Variational multiscale method based on the Crank–Nicolson extrapolation scheme for the non-stationary Navier–Stokes equations

In this report, a variational multiscale (VMS) method based on the Crank–Nicolson extrapolation scheme of time discretization for the turbulent flow is analysed. The flow is modelled by the fully evolutionary Navier–Stokes problem. This method has two differences compared to the standard VMS method: (i) For the trilinear term, we use the extrapolation skill to linearize the scheme; (ii) for the projection term, we lag it onto the previous time level to simplify the construction of the projection. These modifications make the algorithm more efficient and feasible. An unconditionally stability and an a priori error estimate are given for a case with rather general linear (cellwise constant) viscosity of the turbulent models. Moreover, numerical tests for both linear viscosity and nonlinear Smagorinsky-type viscosity are performed, they confirm the theoretical results and indicate the schemes are effective.     

Crank–Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection–diffusion equations

A numerical approach is proposed to examine the singularly perturbed time-dependent convection–diffusion equation in one space dimension on a rectangular domain. The solution of the considered problem exhibits a boundary layer on the right side of the domain. We semi-discretize the continuous problem by means of the Crank–Nicolson finite difference method in the temporal direction. The semi-discretization yields a set of ordinary differential equations and the resulting set of ordinary differential equations is discretized by using a midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost second-order accurate in a coarse mesh and almost first-order accurate in a fine mesh in the spatial direction. The accuracy achieved in the temporal direction is almost second order. An extensive amount of analysis has been carried out in order to prove the uniform convergence of the method. Finally we have found that the resulting method is uniformly convergent with respect to the singular perturbation parameter, i.e. ?-uniform. Some numerical experiments have been carried out to validate the proposed theoretical results.     

The Modified Local Crank–Nicolson method for one- and two-dimensional Burgers’ equations

The Modified Local Crank–Nicolson method is applied to solve one- and two-dimensional Burgers’ equations. New difference schemes that are explicit, unconditionally stable, and easy to compute are obtained. Numerical solutions obtained by the present method are compared with exact solutions, and it is seen that they are in excellent agreement.     

A compact ADI Crank–Nicolson difference scheme for the two-dimensional time fractional subdiffusion equation

In this paper, a compact alternating direction implicit (ADI) Crank–Nicolson difference scheme is proposed and analysed for the solution of two-dimensional time fractional subdiffusion equation. The Riemann–Liouville time fractional derivative is approximated by the weighted and shifted Grünwald difference operator and the spatial derivative is discretized by a fourth-order compact finite difference method. The stability and convergence of the difference scheme are discussed and theoretically proven by using the energy method. Finally, numerical experiments are carried out to show that the numerical results are in good agreement with the theoretical analysis.     

A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations

Recently, Douglas et al. [4] introduced a new, low-order, nonconforming rectangular element for scalar elliptic equations. Here, we apply this element in the approximation of each component of the velocity in the stationary Stokes and Navier–Stokes equations, along with a piecewise-constant element for the pressure. We obtain a stable element in both cases for which optimal error estimates for the approximation of both the velocity and pressure in L ² can be established, as well as one in a broken H ¹-norm for the velocity. Received: January 1999 / Accepted: April 1999     

A unified difference-spectral method for time–space fractional diffusion equations

In this paper we develop a unified difference-spectral method for stably solving time–space fractional sub-diffusion and super-diffusion equations. Based on the equivalence between Volterra integral equations and fractional ordinary differential equations with initial conditions, this proposed method is constructed by combining the spectral Galerkin method in space and the fractional trapezoid formula in time. Numerical experiments are carried out to verify the effectiveness of the method, and demonstrate that the unified method can achieve spectral accuracy in space and second-order accuracy in time for solving two kinds of time–space fractional diffusion equations.     

A stabilized finite element method for stochastic incompressible Navier–Stokes equations

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.     

