference method for solving stationary 2D and 3D Stokes equations with a mixed boundary condition

Abstract: In the present work, a generalized finite difference method (GFDM), a meshless method based on Taylor-series approximations, is proposed to solve stationary 2D and 3D Stokes equations. To overcome the troublesome pressure oscillation in the Stokes problem, a new simple formulation of boundary condition for the Stokes problem is proposed. This numerical approach only adds a mixed boundary condition, the projections of the momentum equation on the boundary outward normal vector, to the Stokes equations, without any other change to the governing equations. The proposed formulation can be easily discretized by the GFDM. The GFDM is evolved from the Taylor series expansions and moving-least squares approximation, and the derivative expressed of unknown variables as linear combinations of function values of neighboring nodes. Numerical examples are utilized to verify the feasibility of the proposed GFDM scheme not only for the Stokes problem, but also for more involved and general problems, such as the Poiseuille flow, the Couette flow and the Navier–Stokes equations in low-Reynolds-number regime. Moreover, numerical results and comparisons show that using the GFDM to solve the proposed formulation of the Stokes equations is more accurate than the classical formulation of the pressure Poisson equation.