Large Courant Numbers Research Articles

Development of a high-order level set method: Compact Conservative Level Set (CCLS)

In this paper a Compact Conservative Level Set (CCLS) method has been developed to improve the accuracy and robustness of Level Set (LS) approach in simulation of incompressible two-phase flows. Since mass conservation error is known as the evident weakness of LS approach, the main aim of this research is to introduce a novel method with an excellent mass conservation property while keeping the simplicity of the approach. Here, the hyperbolic tangent function; used as LS function; is conservatively transported through a High-Order Compact (HOC) finite difference scheme. In order to retain thickness of the coated interface while retrieving the hyperbolic tangent profile, a conservative re-initialization is adopted by introducing a novel Combined Compact Difference (CCD) method. Normals are computed from the distance function, where a fast marching method; an efficient re-distancing method; is applied to re-construct signed distance function from the hyperbolic tangent function. The developed method is employed in a wide range of test cases and its noticeable mass conservation property is revealed even for complex cases, as well as its accuracy and stability in large courant numbers, long time simulation, coarse mesh and even simulations without re-initialization.

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

A new streamfunction/log-conformation formulation of incompressible viscoelastic flows is presented. The log-conformation representation guaranties the positive-definiteness of the conformation tensor and obviates the high Weissenberg number problem. The streamfunction is defined as a vector potential of the velocity field, and provides a pressureless formulation of the conservation laws, which automatically enforces the incompressibility. The resulting numerical method is free from velocity–pressure decoupling errors, and can achieve stable calculations for large Courant numbers, which improve the robustness and the efficiency of the solver. The two-dimensional flow of an Oldroyd-B fluid inside the lid-driven cavity is simulated for a large range of Weissenberg numbers. The numerical results demonstrate the second-order accuracy of our scheme, and our solutions are in good agreement with the available data from the literature for Weissenberg number 3 and below. Finally, the simulations at higher Weissenberg numbers 5 and 10 reveal a structural mechanism that sustains quasi-periodic elastic instabilities arising at the upstream corner of the moving lid.

An Element-Based Spectrally Optimized Approximate Inverse Preconditioner for the Euler Equations

We introduce a method for constructing an element-by-element sparse approximate inverse (SAI) preconditioner designed to be effective in a massively parallel spectral element modeling environment involving nonsymmetric systems. This new preconditioning approach is based on a spectral optimization of a low-resolution preconditioned system matrix. We show that the local preconditioning matrices obtained via this element-based spectrum-optimized (EBSO) approach may be applied to arbitrarily high-resolution versions of the same system matrix without appreciable loss of preconditioner performance. We demonstrate the performance of the EBSO preconditioning approach using two-dimensional spectral element method formulations for a simple linear conservation law and for the fully compressible two-dimensional Euler equations with various boundary conditions. For the latter model running at sufficiently large Courant number, the EBSO preconditioner significantly reduces both iteration count and wall-clock time regar...

On the Use of Temporal Reachout Technique for Characteristics Method with Time-Line Cubic Spline Interpolation

ABSTRACTThe study had indicated that the computational performances of the characteristics method with the time-line cubic spline interpolation are related to the endpoint constraint, especially for large Courant number in which the foot of the characteristic trajectory is located near the endpoint. The first derivative endpoint constraint with higher-order central difference approximation provides better simulation results among various endpoint constraints, but it still induces some degree of numerical error. In this study, by locating the foot of the characteristic trajectory away from the endpoint, the temporal reachout technique is proposed to avoid the effect of endpoint constraint on the time-line cubic spline interpolation. Modeling the transport of a Gaussian concentration distribution in a uniform flow with constant diffusion coefficient and the viscous Burgers equation is used to examine the temporal reachout technique. The outcomes show that the temporal reachout technique yields much better simulation results than the first derivative endpoint constraint with higher-order central difference approximation. The effect of endpoint constraint on the time-line cubic spline interpolation can be greatly diminished by the use of the temporal reachout technique.

