Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy. Received February 2, 1994 / Revised version received December 6, 1996     

Stability of the mixed time partitioning methods in relation to the size of time step

In this paper we focus on stability of a mixed time partitioning methods in relation to time step size which is using in numerical modelling of two-component alloys solidification. We present the numerical integration methods to solve solidification problems in a fast and accurate way. Our approach exploits the fact that physical processes inside a mould are of different nature than those in a solidifying cast. As a result different time steps can be used to run computations within both sub-domains. Because processes that are modeled in the cast sub-domain are more dynamic they require very fine-grained time step. On the other hand a heat transfer within the mould sub-domain is less intense, and thus coarse-grained step is sufficient to guarantee desired precision of computations. We propose using a fixed time step in the cast and its integer multiple in other parts of mould. We use one-step explicit and implicit time integration Θ schemes. These time integration schemes are applied to equations obtained after spatial discretization. The implicit scheme is unconditionally stable, but stability of the explicit scheme depends on the size of time step. Critical time step size can be determined on the basis of eigenvalues of the amplification matrix that depend on the material properties, size and type of the finite element. In this work we present the manner of determining the critical time step and its affect on the course of numerical simulation of solidification. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improved stability techniques for the solution of Navier-Stokes equations

When solving the Navier-Stokes equations for transient incompressible viscous flow problems, one normally encounters a decrease in numerical stability of the time integration algorithm with an increase in Reynolds number. This instability cannot be easily overcome due to the non-linearities present. The present paper, using the finite element method to integrate the equations in the spacial dimension, incorporates a time-staggered semi-implicit fractional step technique to improve stability at the higher Reynolds numbers. Unlike the upwind or directional differencing schemes normally employed to increase numerical stability, the present scheme does not introduce numerical damping or artificial viscocity, and becomes more stable as the Reynolds number increases. Results for this scheme are compared with various explicit integration schemes for the case of flow around a circular cylinder at Reynolds numbers of 100 to 400. For comparable accuracy the time-staggered semi-implicit fractional step technique was found to be up to 25 times more efficient than the other explicit integration schemes.     

Multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem

In this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo-pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely-coupled time-stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely-coupled time-stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.     

On the numerical evaluation of Cauchy transforms

In this paper we consider simple methods for the reconstruction of the Cauchy transform over a curve when an explicit parametrization of the latter is not provided. The methods consist of replacing the parametrization of the curve by piecewise polynomial interpolation followed by the use of Newton-Cotes type formulae for the integration. The order of convergence of the resulting quadrature is higher than would be expected on the basis of considerations involving just interpolation theory, provided that the Cauchy transform is evaluated at known nodes on the curve. These results allow the calculation of the Cauchy transform at other points with the same accuracy if this scheme is followed by an interpolatory formula of sufficiently high accuracy.     

Fourier spectral projection method and nonlinear convergence analysis for Navier-Stokes equations

In this paper, we investigate the convergence rate of the Fourier spectral projection methods for the periodic problem of n-dimensional Navier-Stokes equations. Based on some alternative formulations of the Navier-Stokes equations and the related projection methods, the error estimates are carried out by a global nonlinear error analysis. It simplifies the analysis, relaxes the restriction on the time step size, weakens the regularity requirements on the genuine solution, and leads to some improved convergence results. A new correction technique is proposed for improving the accuracy of the numerical pressure.     

Performance and numerical behavior of the second‐order scheme of precise time‐step integration for transient dynamic analysis

Spurious high‐frequency responses resulting from spatial discretization in time‐step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time‐step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second‐order scheme of the precise integration method (PIM). Taking the Newmark‐β method as a reference, the performance and numerical behavior of the second‐order PIM for elasto‐dynamic impact‐response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock‐excited rod with rectangular, half‐triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine‐like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Integración temporal explícita con grandes pasos de tiempo de la ecuación de transmisión del calor

We present a novel fully explicit time integration method that remains stable for large time steps, requires neither matrix inversions nor solving a system of equations and therefore allows for nearly effort-less parallelization. In this first paper the proposed approach is applied to solve conduction heat transfer problems, showing that it is stable for any time step as is the case with implicit methods but with a much lower computation time.     

A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems

The numerical integration of Hamiltonian systems with multi-frequency and multidimensional oscillatory solutions is encountered frequently in many fields of the applied sciences. In this paper, we firstly summarize the extended Runge–Kutta–Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1873–1887, (2010)) for multi-frequency and multidimensional oscillatory systems and restate the order conditions and symplecticity conditions for the explicit ERKN methods. Secondly, we devote to exploring the explicit symplectic multi-frequency and multidimensional ERKN methods of order five based on the symplecticity conditions and order conditions. A five-stage explicit symplectic multi-frequency and multidimensional ERKN method of order five with some small residuals is proposed and its stability and phase properties are analyzed. It is shown that the new method is dispersive of order six. Numerical experiments are carried out and the numerical results demonstrate that the new method is much more efficient than the methods appeared in the scientific literature.     

