Conservative Advection Scheme Research Articles

The water cycle in the general circulation model of the martian atmosphere

Within the numerical general-circulation model of the Martian atmosphere MAOAM (Martian Atmosphere: Observation and Modeling), we have developed the water cycle block, which is an essential component of modern general circulation models of the Martian atmosphere. The MAOAM model has a spectral dynamic core and successfully predicts the temperature regime on Mars through the use of physical parameterizations typical of both terrestrial and Martian models. We have achieved stable computation for three Martian years, while maintaining a conservative advection scheme taking into account the water–ice phase transitions, water exchange between the atmosphere and surface, and corrections for the vertical velocities of ice particles due to sedimentation. The studies show a strong dependence of the amount of water that is actively involved in the water cycle on the initial data, model temperatures, and the mechanism of water exchange between the atmosphere and the surface. The general pattern and seasonal asymmetry of the water cycle depends on the size of ice particles, the albedo, and the thermal inertia of the planet’s surface. One of the modeling tasks, which results from a comparison of the model data with those of the TES experiment on board Mars Global Surveyor, is the increase in the total mass of water vapor in the model in the aphelion season and decrease in the mass of water ice clouds at the poles. The surface evaporation scheme, which takes into account the turbulent rise of water vapor, on the one hand, leads to the most complete evaporation of ice from the surface in the summer season in the northern hemisphere and, on the other hand, supersaturates the atmosphere with ice due to the vigorous evaporation, which leads to worse consistency between the amount of the precipitated atmospheric ice and the experimental data. The full evaporation of ice from the surface increases the model sensitivity to the size of the polar cap; therefore, the increase in the latter leads to better results. The use of a more accurate dust scenario changes the model temperatures, which also strongly affects the water cycle.

Mass-Conserving Eulerian Liquid Simulation

We present a GPU friendly, Eulerian, free surface fluid simulation method that conserves mass locally and globally without the use of Lagrangian components. Local mass conservation prevents small-scale details of the free surface from disappearing, a problem that plagues many previous approaches, while global mass conservation ensures that the total volume of the liquid does not decrease over time. Our method handles moving solid boundaries as well as cells that are partially filled with solids. Due to its stability, it allows the use of large time steps that makes it suitable for both offline and real-time applications. We achieve this by using density-based surface tracking with a novel, unconditionally stable, conservative advection scheme. We also propose mass conserving methods to sharpen the interface and to reveal subgrid features of the liquid. While our approach conserves mass, volume loss is still possible but only temporarily. With constant mass, local volume loss causes a local increase of the density used for surface tracking which we detect and correct over time. We show the effectiveness of the proposed methods in several practical examples all running either at interactive rates or in real time.

Fully conservative leak-proof treatment of thin solid structures immersed in compressible fluids

We propose a novel high resolution conservative advection scheme that is suitable for thin, embedded moving solid structures. The scheme works by coupling together a high order flux-based method with a conservative semi-Lagrangian solver that is similar in spirit to that of Lentine et al. [26], but modified to treat the cut cells and partial volumes that arise near a thin solid structure. The conservative semi-Lagrangian scheme is unconditionally stable, and so unlike previous methods no cell merging is required to compensate for the small cell volumes that arise. Furthermore, as the semi-Lagrangian scheme works via tracing characteristic curves, no special treatment is required either to enforce non-penetration through thin, moving solid structures, or to populate swept or uncovered degrees of freedom. For the flux-based solver, we use finite-difference ENO with Lax–Friedrich’s diffusion (although any flux-based scheme works), and in doing so we found that a modification to the diffusion calculation leads to improved stability in its third order accurate variant. We integrate this novel hybrid advection scheme into a semi-implicit compressible flow solver, and modify the implicit pressure solver to work with cells of variable size. In addition, we propose an improvement to the semi-implicit compressible flow solver via a new method for computing a post-advected pressure. Finally, this hybrid conservative advection scheme is integrated into a semi-implicit fluid–structure solver, and a number of one-dimensional and two-dimensional examples are considered—in particular, showing that we can handle thin solid structures moving through the grid in a fully conservative manner, preventing fluid from leaking from one side of the structure to the other and without the need for cell merging or other special treatment of cut cells and partial volumes.

