The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

Decoupled two‐grid finite element method for the time‐dependent natural convection problem I: Spatial discretization

In this article, a decoupled two grid finite element method (FEM) is proposed and analyzed for the nonsteady natural convection problem using the coarse grid numerical solutions to decouple the nonlinear coupled terms, and the corresponding optimal error estimates are derived. Compared with the standard Galerkin FEM and the usual two‐grid FEM, our algorithm not only keeps good accuracy but also saves a lot of computational cost. Some numerical examples are provided to verify the performances of the decoupled two‐grid FEM. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the decoupled two‐grid FEM for the nonsteady natural convection problem. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2135–2168, 2015     

Numerical analysis of the Crank–Nicolson extrapolation time discrete scheme for magnetohydrodynamics flows

In this article, we consider the time‐discrete method for three‐dimensional incompressible magnetohydrodynamics (MHD) equations. The Crank–Nicolson extrapolation scheme is used for time discretization. From the previous articles, under the assumption that the solution has high regularity which cannot be realistically assumed, the convergence of this scheme is optimal two‐order. In this article, under modest assumptions of initial values and the body force, we prove some new regularity results of the MHD equations. In addition, we derive the unconditional convergence of our scheme, but the convergent order is not optimal. Furthermore, we provide another conditional convergence estimation to increase the order. It is shown that the convergent rate increase half order in ‐norm, and at least a quarter order increased in ‐norm than the uncondtional results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2169–2208, 2015     

Numerical study of Sivashinsky equation using a splitting scheme based on Crank‐Nicolson method

The mathematical modeling of a planar solid‐liquid interface in the solidification of a dilute binary alloy is formulating by one of nonintegrable, nonlinear evolution equation known as Sivashinsky equation. In the first part of this paper, the mathematical modeling of Sivashinsky equation is briefly discussed. Since, the exact solutions of this equation is yet unknown, obtaining its numerical solution plays an important role to simulate its behavior. Therefore, in the second part, a second‐order splitting finite difference scheme, based on Crank‐Nicolson method, is investigated to approximate the solution of the Sivashinsky equation with homogeneous boundary conditions. We prove the solvability of the present scheme and establish the error estimate of the numerical scheme.     

Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods

In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose in the first algorithm and in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly.     

Second order fully discrete defect‐correction scheme for nonstationary conduction‐convection problem at high Reynolds number

This survey enfolds rigorous analysis of the defect‐correction finite element (FE) method for the time‐dependent conduction‐convection problem which based on the Crank‐Nicolson scheme. The method consists of two steps: solve a nonlinear problem with an added artificial viscosity term on a FE grid and correct the solutions on the same grid using a linearized defect‐correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. As a consequence, the effectiveness of the method to deal with high Reynolds number is illustrated in several numerical experiments. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 681–703, 2017     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

On the long‐time stability of the Crank–Nicolson scheme for the 2D Navier–Stokes equations

In this article we study the stability for all positive time of the Crank–Nicolson scheme for the two‐dimensional Navier–Stokes equations. More precisely, we consider the Crank–Nicolson time discretization together with a general spatial discretization, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable, provided a CFL‐type condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A Crank‐Nicolson and ADI Galerkin method with quadrature for hyperbolic problems

We propose, analyze, and implement fully discrete two‐time level Crank‐Nicolson methods with quadrature for solving second‐order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265–1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564–576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577–598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A new two grid variational multiscale method for steady‐state natural convection problem

A two‐grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method. Copyright © 2016 John Wiley & Sons, Ltd.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

An energy‐preserving Crank–Nicolson Galerkin method for Hamiltonian partial differential equations

A semidiscretization based method for solving Hamiltonian partial differential equations is proposed in this article. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy‐preserving integrator. The relay leads to a fully discretized and energy‐preserved scheme. This strategy is fully realized for solving a nonlinear Schrödinger equation through a combination of the Galerkin discretization in space and a Crank–Nicolson scheme in time. The order of convergence of our new method is if the discrete L²‐norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy‐preserving scheme constructed.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1485–1504, 2016     

Stability and convergence of the mark and cell finite difference scheme for Darcy‐Stokes‐Brinkman equations on non‐uniform grids

In this paper, we consider the mark and cell (MAC) method for Darcy‐Stokes‐Brinkman equations and analyze the stability and convergence of the method on nonuniform grids. Firstly, to obtain the stability for both velocity and pressure, we establish the discrete inf‐sup condition. Then we introduce an auxiliary function depending on the velocity and discretizing parameters to analyze the super‐convergence. Finally, we obtain the second‐order convergence in L2 norm for both velocity and pressure for the MAC scheme, when the perturbation parameter ? is not approaching 0. We also obtain the second‐order convergence for some terms of ∥·∥_? norm of the velocity, and the other terms of ∥·∥_? norm are second‐order convergence on uniform grid. Numerical experiments are carried out to verify the theoretical results.     

Two‐grid methods for time‐harmonic Maxwell equations

In this paper, we develop several two‐grid methods for the Nédélec edge finite element approximation of the time‐harmonic Maxwell equations. We first present a two‐grid method that uses a coarse space to solve the original problem and then use a fine space to solve a corresponding symmetric positive definite problem. Then, we present two types of iterative two‐grid methods, one is to add the kernel of the curl ‐operator in the fine space to a coarse mesh space to solve the original problem and the other is to use an inner iterative method for dealing with the kernel of the curl ‐operator in the fine space and the coarse space, separately. We provide the error estimates for the first two methods and present numerical experiments to show the efficiency of our methods.Copyright © 2012 John Wiley & Sons, Ltd.     

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

In this article, a finite element scheme based on the Newton's method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank‐Nicolson method is used for time discretization. Well‐posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.     

A two‐grid stabilization method for solving the steady‐state Navier‐Stokes equations

We formulate a subgrid eddy viscosity method for solving the steady‐state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H) and a linear system on the fine‐grid space. An optimal error estimate in H¹ ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H²). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

An a posteriori upper bound is derived for the nonstationary convection–diffusion problem using the Crank–Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.A space and time adaptive algorithm is developed to ensure the control of the relative error in the L²(H¹) norm. Numerical experiments illustrating the efficiency of this approach are reported; it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step, even in the convection dominated regime and in the presence of boundary layers.     

A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

In this paper, we first establish the Crank–Nicolson collocation spectral (CNCS) method for two‐dimensional (2D) viscoelastic wave equation by means of the Chebyshev polynomials. And then, we analyze the existence, uniqueness, stability, and convergence of the CNCS solutions. Finally, we use some numerical experiments to verify the correctness of theoretical analysis. This implies that the CNCS model is very effective for solving the 2D viscoelastic wave equations.