Improving the time-splitting errors of one-dimensional advection schemes in multidimensional applications

A method is presented to reduce the time-splitting errors of one-dimensional flux form advection schemes. In multidimensional applications deformational correction terms are included in each one-dimensional advection step whereby all correction terms add up to zero. With the correction terms a consistent form of a given one-dimensional advection scheme is achieved when time-splitting is performed. Positive definiteness and shape preservation of the scheme are obtained by nonlinear limitation of the advective fluxes and the deformational correction terms. Large Courant numbers are treated by applying local timestep reduction. The resulting scheme is strictly one-dimensional, mass conservative and positive definite. Shape preservation of the method is nearly achieved. Two- and three-dimensional test runs are presented demonstrating that the method yields satisfactory results. The horizontal flow in a terrain with a steep mountain ridge is investigated by utilizing a terrain following vertical coordinate. It is shown that even in steep orographical terrain the method works quite well, however, small errors occur in the shape preservation of the scheme. These errors increase with increasing spatial variation of the vertical grid mesh.

A Technique for High-Order Treatment of Diffusion Terms in Semi-Lagrangian Schemes

We consider in this paper a high-order, semi-Lagrangian technique to treat possibly degenerate advection–diffusion equations, which has been proposed in similar forms by various authors. The scheme is based on a stochastic representation formula for the solution, which allows to avoid the splitting between advective and diffusive part of the evolution operator. A general theoretical analysis is carried out in the paper, with a special emphasis on the possibility of using large Courant numbers, and numerical tests in one and two space dimensions are presented. AMS subject classifications: 65M25, 65M12, 60H30

Particle trajectory calculations with a two‐step three‐time level semi‐Lagrangian scheme well suited for curved flows

AbstractThis study proposes a new two‐step three‐time level semi‐Lagrangian scheme for calculation of particle trajectories. The scheme is intended to yield accurate determination of the particle departure position, particularly in the presence of significant flow curvature. Experiments were performed both for linear and non‐linear idealized advection problems, with different flow curvatures. Results for simulations with the proposed scheme, and with three other semi‐Lagrangian schemes, and with an Eulerian method are presented. In the linear advection problem the two‐step three‐time level scheme produced smaller root mean square errors and more accurate replication of the angular displacement of a Gaussian hill than the other schemes. In the non‐linear advection experiments the proposed scheme produced, in general, equal or better conservation of domain‐averaged quantities than the other semi‐Lagrangian schemes, especially at large Courant numbers. In idealized frontogenesis simulations the scheme performed equally or better than the other schemes in the representation of sharp gradients in a scalar field. The two‐step three‐time level scheme has some computational overhead as compared with the other three semi‐Lagrangian schemes. Nevertheless, the additional computational effort was shown to be worthwhile, due to the accuracy obtained by the scheme in the experiments with large time steps. The most remarkable feature of the scheme is its robustness, since it performs well both for small and large Courant numbers, in the presence of weak as well strong flow curvatures. Copyright © 2009 John Wiley & Sons, Ltd.

A Stability Analysis of Finite-Volume Advection Schemes Permitting Long Time Steps

Abstract Finite-volume schemes developed in the meteorological community that permit long time steps are considered. These include Eulerian flux-form schemes as well as fully two-dimensional and cascade cell-integrated semi-Lagrangian (CISL) schemes. A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given. Contrary to previous analysis, no simplifications in terms of reducing the formal order of the schemes, which makes the analysis mathematically less complex, have been applied. An interscheme comparison of both dissipation and dispersion properties is given. The main finding is that the dissipation and dispersion properties of Eulerian flux-form schemes are sensitive to the choice of inner and outer operators applied in the scheme that can lead to increased numerical damping for large Courant numbers. This spurious dependence on the integer value of the Courant number disappears if the inner and outer operators are identical, in which case, under the assumptions used in the stability analysis, the Eulerian flux-form scheme becomes identical to the cascade scheme. To explain these properties a conceptual interpretation of the flux-based Eulerian schemes is provided. Of the two CISL schemes, the cascade scheme has superior stability properties.

