Achieving high-order fluctuation splitting schemes by extending the stencil

An extension to the fluctuation splitting approach for approximating hyperbolic conservation laws is described, which achieves higher than second-order accuracy in both space and time by extending the range of the distribution of the fluctuations. Initial results are presented for a simple linear scheme which is third-order accurate in both space and time on uniform triangular grids. Numerically induced oscillations are suppressed by applying the flux-corrected transport algorithm. These schemes are evaluated in the context of existing fluctuation splitting approaches to modelling time-dependent flows and some suggestions for their future development are made.     

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.     

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L²- and H¹-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.     

Navier-Stokes Solution by New Compact Scheme for Incompressible Flows

A new high spectral accuracy compact difference scheme is proposed here. This has been obtained by constrained optimization of error in spectral space for discretizing first derivative for problems with non-periodic boundary condition. This produces a scheme with the highest spectral accuracy among all known compact schemes, although this is formally only second-order accurate. Solution of Navier-Stokes equation for incompressible flows are reported here using this scheme to solve two fluid flow instability problems that are difficult to solve using explicit schemes. The first problem investigates the effect of wind-shear past bluff-body and the second problem involves predicting a vortex-induced instability.     

Energy stable and large time-stepping methods for the Cahn–Hilliard equation

We present the numerical methods for the Cahn–Hilliard equation, which describes phase separation phenomenon. The goal of this paper is to construct high-order, energy stable and large time-stepping methods by using Eyre's convex splitting technique. The equation is discretized by using a fourth-order compact difference scheme in space and first-order, second-order or third-order implicit–explicit Runge–Kutta schemes in time. The energy stability for the first-order scheme is proved. Numerical experiments are given to demonstrate the performance of the proposed methods.     

A Spectral Method for the Time Evolution in Parabolic Problems

Numerical time propagation of linear parabolic problems is commonly performed by Taylor expansion based schemes, such as Runge–Kutta. However, explicit schemes of this type impose a stringent stability restriction on the time step when the space discretization matrix is poorly conditioned. Thus the computational work required for integration over a long and fixed time interval is controlled by stability rather than by accuracy of the scheme. We develop an improved time evolution scheme based on a new Chebyshev series expansion for solving time-dependent inhomogeneous parabolic initial-boundary value problems in which the stability condition is relaxed. Spectral accuracy of the time evolution scheme is achieved. Additionally, the approximation derived here can be useful for solving quasi-linear parabolic evolution problems by exponential time differencing methods     

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

In this paper, a Cartesian grid method with cut cell approach has been developed to simulate two dimensional unsteady viscous incompressible flows with rigid bodies of arbitrary shape. A collocated finite volume method with nominally second-order accurate schemes in space is used for discretization. A pressure-free projection method is used to solve the equations governing incompressible flows. For fixed-body problems, the Adams-Bashforth scheme is employed for the advection terms and the Crank-Nicholson scheme for the diffusion terms. For moving-body problems, the fully implicit scheme is employed for both terms. The present cut cell approach with cell merging process ensures global mass/momentum conservation and avoid exceptionally small size of control volume which causes impractical time step size. The cell merging process not only keeps the shape resolution as good as before merging, but also makes both the location of cut face center and the construction of interpolation stencil easy and systematic, hence enables the straightforward extension to three dimensional space in the future. Various test examples, including a moving-body problem, were computed and validated against previous simulations or experiments to prove the accuracy and effectiveness of the present method. The observed order of accuracy in the spatial discretization is superlinear.     

An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem—that standard schemes are not stable on the arbitrarily small cut cells—while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in \(L^1\) and between first- and second-order accurate along the embedded boundary in two and three dimensions.     

Moment preserving schemes for Euler equations

A high order accurate finite difference scheme is proposed for one-dimensional Euler equations. In the scheme a set of first three moments of each signal are preserved during the updating. The scheme is one of 5th order in space and 4th order in time. This feature is different from that in typical existing methods in which the use of the first three polynomials results in only 3rd order accuracy in space. The scheme has different features from the existing high order schemes, and the most noticeable are the simultaneous discretization both in space and time, and the use of moments of Riemann invariants instead of primitive physical variables. Numerical examples are given to show the accuracy of the scheme and its robustness for the flows involving shocks.     

Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification

Two different explicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order, 5-point Forward Time Centred Space (FTCS) explicit formula, and the (1,9) FTCS explicit scheme which is generally second-order, but is fourth order when the diffusion number takes the value s = (1/6). These schemes are economical to use, are second-order and have bounded range of stability. The range of stability for the 5-point formula is less restrictive than the (1,9) FTCS explicit scheme. The results of numerical experiments are presented, and accuracy and Central Processor (CPU) times needed for each of the methods are discussed. These schemes use less central processor times than the second-order fully implicit method for two-dimensional diffusion with temperature overspecification. We also give error estimates in the maximum norm for each of these methods.     

A comparison of second- and sixth-order methods for large-eddy simulations

Large-eddy simulations of spatially developing planar turbulent jets are performed using a compact finite-difference scheme of sixth-order and an advective upstream splitting method-based method of second-order accuracy. The applicability of these solution schemes with different subgrid scale models and their performance for realistic turbulent flow problems are investigated. Solutions of the turbulent channel flow are used as an inflow condition for the turbulent jets. The results compare well with each other and with analytical and experimental data. For both solution schemes, however, the influence of the subgrid scale model on the time averaged turbulence statistics is small. This is known to be the case for upwind schemes with a dissipative truncation error, but here it is also observed for the high-order compact scheme. The reason is found to be the application of a compact high-frequency filter, which has to be used with strongly stretched computational grids to suppress high-frequency oscillations. The comparison of the results of the two schemes shows hardly any difference in the quality of the solutions. The second-order scheme, however, is computationally more efficient.     

An interpretation and derivation of the lattice Boltzmann method using Strang splitting

The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate approximation to evolution under the combination of two non-commuting operators, here identified with the streaming and collision terms in the discrete Boltzmann partial differential equation. Strang splitting achieves second-order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. We show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second-order accurate and explicit scheme from an integration of the discrete Boltzmann equation along characteristics. This approach extends easily to include general matrix collision operators, and also body forces. Finally, we show that the validity of the lattice Boltzmann discretisation for grid-scale Reynolds numbers larger than unity depends crucially on the use of a Crank–Nicolson approximation to discretise the collision operator. Replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive, due to the splitting error, unless the grid-scale Reynolds number remains well below unity.     

Iterative inflow-implicit outflow-explicit finite volume scheme for level-set equations on polyhedron meshes

In this paper, we propose a cell-centered finite volume method for advective and normal flows on polyhedron meshes which is second-order accurate in space and time for smooth solutions. In order to overcome a time restriction caused by CFL condition, an implicit time discretization of inflow fluxes and an explicit time discretization of outflow fluxes are used in an iterative procedure. For an efficient computation, an 1-ring face neighborhood structure is introduced. Since it is limited to access unknown variables in an 1-ring face neighborhood structure, an iterative procedure is proposed to resolve the limitation of assembled linear system. Two types of gradient approximations, an inflow-based gradient and an average-based gradient, are studied and compared from the point of numerical accuracy. Numerical schemes are tested for an advective and a normal flow of level-set functions illustrating a behavior of the proposed method for an implicit tracking of a smooth and a piecewise smooth interface.     

Numerical investigation of fully developed channel flow using shock-capturing schemes

The ability to simulate wall-bounded channel flows with second- and third-order shock-capturing schemes is tested on both subsonic and supersonic flow regimes, respectively at Mach 0.5 and 1.5. Direct numerical simulations (DNSs) and large-eddy simulations (LESs) are performed at Reynolds number 3000.In both flow regimes, results are compared with well-documented DNS, LES or experimental data.At Ma₀=0.5, a simple second-order centred scheme provides results in excellent agreement with incompressible DNS databases, while the addition of artificial or subgrid-scale (SGS) dissipation decreases the resolution accuracy giving just satisfactory results. At Ma₀=1.5, the second-order space accuracy is just sufficient to well resolve small turbulence scales on the chosen grid: without any dissipation models, such accuracy provides results in good agreement with reference data, while the addition of dissipation models considerably reduces the turbulence level and the flow appears almost laminar. Moreover, the use of explicit dissipative SGS models reduces the results accuracy.In both flow regimes, the numerical dissipation due to the discretization of the convective terms is also interpreted in terms of SGS dissipation in an LES context, yielding a generalised dynamic coefficient, equivalent to the dynamic coefficient of the Germano et al. [Phys. Fluids A 3(7) (1991) 1760] SGS model. This new generalised coefficient is thus developed to compare the order of magnitude of the intrinsic numerical dissipation of a shock-capturing scheme with respect to the SGS dissipation.     

