Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of central differencing and low‐dissipation weights in a weighted compact nonlinear scheme

This paper proposes WCNS‐CU‐Z, a weighted compact nonlinear scheme, that incorporates adapted central difference and low‐dissipative weights together with concepts of the adaptive central‐upwind sixth‐order weighted essentially non‐oscillatory scheme (WENO‐CU) and WENO‐Z schemes. The newly developed WCNS‐CU‐Z is a high‐resolution scheme, because interpolation of this scheme employs a central stencil constructed by upwind and downwind stencils. The smoothness indicator of the downwind stencil is calculated using the entire central stencil, and the downwind stencil is stopped around the discontinuity for stability. Moreover, interpolation of the sixth‐order WCNS‐CU‐Z exhibits sufficient accuracy in the smooth region through use of low‐dissipative weights. The sixth‐order WCNS‐CU‐Zs are implemented with a robust linear difference formulation (R‐WCNS‐CU6‐Z), and the resolution and robustness of this scheme were evaluated. These evaluations showed that R‐WCNS‐CU6‐Z is capable of achieving a higher resolution than the seventh‐order classical robust weighted compact nonlinear scheme and can provide a crisp result in terms of discontinuity. Among the schemes tested, R‐WCNS‐CU6‐Z has been shown to be robust, and variable interpolation type R‐WCNS‐CU6‐Z (R‐WCNS‐CU6‐Z‐V) provides a stable computation by modifying the first‐order interpolation when negative density or negative pressure arises after nonlinear interpolation. Copyright © 2016 John Wiley & Sons, Ltd.     

High‐order upwind compact scheme and simulation of turbulent premixed V‐flame

A high‐order accurate upwind compact difference scheme with an optimal control coefficient is developed to track the flame front of a premixed V‐flame. In multi‐dimensional problems, dispersion effect appears in the form of anisotropy. By means of Fourier analysis of the operators, anisotropic effects of the upwind compact difference schemes are analysed. Based on a level set algorithm with the effect of exothermicity and baroclinicity, the flame front is tracked. The high‐order accurate upwind compact scheme is employed to approximate the level set equation. In order to suppress numerical oscillations, the group velocity control technique is used and the upwind compact difference scheme is combined with the random vortex method to simulate the turbulent premixed V‐flame. Distributions of velocities and flame brush thickness are obtained by this technique and found to be comparable with experimental measurement. Copyright © 2005 John Wiley & Sons, Ltd.     

A pressure correction‐volume of fluid method for simulation of two‐phase flows

A pressure correction method coupled with the volume of fluid (VOF) method is developed to simulate two‐phase flows. A volume fraction function is introduced in the VOF method and is governed by an advection equation. A modified monotone upwind scheme for a conservation law (modified MUSCL) is used to solve the solution of the advection equation. To keep the initial sharpness of an interface, a slope modification scheme is introduced. The continuum surface tension (CST) model is used to calculate the surface tension force. Three schemes, central‐upwind, Parker–Youngs, and mixed schemes, are introduced to compute the interface normal vector and the gradient of the volume fraction function. Moreover, a height function technique is applied to compute the local curvature of the interface. Several basic test problems are performed to check the order of accuracy of the present numerical schemes for computing the interface normal vector and the gradient of the volume fraction function. Three physical problems, two‐dimensional broken dam problem, static drop, and spurious currents, and three‐dimensional rising bubble, are performed to demonstrate the efficiency and accuracy of the pressure correction method. Copyright © 2010 John Wiley & Sons, Ltd.     

Multigrid solutions of the Euler and Navier–Stokes equations on unstructured grids

A computationally efficient multigrid algorithm for upwind edge‐based finite element schemes is developed for the solution of the two‐dimensional Euler and Navier–Stokes equations on unstructured triangular grids. The basic smoother is based upon a Galerkin approximation employing an edge‐based formulation with the explicit addition of an upwind‐type local extremum diminishing (LED) method. An explicit time stepping method is used to advance the solution towards the steady state. Fully unstructured grids are employed to increase the flexibility of the proposed algorithm. A full approximation storage (FAS) algorithm is used as the basic multigrid acceleration procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

An upwind compact mpact approach with group velocity control for compressible flow fields

In this paper we construct an upwind compact finite difference scheme with group velocity control for better simulation of compressible flow fields. Compared with traditional difference schemes, compact schemes have higher accuracy for the same stencil width. By means of the characteristic analysis of the operators, the group velocity of wave packets will be controlled to suppress the non‐physical oscillations in numerical solutions. In numerical simulation of the 3D compressible flow fields the third‐order accurate upwind compact operator is used to approximate the derivatives in the convection terms of the compressible N–S equations, the traditional finite difference scheme is used to approximate the viscous terms. Numerical solutions indicate that the method is satisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Arnoldi and Crank–Nicolson methods for integration in time of the transport equation