Unconditional convergence and optimal L2 error estimates of the Crank–Nicolson extrapolation FEM for the nonstationary Navier–Stokes equations

In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank–Nicolson extrapolation scheme for the nonstationary Navier–Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank–Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the $L^2$, $H^1$-unconditionally optimal error estimates for the velocity, and the $L^2$-unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditional$L^2$-stability and convergent rates of this method are shown.     

Third-order temporal discrete scheme for the non-stationary Navier–Stokes equations

In this article, we suggest a new third-order time discrete scheme for the two-dimensional non-stationary Navier–Stokes equations. After presenting the Galerkin finite element approximation for the spatial discretization, we consider an implicit/explicit time discrete scheme for the problem, which is based on the two-step Adams–Moulton scheme (implicit scheme) for the linear term and the three-step Adams–Bashforth scheme (explicit scheme) for the nonlinear term. In this method, we only need to solve a linearized discrete system at each time step, so the scheme can converge fast and the computational cost can be reduced. Moreover, under some assumptions, we deduce the stability and optimal error estimate for the velocity in L ²-norm.     

A characteristic stabilized finite element method for the non-stationary Navier–Stokes equations

This paper is concerned with the analysis of a new stabilized method based on the local pressure projection. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures. Error estimates of the velocity and the pressure are obtained for both the continuous and the fully discrete versions. Finally, some numerical experiments show that this method is highly efficient for the non-stationary Navier–Stokes equations.     

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier–Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier–Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity–stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.     

Decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data

Based on the mixed finite element method, we consider the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data in this paper. The temporal treatment of the spatial discrete Boussinesq equations is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms. Thanks to the decoupled method, the considered problem is split into two subproblems and these subproblems can be solved in parallel. Under some restriction on the time step, we present the stability and convergence results of numerical solutions, Finally, some numerical experiments are provided to test the performance of the developed numerical scheme and verify the established theoretical findings.     

Finite difference method for time–space linear and nonlinear fractional diffusion equations

ABSTRACT

In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.     

Characteristic stabilized finite element method for the transient Navier–Stokes equations

Based on the lowest equal-order conforming finite element subspace (X_h, M_h) (i.e. P₁–P₁ or Q₁–Q₁ elements), a characteristic stabilized finite element method for transient Navier–Stokes problem is proposed. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, and averting the difficulties caused by trilinear terms. Existence,uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method. Finally, a serious of numerical experiments are given to show that this method is highly efficient for transient Navier–Stokes problem.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.     

An accelerated algorithm for Navier–Stokes equations

In this paper, the numerical solution of the Navier–Stokes equations by the Characteristic-Based-Split (CBS) scheme is accelerated with the Minimum Polynomial Extrapolation (MPE) method to obtain the steady state solution for evolution incompressible and compressible problems.The CBS is essentially a fractional time-stepping algorithm based on an original finite difference velocity-projection scheme where the convective terms are treated using the idea of the Characteristic-Galerkin method. In this work, the semi-implicit version of the CBS with global time-stepping is used for incompressible problems whereas the fully-explicit version is used for compressible flows.At the other end, the MPE is a vector extrapolation method that transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator.The developed algorithm, tested on two-dimensional benchmark problems, demonstrates the new computational features arising from the introduction of the extrapolation procedure to the CBS scheme. In particular, the results show a remarkable reduction of the computational cost of the simulation.     

A space–time spectral collocation method for the two-dimensional variable-order fractional percolation equations

In this article, we introduce a space–time spectral collocation method for solving the two-dimensional variable-order fractional percolation equations. The method is based on a Legendre–Gauss–Lobatto (LGL) spectral collocation method for discretizing spatial and the spectral collocation method for the time integration of the resulting linear first-order system of ordinary differential equation. Optimal priori error estimates in $L^2$ norms for the semi-discrete and full-discrete formulation are derived. The method has spectral accuracy in both space and time. Numerical results confirm the exponential convergence of the proposed method in both space and time.