A linear,symmetric and energy-conservative scheme for the space-fractional Klein–Gordon–Schrödinger equations

In this paper, we propose an efficient numerical scheme for the space-fractional Klein–Gordon–Schrödinger (SFKGS) equations. Motivated by the “Invariant Energy Quadratization” (IEQ) approach, we introduce two auxiliary variables to transform the SFKGS system into a new equivalent system in which the time derivative is discretized by the Crank–Nicolson method, and the space discretization is based on the Fourier spectral method. Consequently, the numerical scheme shares two good features. The first feature is that the nonlinear terms are treated semi-explicitly and a linear symmetric system is solved at each time step. The second feature is the energy conservation at the discrete level. These two advantages are proved by the theoretical analysis and illustrated by a given numerical example.     

ANALYSIS ON A NUMERICAL SCHEME WITH SECOND-ORDER TIME ACCURACY FOR NONLINEAR DIFFUSION EQUATIONS

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization (TLCD) at each time step.It does not stir numerical oscillation,while per-mits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson (CN)scheme and the backward difference formula second-order (BDF2) scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.     

A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.     

NONLINEAR GALERKIN METHOD AND CRANK-NICOLSONMETHOD FOR VISCOUS INCOMPRESSIBLE FLOW

1.Introduction'NonlinearGalerkinmethodisnumericalmethodfordissipativeevolutionpartialdifferentialequationswherethespatialdiscretizationreliesonanonlinearmanifoldinsteadofalinearspaceasintheclassicalGalerkinmethod.Morepreciselygoneconsidersafinitedimension…     

An asymptotic preserving scheme on staggered grids for the barotropic Euler system in low Mach regimes

We present a new scheme for the simulation of the barotropic Euler equation in low Mach regimes. The method uses two main ingredients. First, the system is treated with a suitable time splitting strategy, directly inspired from the previous study that separates low and fast waves. Second, we adapt a numerical scheme where the discrete densities and velocities are stored on staggered grids, in the spirit of MAC methods, and with numerical fluxes derived from the kinetic approach. We bring out the main properties of the scheme in terms of consistency, stability, and asymptotic behavior, and we present a series of numerical experiments to validate the method.     

A Linearized Spectral-Galerkin Method for Three-Dimensional Riesz-Like Space Fractional Nonlinear Coupled Reaction-Diffusion Equations

In this paper, we establish a novel fractional model arising in the chemical reaction and develop an efficient spectral method for the three-dimensional Riesz-like space fractional nonlinear coupled reaction-diffusion equations. Based on the backward difference method for time stepping and the Legendre-Galerkin spectral method for space discretization, we construct a fully discrete numerical scheme which leads to a linear algebraic system. Then a direct method based on the matrix diagonalization approach is proposed to solve the linear algebraic system, where the cost of the algorithm is of a small multiple of $N^4$ ($N$ is the polynomial degree in each spatial coordinate) flops for each time level. In addition, the stability and convergence analysis are rigorously established. We obtain the optimal error estimate in space, and the results also show that the fully discrete scheme is unconditionally stable and convergent of order one in time. Furthermore, numerical experiments are presented to confirm the theoretical claims. As the applications of the proposed method, the fractional Gray-Scott model is solved to capture the pattern formation with an analysis of the properties of the fractional powers.     

An unconditionally stable finite difference scheme for solving a 3D heat transport equation in a sub-microscale thin film

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a finite difference scheme with two levels in time for the 3D heat transport equation in a sub-microscale thin film. It is shown by the discrete energy method that the scheme is unconditionally stable. The 3D implicit scheme is then solved by using a preconditioned Richardson iteration, so that only a tridiagonal linear system is solved for each iteration. The numerical procedure is employed to obtain the temperature rise in a gold sub-microscale thin film.     

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay

In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L² dissipative for any time step size and H¹ dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017     

Analysis of a meshless method for the time fractional diffusion-wave equation

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.     

Solving complex PDE systems for pricing American options with regime‐switching by efficient exponential time differencing schemes

In this article, we study the numerical solutions of a class of complex partial differential equation (PDE) systems with free boundary conditions. This problem arises naturally in pricing American options with regime‐switching, which adds significant complexity in the PDE systems due to regime coupling. Developing efficient numerical schemes will have important applications in computational finance. We propose a new method to solve the PDE systems by using a penalty method approach and an exponential time differencing scheme. First, the penalty method approach is applied to convert the free boundary value PDE system to a system of PDEs over a fixed rectangular region for the time and spatial variables. Then, a new exponential time differncing Crank–Nicolson (ETD‐CN) method is used to solve the resulting PDE system. This ETD‐CN scheme is shown to be second order convergent. We establish an upper bound condition for the time step size and prove that this ETD‐CN scheme satisfies a discrete version of the positivity constraint for American option values. The ETD‐CN scheme is compared numerically with a linearly implicit penalty method scheme and with a tree method. Numerical results are reported to illustrate the convergence of the new scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

非局部Allen-Cahn模型全离散数值格式的误差估计

在这项工作中,我们研究了求解非局部体积守恒Allen-Cahn (AC)方程的全离散傅里叶伪谱数值格式的误差估计.该数值格式的时间行军方法基于著名的不变能量平方法(IEQ). 我们证明了所提出的全离散数值方法是唯一可解,无条件能量稳定的,并获得了该方案在空间和时间上的最优误差估计.此外,我们还进行了一些数值检验来验证理论结果.     