Cloud Modeling Tests of the ULTIMATE–MACHO Scalar Advection Scheme

Abstract Numerical diffusion can be minimized using fine grid spacing and/or higher-order numerical schemes. In this study, the authors focus on higher-order scalar advection schemes and their effects on simulated cloud fields. A monotonic multidimensional odd-order conservative advection scheme has been implemented, following the approach of Leonard. It has been tested in simulations of idealized scalar fields advected by simple prescribed motion, as well as turbulence fields; large-eddy simulations of turbulent stratocumulus clouds; and simulations of deep convective clouds. New third-, fifth-, and seventh-order schemes are compared with the second-order scheme originally used in the model. For the deep cumulus case, a high-resolution large-eddy simulation with the same domain size is used as a benchmark. The fifth-order scheme shows much less numerical diffusion than the lower-order scheme. The additional improvement with the seventh-order scheme is minor. The higher-order scheme generally produces simulated cloud fields similar to those obtained with a lower-order scheme with a finer grid spacing. This effect is especially noticeable for the updraft-core statistics of the deep cumulus simulation, as compared with the benchmark simulation. The fifth-order scheme with coarse horizontal resolution produces results close to those of the benchmark simulation. Compared to a high-resolution simulation with the low-order scheme, the numerical cost of the fifth-order simulation is smaller than a factor of 10.

A Class of Conservative Fourth-Order Advection Schemes and Impact of Enhanced Formal Accuracy on Extended-Range Forecasts

Starting from three Eulerian second-order nonlinear advection schemes for semi-staggered Arakawa grids B/E, advection schemes of fourth order of formal accuracy were developed. All three second-order advection schemes control the nonlinear energy cascade in case of nondivergent flow by conserving quadratic quantities. Linearization of all three schemes leads to the same second-order linear advection scheme. The second-order term of the truncation error of the linear advection scheme has a special form so that it can be eliminated by modifying the advected quantity while still preserving consistency. Tests with linear advection of a cone confirm the advantage of the fourth-order scheme. However, if a localized, large amplitude and high wavenumber pattern is present in initial conditions, the clear advantage of the fourth-order scheme disappears. The new nonlinear fourth-order schemes are quadratic conservative and reduce to the Arakawa Jacobian for advected quantities in case of nondivergent flow. In case of general flow the conservation properties of the new momentum advection schemes impose stricter constraint on the nonlinear cascade than the original second-order schemes. However, for nondivergent flow, the conservation properties of the fourth-order schemes cannot be proven in the same way as those of the original second-order schemes. Therefore, demanding long-term and low-resolution nonlinear tests were carried out in order to investigate how well the fourth-order schemes control the nonlinear energy cascade. All schemes were able to maintain meaningful solutions throughout the test. Finally, the impact was examined of the fourth-order momentum advection on global medium-range forecasts. The 500-hPa anomaly correlation coefficient obtained using the best performing fourth-order scheme did not show an improvement compared to the tests using its second-order counterpart.

CABARET in the ocean gyres

A new high-resolution Eulerian numerical method is proposed for modelling quasigeostrophic ocean dynamics in eddying regimes. The method is based on a novel, second-order non-dissipative and low-dispersive conservative advection scheme called CABARET. The properties of the new method are compared with those of several high-resolution Eulerian methods for linear advection and gas dynamics. Then, the CABARET method is applied to the classical model of the double-gyre ocean circulation and its performance is contrasted against that of the common vorticity-preserving Arakawa method. In turbulent regimes, the new method permits credible numerical simulations on much coarser computational grids.

An Upwind-Biased Conservative Advection Scheme for Spherical Hexagonal–Pentagonal Grids

Abstract A discrete form of the flux-divergence operator is developed to compute advection of tracers on spherical hexagonal–pentagonal grids. An upwind-biased advection scheme based on a piecewise linear approximation for one-dimensional regular grids is extended simply for spherical hexagonal–pentagonal grids. The distribution of a tracer over the upwind side of a cell face is linearly approximated using a nodal value and a gradient at a computational node on the upwind side. A piecewise linear approximation is relaxed to a local linear approximation, and the relaxation precludes the complicated conditional branching present in remapping schemes. Results from a cosine bell advection test show that the new scheme compares favorably with other upwind-biased schemes for spherical hexagonal–pentagonal grids.