A comparison is made between the Arnoldi reduction method and the Crank–Nicolson method for the integration in time of the advection–diffusion equation. This equation is first discretized in space by the classic finite element (FE) approach, leading to an unsymmetric first‐order differential system, which is then solved by the aforementioned methods. Arnoldi reduces the native FE equations to a much smaller set to be efficiently integrated in the Arnoldi vector space by the Crank–Nicolson scheme, with the solution recovered back by a standard Rayleigh–Ritz procedure. Crank–Nicolson implements a time marching scheme directly on the original first‐order differential system. The computational performance of both methods is investigated in two‐ and three‐dimensional sample problems with a size up 30 000. The results show that in advection‐dominated problems less then 100 Arnoldi vectors generally suffice to give results with a 10⁻³–10⁻⁴ difference relative to the direct Crank–Nicolson solution. However, while the CPU time with the Crank–Nicolson starts from zero and increases linearly with the number of time steps used in the simulation, the Arnoldi requires a large initial cost to generate the Arnoldi vectors with subsequently much less expensive dynamics for the time integration. The break‐even point is problem‐dependent at a number of time steps which may be for some problems up to one order of magnitude larger than the number of Arnoldi vectors. A serious limitation of Arnoldi is the requirement of linearity and time independence of the flow field. It is concluded that Arnoldi can be cheaper than Crank–Nicolson in very few instances, i.e. when the solution is needed for a large number of time values, say several hundreds or even 1000, depending on the problem. Copyright © 2001 John Wiley & Sons, Ltd.     

对流扩散方程的迎风变换及相应有限差分方法

本文提出所谓迎风变换,将对流扩散方程分解为对流迎风函数和扩散方程,并构造相应的有限差分格式。对流迎风函数以简明的指数解析形式反映对流扩散现象的迎风效应,原则上消除了源于不对称对流算子的困难,能够便利对流扩散方程的数值求解。有限差分格式具有二阶精度和无条件稳定性,算例表明其准确性、收敛速度及对边界层效应的适应能力均明显优于中心差分格式和迎风差分格式。     

The Lagrangian approach of advective term treatment and its application to the solution of Navier—Stokes equations

A one-dimensional transport test applied to some conventional advective Eulerian schemes shows that linear stability analyses do not guarantee the actual performances of these schemes. When adopting the Lagrangian approach, the main problem raised in the numerical treatment of advective terms is a problem of interpolation or restitution of the transported function shape from discrete data. Several interpolation methods are tested. Some of them give excellent results and these methods are then extended to multi-dimensional cases. The Lagrangian formulation of the advection term permits an easy solution to the Navier-Stokes equations in primitive variables V, p, by a finite difference scheme, explicit in advection and implicit in diffusion. As an illustration steady state laminar flow behind a sudden enlargement is analysed using an upwind differencing scheme and a Lagrangian scheme. The importance of the choice of the advective scheme in computer programs for industrial application is clearly apparent in this example.     

On the properties of discrete adjoints of numerical methods for the advection equation

This paper discusses several aspects related to the consistency of the discrete adjoints of upwind numerical schemes. Both linear (finite differences, finite volumes) and nonlinear (slope and flux‐limited) discretizations of the one‐dimensional advection equation are considered. The analysis is focused on uniform meshes and on explicit numerical schemes. We show that the discrete adjoints may lose consistency near the points where upwinding changes, near inflow boundaries where another numerical scheme is employed, and near the locations where the slope/flux limiter is active in the forward simulation. Numerical results are presented to support the theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution

The simulation of transient flows is relevant in several applications involving viscoelastic fluids. In the last decades, much effort has been spent on deriving time-marching schemes able to efficiently solve the governing equations at low computational cost. In this direction, decoupling schemes, where the global system is split into smaller subsystems, have been particularly successful. However, most of these techniques only work if inertia and/or a large Newtonian solvent contribution is included in the modeling. This is not the case for polymer melts or concentrated polymer solutions.In this work, we propose two second-order time-integration schemes for discretizing the momentum balance as well as the constitutive equation, based on a Gear and a Crank–Nicolson scheme. The solution of the momentum and continuity equations is decoupled from the constitutive one. The stress tensor term in the momentum balance is replaced by its space-continuous but time-discretized form of the constitutive equation through an Euler scheme implicit in the velocity. This adds velocity unknowns in the momentum equation thus an updating of the velocity field is possible even if inertia and solvent viscosity are not included in the model. To further reduce computational costs, the non-linear relaxation term in the constitutive equation is taken explicitly leading to a linear system of equations for each stress component.Four benchmark problems are considered to test the numerical schemes. The results show that a Crank–Nicolson based discretization for the momentum equation produces oscillations when combined with a Crank–Nicolson based scheme for the constitutive equation whereas, if a Gear based scheme is implemented for the constitutive equation, the stability is found to be dependent on the specific problem. However, the Gear based scheme applied to the momentum balance combined with both second-order methods used for the constitutive equation is stable and accurate and performs much better than a first-order Euler scheme. Finally, a numerical proof of the second-order convergence is also carried out.     

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.     