Analysis of the response to orographic forcing of a time‐staggered semi‐Lagrangian discretization of the rotating shallow‐water equations

AbstractA recently proposed time‐staggered semi‐Lagrangian discretization of the unforced rotating shallow‐water equations is extended to include orographic forcing. Linear analysis shows that, as for traditional semi‐implicit semi‐Lagrangian schemes, it is also susceptible to spurious orographic resonance for large Courant numbers. A solution is proposed which, as shown by further linear analysis, addresses spurious orographic resonance whilst also maintaining computational stability. © Crown copyright, 2006. Royal Meteorological Society

A semi‐Lagrangian Crank‐Nicolson algorithm for the numerical solution of advection‐diffusion problems

We present a hybrid method for the numerical solution of advection‐diffusion problems that combines two standard algorithms: semi‐Lagrangian schemes for hyperbolic advection‐reaction problems and Crank‐Nicolson schemes for purely diffusive problems. We show that the hybrid scheme is identical to the two end‐member schemes in the limit of infinite and zero Peclet number and remains accurate over a wide range of Peclet numbers. This scheme does not have a CFL stability criterion allowing the choice of time step to be decoupled from the spatial resolution. We compare numerical results with an analytic solution and test both an operator split version of our method and a combined version that solves advection and diffusion simultaneously. We also compare results of simple explicit and implicit numerical schemes and show that the semi‐Lagrangian Crank‐Nicolson (SLCN) scheme is both faster and more accurate on the same problem. We then apply the combined SLCN scheme to a more geologically relevant benchmark for calculating the thermal structure of a subduction zone. This problem demonstrates that the SLCN scheme can remain stable and accurate at large Courant numbers even in flows with highly curved streamlines. Finally, we introduce a variable order interpolation scheme for the semi‐Lagrangian schemes that reduces interpolation artifacts for sharp fronts without introducing numerical diffusion.

Examination of characteristics method with cubic interpolation for advection–diffusion equation

In this study, the use of the characteristics method integrated with the Hermite cubic interpolation or the cubic-spline interpolation on the space line or the time line, i.e., the HCSL scheme, the CSSL scheme, the HCTL scheme, and the CSTL scheme, respectively, for solving the advection–diffusion equation is examined. The advection and diffusion of a Gaussian concentration distribution in a uniform flow with constant diffusion coefficient is used to conduct this investigation. The effects of parameters, such as Peclet number, Courant number, and the reachback number, on these four schemes used herein for solving the advection–diffusion equation are investigated. The simulated results show that the CSSL scheme is comparable to the HCSL scheme, and the two schemes seem insensitive to Courant number as compared with the HCTL scheme and the CSTL scheme. With large Peclet number, for small Courant number the HCTL scheme is more accurate than the HCSL scheme and the CSSL scheme. However, for large Courant number the HCTL scheme has worse computed results in comparison with the HCSL scheme and the CSSL scheme. With small Peclet number, the HCTL scheme, the HCSL scheme, and the CSSL scheme have close simulated results. Despite Peclet number, for small Courant number the CSTL scheme is comparable to the HCTL scheme, but for large Courant number the former scheme provides unacceptable simulated results in which very large numerical diffusion is induced due to the effect of the natural endpoint constraint. For large Peclet number the HCSL scheme and the CSSL scheme integrated with the reachback technique can improve simulated results, but for small Peclet number the HCSL scheme and the CSSL scheme seem not to be influenced by increasing the reachback number.

A higher-order implicit IDO scheme and its CFD application to local mesh refinement method

The Interpolated Differential Operator (IDO) scheme has been developed for the numerical solution of the fluid motion equations, and allows to produce highly accurate results by introducing the spatial derivative of the physical value as an additional dependent variable. For incompressible flows, semi-implicit time integration is strongly affected by the Courant and diffusion number limitation. A high-order fully-implicit IDO scheme is presented, and the two-stage implicit Runge-Kutta time integration keeps over third-order accuracy. The application of the method to the direct numerical simulation of turbulence demonstrates that the proposed scheme retains a resolution comparable to that of spectral methods even for relatively large Courant numbers. The scheme is further applied to the Local Mesh Refinement (LMR) method, where the size of the time step is often restricted by the dimension of the smallest meshes. In the computation of the Karman vortex street problem, the implicit IDO scheme with LMR is shown to allow a conspicuous saving of computational resources.

Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio.

We have developed a lattice Boltzmann model for the compressible Navier-Stokes equations with a flexible specific-heat ratio. Several numerical results are presented, and they agree well with the corresponding solutions of the Navier-Stokes equations. In addition, an explicit finite-difference scheme is proposed for the numerical calculation that can make a stable calculation with a large Courant number.