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.     

Positivity-preserving flux-limited method for compressible fluid flow

The extension of a flux discretization method to second-order accuracy can lead to some difficulties in maintaining positivity preservation. While the MUSCL-TVD scheme maintains the positivity preservation property of the underlying 1st-order flux discretization method, a flux-limited-TVD scheme does not. A modification is here proposed to the flux-limited-TVD scheme to make it positivity-preserving when used in conjunction with the Steger-Warming flux vector splitting method. The proposed algorithm is then compared to MUSCL for several test cases. Results obtained indicate that while the proposed scheme is more dissipative in the vicinity of contact discontinuities, it performs significantly better than MUSCL when solving strong shocks in hypersonic flowfields: the amount of pressure overshoot downstream of the shock is minimized and the time step can be set to a value typically two or three times higher. While only test cases solving the one-dimensional Euler equations are here presented, the proposed scheme is written in general form and can be extended to other physical models.     

An adaptive multiresolution method for combustion problems: application to flame ball-vortex interaction

In this article, we present numerical simulations of flame ball-vortex interactions using adaptive multiresolution methods. The numerical scheme for the convection-diffusion-reaction equations modelling this problem is based on a finite volume discretization coupled with a discrete multiresolution analysis. The grid in physical space is adapted dynamically to track the evolution of the solution in scale and space. Time integration is done by an explicit Runge-Kutta scheme. In space we use second-order centered schemes. The implementation is based on a graded tree data structure, which improves both CPU and memory performances, as no fine gridding is required in regions where the solution is smooth. To illustrate the features and the efficiency of the method, we compute several flame ball-vortex interactions and study the role played by the fluid flow on the evolution of the flame ball. We observe the roll-up of the flame ball around the vortex into a snail-like structure. We also put into evidence the flammability limit of the flame ball in function of both vortex and radiation intensities.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

Aeroacoustic modeling of complex flow problems: I-Domain decomposition method for the reduced wave equation

An accurate and efficient method for solving the wave equation on multi-domains is developed for two-dimensional geometries. In this work we treat Cartesian geometries, but the method may be directly extended to more general geometries. As a first step, the one-dimensional problem is investigated. The wave equation is solved in the Fourier space. Three different numerical discretizations are tested, a Pointwise second-order accurate discretization (PT), and two fourth-order schemes: a Padè approximation (HO), and an Equation Based scheme (EB). A consistent discretization of the non reflecting boundary conditions is proposed, which preserves the overall accuracy of the corresponding interior scheme. For the solution of the linear system, it is shown that the preconditioned ILUT-GMRES method is an appropriate choice. In the multi-domain method, an optimal iterative procedure is described, specifying the correct form of the transmission conditions at the interfaces. The numerical tests confirm that the present multi-domain technique retains the same numerical properties of the single domain method. Finally the single and multi domain methods are extended to the two-dimensional case. Received: 31 January 2001 / Accepted: 30 September 2001     

Dispersion-bounded numerical integration of the elastodynamic equations with cost-effective staggered schemes

Known procedures for designing numerical schemes for the integration of elastodynamic equations with explicit control over numerical dispersion are reviewed. In the literature, the analysis of such schemes has concentrated on the discrete space differentiators, and has neglected the role played by time discretization in the overall accuracy. In this paper we define a computational cost for a given dispersion error bound which fully includes the effect of temporal differencing. For some representative schemes based on leap-frog time marching, we provide an optimal operating point (time sampling rate and number of grid points per shortest wavelength) which minimizes the computational cost for a given dispersion error threshold. Based on this notion of cost, we introduce new optimal operators for staggered grids. Additionally, we introduce the notion of composite differentiators to design still more cost-effective schemes. The cost of the proposed schemes is shown to be less than that of known finite difference (FD) operators and compares favorably with pseudo-spectral (PS) algorithms. Numerical simulations are presented to illustrate the effectiveness of the new operators.