Stability of a Crank–Nicolson pressure correction scheme based on staggered discretizations

In the context of LES of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. Designing such an algorithm, that is, as less dissipative as possible while being simple, for the resolution of variable density Navier–Stokes equations is the aim of the present work. The developed numerical scheme, based on a pressure correction technique, uses a Crank–Nicolson time discretization and a staggered space discretization relying on the Rannacher–Turek finite element. For the inertia term in the momentum balance equation, we propose a finite volume discretization, for which we derive a discrete analogue of the continuous kinetic energy local conservation identity. Contrary to what was obtained for the backward Euler discretization, the dissipation defect term associated to the Crank–Nicolson scheme is second order in time. This behavior is evidenced by numerical simulations. Copyright © 2013 John Wiley & Sons, Ltd.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Accurate multi‐level schemes for advection

The upwind leapfrog method for the advection equation, which is non‐dissipative and very accurate, is extended to higher‐order and multiple dimensions. The higher‐order version is developed by extending the stencil into space and time, and an analysis of the phase error is given. The schemes are then successfully applied to the classical test cases of rotating flow, and to a more realistic problem of non‐uniform advection. Copyright © 2003 John Wiley & Sons, Ltd.     

Developing a physical influence upwind scheme for pressure-based cell-centered finite volume methods

In the framework of a cell-centered finite volume method (FVM), the advection scheme plays the most important role in developing FVMs to solve complicated fluid flow problems for a wide range of Reynolds numbers. Advection schemes have been widely developed for FVMs employing pressure-velocity coupling methodology in the incompressible flow limit. In this regard, the physical influence upwind scheme (PIS) is developed for a cell-centered finite volume coupled solver (FVCS) using a pressure-weighted interpolation method for linking the pressure and velocity fields. The well-known exponential differencing scheme and skew upwind differencing scheme are also deployed in the current FVCS and their numerical results are presented. The accuracy and convergence of the present PIS are evaluated solving flow in a lid-driven square cavity, a lid-driven skewed cavity, and over a backward-facing step (BFS). The flow within the lid-driven square cavity is numerically solved at Reynolds numbers from 400 to 10 000 on a relatively coarse mesh with respect to other reported solutions. The lid-driven skewed cavity test case at Reynolds number of 1000 demonstrates the numerical performance of the present PIS on nonorthogonal grids. The flow over a BFS at Reynolds number of 800 is numerically solved to examine capabilities of current FVCS employing the current PIS in inlet-outlet flow computations. The numerical results obtained by the current PIS are in excellent agreement with those of benchmark solutions of corresponding test cases. Incorporating implicit role of pressure terms in a pressure-weighted interpolation method and development of PIS provides satisfactory solution convergence alongside the numerical accuracy for the current FVCS. A particular numerical verification is performed for the V velocity calculation within the BFS flow field, which confirms the reliability of present PIS.     

Fourier analysis of several finite difference schemes for the one‐dimensional unsteady convection–diffusion equation

This paper reports a comparative study on the stability limits of nine finite difference schemes to discretize the one‐dimensional unsteady convection–diffusion equation. The tested schemes are: (i) fourth‐order compact; (ii) fifth‐order upwind; (iii) fourth‐order central differences; (iv) third‐order upwind; (v) second‐order central differences; and (vi) first‐order upwind. These schemes were used together with Runge–Kutta temporal discretizations up to order six. The remaining schemes are the (vii) Adams–Bashforth central differences, (viii) the Quickest and (ix) the Leapfrog central differences. In addition, the dispersive and dissipative characteristics of the schemes were compared with the exact solution for the pure advection equation, or simple first or second derivatives, and numerical experiments confirm the Fourier analysis. The results show that fourth‐order Runge–Kutta, together with central schemes, show good conditional stability limits and good dispersive and dissipative spectral resolution. Overall the fourth‐order compact is the recommended scheme. Copyright © 2001 John Wiley & Sons, Ltd.     

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.     

A new numerical model for simulations of wave transformation,breaking and long‐shore currents in complex coastal regions

In this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations. The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Accuracy and stability analysis of a second‐order time‐accurate loosely coupled partitioned algorithm for transient conjugate heat transfer problems

In this paper, a second‐order time‐accurate loosely coupled partitioned algorithm is presented for solving transient thermal coupling of solids and fluids, also referred to by conjugate heat transfer. The Crank–Nicolson scheme is used for time integration. The accuracy and stability of the loosely coupled solution algorithm are analyzed analytically. Based on the accuracy analysis, the design order of the time integration scheme is preserved by following a predictor (implicit)–corrector (explicit) approach. Hence, the need to perform an additional implicit solve (a subiteration) at each time step is avoided. The analytical stability analysis shows that by using the Crank–Nicolson scheme for time integration, the partitioned algorithm is unstable for large Fourier numbers, unlike the monolithic approach. Accordingly, using the stability analysis, a stability criterion is obtained for the Crank–Nicolson scheme that imposes restriction on Δt given the material properties and mesh spacings of the coupled domains. As the ratio of the thermal effusivities of the coupled domains reaches unity, the stability of the algorithm reduces. To demonstrate the applicability of the algorithm, a numerical example is considered (an unsteady conjugate natural convection in an enclosure). Copyright © 2013 John Wiley & Sons, Ltd.