Conservative CIP Transport in Meteorological Models

In order to improve the model representation, we have implemented the CIP-CSLR in meteorological models. Real-case simulations and idealized tests were carried out on the Earth Simulator with atmospheric general circulation models that only the watervapor and liquid water were transported with the CIP-CSLR method. Numerical experiments show that the CIP-CSLR scheme substantially improved the model outputs for both pure advection tests and the long-term climate simulations. Reasonable tropical precipitation is shown with the CIP-CSLR scheme, which is largely improved in comparison with the original spectral method. Using the CIP-CSLR on a new grid (Yin-Yang) system, we also achieved conservative and more accurate results of passive tracer advection with relatively fewer grid points compared with the latitude-longitude grid. Without polar singularity, the splitting procedure of CIP-CSLR achieves promising transport in spherical geometry even if large Courant number is specified.

Extension of a Conservative Cascade Scheme on the Sphere to Large Courant Numbers

The conservative cascade scheme (CCS) combines a one-dimensional mass-conserving finite-volume method with an efficient semi-Lagrangian scheme over the sphere. Major limitations of this scheme are a breakdown in the polar region due to the coordinate singularity and a restriction to polar meridional Courant number Cθ ≤ 1. The proposed scheme is an extension of the CCS over the sphere for Cθ > 1. This is achieved by isolating a region near the pole where the CCS fails and applying a two-dimensional remapping based on the cell-integrated semi-Lagrangian (CISL) scheme. The interface between these two schemes is the singular polar region where the total mass is computed and redistributed in a conservative manner. The resulting scheme is applicable to large polar Courant number and is tested using solid-body rotation and a deformational flow.

An efficient PML implementation for the ADI-FDTD method

A novel implementation of the perfectly matched layer (PML) absorber for the alternating-direction implicit finite-difference time-domain method is proposed and implemented. It is shown that, compared to the traditional PML implementation, the performance of the proposed PML is more efficient for large Courant numbers.

Error bounds for the large time step Glimm scheme applied to scalar conservation laws

In this paper we derive an \(L^1\) error bound for the large time step, i.e. large Courant number, version of the Glimm scheme when used for the approximation of solutions to a genuinely nonlinear, i.e. convex, scalar conservation law for a generic class of piecewise constant data. We show that the error is bounded by \(O(\Delta x^{1/2}\vert \log\Delta x\vert )\) for Courant numbers up to 1. The order of the error is the same as that given by Hoff and Smoller [5] in 1985 for the Glimm scheme under the restriction of Courant numbers up to 1/2.

F-1520 定常超音速流れの数値計算における陽的時間積分法について(S55-3 高温/高速現象と可視化(3))(S55 高速/高温現象と可視化)

In the present work, a robust explicit time-integration scheme named the rational Runge-Kutta (RRK) method is applied to steady supersonic flow simulations with unstructured solution adaptive grids. The results from the present numerical tests show that the RRK method allows larger Courant numbers than the modified Euler explicit method. Also, it is found that the maximal value of allowed Courant numbers in the simulations with the RRK method strongly depends on the size of the smallest edge length in the computational grids.

Conservative Semi-Lagrangian Algorithm for Pollutant Transport in Rivers

A conservative semi-Lagrangian numerical method for solute transport in steady nonuniform flows is presented. The method is an extension of earlier work on the authors' DISCUS method. Numerical results are compared against an exact solution for solute transport in a nonuniform flow with a linearly varying velocity coefficient and a quadratically varying dispersion coefficient. The method is stable and fully conservative at large Courant and grid Peclet numbers. Accuracy is also good and appears to be primarily related to spatial resolution and grid Peclet number. The method is significantly more computationally efficient than Eulerian numerical methods.

Two-Dimensional Flow Model for Trapezoidal High-Velocity Channels

A two-dimensional numerical flow model for trapezoidal high-velocity channels is developed. The model is designed specifically for simulation of flow in channels having sloping sidewalls in which the depth is an unknown variable in the governing equations and therefore the plan view of the flow domain is not known a priori. Solutions are obtained by time stepping from specified initial conditions using an implicit Petrov-Galerkin moving finite-element representation of the governing equations. The moving finite-element model produces a simultaneous solution for the boundary displacement and flow variables. This implementation provides stable solutions for supercritical flow even at relatively large Courant numbers. The model is tested by comparison of simulation results with laboratory data. These data sets serve as a basis for evaluation of the numerical model and should also prove useful to researchers in testing other numerical flow models applied to supercritical flow in channels having sloping sidewalls.