Multirate Runge–Kutta schemes for advection equations

Explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena. The wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step. Multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps, resulting in a moderate to substantial reduction of the computational cost, depending on the scenario to simulate [S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput. 41 (1983) 321–336; H. Tang, G. Warnecke, A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and pace grids, SIAM J. Sci. Comput. 26 (4) (2005) 1415–1431; E. Constantinescu, A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, SIAM J. Sci. Comput. 33 (3) (2007) 239–278]. In air pollution modeling the advection part is usually integrated explicitly in time, where the time step is constrained by a locally varying Courant–Friedrichs–Lewy (CFL) number. Multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction. Therefore it is of major interest to apply multirate schemes to the advection equation. We introduce a generic recursive multirate Runge–Kutta scheme that can be easily adapted to an arbitrary number of refinement levels. It preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit Runge–Kutta methods as base method.     

A particle method for some parabolic equations

We present a quasi-Monte-Carlo particle simulation of some multidimensional linear parabolic equations with constant coefficients. We approximate the elliptic operator in space by a finite-difference operator. We discretize time into intervals of length Δt. The discrete representation of the solution at time t_n = nΔt is a sum of Dirac delta measures. Using the explicit Euler scheme, the resulting approximation at time t_n+1 is recovered by a quasi-Monte-Carlo integration. We make use of a technique involving renumbering the simulated particles in every time step. We state and prove a convergence theorem for the method. Experimental results are presented for some model problems. The results suggest that the quasi-Monte-Carlo simulation tends to give more accurate solutions than a Monte-Carlo simulation, when the correct renumbering technique is used. Other choices can result in significant loss of efficiency.     

On unconditionally positivity preserving DG schemes for shallow water flows with shock capturing by adaptive filtering procedures

In this work, we present an unconditionally positivity preserving implicit time integration scheme for the DG method applied to shallow water flows. For locally refined grids with very small elements, the ODE resulting from space discretization is stiff and requires implicit or partially implicit time stepping. However, for simulations including wetting and drying or regions with small water height, severe time step restrictions have to be imposed due to positivity preservation. Nevertheless, as implicit time stepping demands a significant amount of computational time in order to solve a large system of nonlinear equations for each time step we need large time steps to obtain an efficient scheme. In this context, we propose a novel approach to the strategy of positivity preservation. This new technique is based on the so-called Patankar trick and guarantees non-negativity of the water height for any time step size while still preserving conservativity. Due to this modification, the implicit scheme can take full advantage of larger time steps and is therefore able to beat explicit time stepping in terms of CPU time. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometric multigrid for an implicit-time immersed boundary method

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.     

Analysis of estimation accuracy of the first moments of a Monte Carlo solution to an SDE with Wiener and Poisson components

In this paper, the estimation accuracy of the first moments of a numerical solution to an SDE with Wiener and Poisson components is investigated by a generalized explicit Euler method. Exact expressions for the mathematical expectation and variance of a test SDE solution are obtained. These expressions allow us to investigate the estimation accuracy obtained by a Monte Carlo method versus the SDE parameters, the integration step, and the size of the ensemble of simulated trajectories of the solution. The results of test numerical experiments are presented.     

Actual time integration methods for elastic wave propagation analysis

For the simulation of the interaction of elastic waves in CFRP plates with inhomogeneities and defects the spectral finite element method (SEM) is under investigation. The SEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature (GLq) points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. In this paper we analyse how actual time integration methods perform in combination with the SEM. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit multistep method for the numerical solution of stiff differential equations

An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods.     

A parabolic acceleration time integration method for structural dynamics using quartic B-spline functions

In this paper, an explicit time integration method is proposed for structural dynamics using periodic quartic B-spline interpolation polynomial functions. In this way, at first, by use of quartic B-splines, the authors have proceeded to solve the differential equation of motion governing SDOF systems and later the proposed method has been generalized for MDOF systems. In the proposed approach, a straightforward formulation was derived in a fluent manner from the approximation of response of the system with B-spline basis. Because of using a quartic function, the system acceleration is approximated with a parabolic function. For the aforesaid method, a simple step-by-step algorithm was implemented and presented to calculate dynamic response of MDOF systems. The proposed method has appropriate convergence, accuracy and low time consumption. Accuracy and stability analyses have been done perfectly in this paper. The proposed method benefits from an extraordinary accuracy compared to the existing methods such as central difference, Runge–Kutta and even Duhamel integration method. The validity and effectiveness of the proposed method is demonstrated with four examples and the results of this method are compared with those from some of the existent numerical methods. The high accuracy and less time consumption are only two advantages of this method.     

An asymptotically stable Particle-in-Cell (PIC) scheme for collisionless plasma simulations near quasineutrality

We propose a new Particle-in-Cell scheme for the Vlasov–Poisson equation. This scheme remains stable when the Debye length and plasma period tend to zero without any restriction on the size of the time and length step. It relies on a semi-implicit integration of the particle trajectories. The numerical integration cost is that of the standard explicit method thanks to the use of a reformulation of the Poisson equation. To cite this article: P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.