A new reliable algorithm based on the sinc function for the time fractional diffusion equation

This paper deals with the numerical solution of time fractional diffusion equation. In this work, we consider the fractional derivative in the sense of Riemann-Liouville. At first, the time fractional derivative is discretized by integrating both sides of the equation with respect to the time variable and we arrive at a semi–discrete scheme. The stability and convergence of time discretized scheme are proven by using the energy method. Also we show that the convergence order of this scheme is O(τ2?α). Then we use the sinc collocation method to approximate the solution of semi–discrete scheme and show that the problem is reduced to a Sylvester matrix equation. Besides by performing some theorems, the exponential convergence rate of sinc method is illustrated. The numerical experiments are presented to show the excellent behavior and high accuracy of the proposed hybrid method in comparison with some other well known methods.     

Nonstandard methods for the convective transport equation with nonlinear reactions

A new nonstandard Lagrangian method is constructed for the one-dimensional, transient convective transport equation with nonlinear reaction terms. An “exact” time-stepping scheme is developed with zero local truncation error with respect to time. The scheme is based on nonlocal treatment of nonlinear reactions, and when applied at each spatial grid point gives the new fully discrete numerical method. This approach leads to solutions free from the numerical instabilities that arise because of incorrect modeling of derivatives and nonlinear reaction terms. Algorithms are developed that preserve the properties of the numerical solution in the case of variable velocity fields by using nonuniform spatial grids. Effects of different interpolation techniques are examined and numerical results are presented to demonstrate the performance of the proposed new method. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 467–485, 1998     

Numerical time-delayed feedback control in a neural network with delay

In this article, by a nonstandard finite-difference method we obtain the general time delayed feedback control numerical discrete scheme for a delayed neural network model. Firstly, the local stability of the equilibria point is discussed according to the Neimark–Sacker bifurcation theory. Then, from the point of view of control, for any step-size, a general time delayed feedback control numerical algorithm is introduced to delay the onset of the Neimark–Sacker bifurcation at a desired point by choosing appropriate control parameters. This controller can deal with the general system that the natural equilibrium cannot be given by analytic expression. Finally, numerical examples are provided to illustrate the theoretical results. The results show that the time delayed feedback numerical scheme is better than a polynomial function time delayed feedback method.     

Dynamics of a nonstandard finite-difference scheme for Mackey-Glass system

In this article, we study the dynamics of Mackey-Glass system applied a nonstandard finite-difference scheme. For the discrete system we show that a sequence of Hopf bifurcations occur at the positive fixed point as the delay increasing, analyze the stability of the fixed point and calculate the direction of the Hopf bifurcations. At last, by giving some numerical experiments, we illustrate the relation between the time of producing blood cells and symptom.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

A splitting positive definite mixed element method for second‐order hyperbolic equations

In this article, we establish a new mixed finite element procedure, in which the mixed element system is symmetric positive definite, to solve the second‐order hyperbolic equations. The convergence of the mixed element methods with continuous‐ and discrete‐time scheme is proved. And the corresponding error estimates are given. Finally some numerical results are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An efficient linear second order unconditionally stable direct discretization method for the phase-field crystal equation on surfaces

We develop an unconditionally stable direct discretization scheme for solving the phase-field crystal equation on surfaces. The surface is discretized by using an unstructured triangular mesh. Gradient, divergence, and Laplacian operators are defined on triangular meshes. The proposed numerical method is second-order accurate in space and time. At each time step, the proposed computational scheme results in linear elliptic equations to be solved, thus it is easy to implement the algorithm. It is proved that the proposed scheme satisfies a discrete energy-dissipation law. Therefore, it is unconditionally stable. A fast and efficient biconjugate gradients stabilized solver is used to solve the resulting discrete system. Numerical experiments are conducted to demonstrate the performance of the proposed algorithm.     

Locally linearized fractional step methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. Via the method of lines approach, we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme. The system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method. For that purpose, we locally decompose the discrete nonlinear diffusion operator using suitable Taylor expansions and a domain decomposition splitting technique. The totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones. As a result, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems. The convergence of the proposed methods is illustrated by numerical experiments.