636 非構造格子とCIVA法に基づく保存形スキームの開発(S04-1 ここまで来たCIP法とその将来(1),S04 ここまで来たCIP法とその将来)

636 非構造格子とCIVA法に基づく保存形スキームの開発(S04-1 ここまで来たCIP法とその将来(1),S04 ここまで来たCIP法とその将来)

Laminated edge filter structure

We propose a novel high resolution conservative advection scheme that is suitable for thin, embedded moving solid structures. The scheme works by coupling together a high order flux-based method with a conservative semi-Lagrangian solver that is similar in spirit to that of Lentine et al. [26], but modified to treat the cut cells and partial volumes that arise near a thin solid structure. The conservative semi-Lagrangian scheme is unconditionally stable, and so unlike previous methods no cell merging is required to compensate for the small cell volumes that arise. Furthermore, as the semi-Lagrangian scheme works via tracing characteristic curves, no special treatment is required either to enforce non-penetration through thin, moving solid structures, or to populate swept or uncovered degrees of freedom. For the flux-based solver, we use finite-difference ENO with Lax–Friedrich’s diffusion (although any flux-based scheme works), and in doing so we found that a modification to the diffusion calculation leads to improved stability in its third order accurate variant. We integrate this novel hybrid advection scheme into a semi-implicit compressible flow solver, and modify the implicit pressure solver to work with cells of variable size. In addition, we propose an improvement to the semi-implicit compressible flow solver via a new method for computing a post-advected pressure. Finally, this hybrid conservative advection scheme is integrated into a semi-implicit fluid–structure solver, and a number of one-dimensional and two-dimensional examples are considered—in particular, showing that we can handle thin solid structures moving through the grid in a fully conservative manner, preventing fluid from leaking from one side of the structure to the other and without the need for cell merging or other special treatment of cut cells and partial volumes.

The Mass-Conservative Cell-Integrated Semi-Lagrangian Advection Scheme on the Sphere

A mass-conservative cell-integrated semi-Lagrangian (CISL) scheme is presented and tested for 2D transport on the sphere. The total mass is conserved exactly and the mass of each individual grid cell is conserved in general. The scheme is based on a general scheme developed by Machenhauer and Olk that has increased cost effectiveness without loss of accuracy, compared to the CISL scheme of Rančić. A regular latitude–longitude grid is used on the sphere and upstream trajectories from the corner points of the regular grid cells (the Eulerian cells) define the corner points of the departure cells. The sides in these so-called Lagrangian cells are generally defined as straight lines in a (λ, μ) plane, where λ is the longitude and μ is the sine of the latitude. The mass distribution within each Eulerian grid cell is defined by quasi-biparabolic functions, which are used to integrate analytically the mass in each Lagrangian computational cell. The auxiliary computational cells are polygons with each side parallel to the coordinate axis. Also, the computational cells have the same area as the Lagrangian cells they approximate. They were introduced in order to simplify the analytical integrals of mass. Near the poles, the east and west sides of certain Lagrangian cells cannot be approximated by straight lines in the (λ, μ) plane, and are instead represented by straight lines in polar tangent plane coordinates. Each of the latitudinal belts of Lagrangian cells in the polar caps are split up into several latitudinal belts of subcells, which can be approximated by computational cells as in the case of cells closer to the equator. One latitudinal belt in each hemisphere, which encloses the Eulerian pole (singular belt), is treated in a special way. First the total mass in the singular belt is determined and then it is redistributed among the cells in the belt using weights determined by a traditional SL scheme at the midpoints of the cells. By this procedure the total mass is still conserved while the conservation is only approximately maintained for the individual cells in the singular belt. These special treatments in the polar caps fit well into the general structure of the code and can be implemented with minor modifications in the code used for the rest of the sphere. Compared to two other conservative advection schemes implemented on the sphere the CISL scheme used here was found to be competitive in terms of accuracy for the same resolution. In addition the CISL scheme has the advantage over these schemes that it is applicable for Courant numbers larger than one. In plane geometry the scheme of Rančić had an overhead factor of 2.5 in CPU time compared to a traditional bicubic semi-Lagrangian scheme. This factor is reduced to 1.1 for the Machenhauer and Olk scheme on the plane while on the sphere the factor is found to be 1.28 for the present scheme. This overhead seems to be a reasonable price to pay for increased accuracy and exact mass conservation.

Consistency with continuity in conservative advection schemes for free‐surface models

AbstractThe consistency of the discretization of the scalar advection equation with the discretization of the continuity equation is studied for conservative advection schemes coupled to three‐dimensional flows with a free‐surface. Consistency between the discretized free‐surface equation and the discretized scalar transport equation is shown to be necessary for preservation of constants. In addition, this property is shown to hold for a general formulation of conservative schemes. A discrete maximum principle is proven for consistent upwind schemes. Various numerical examples in idealized and realistic test cases demonstrate the practical importance of the consistency with the discretization of the continuity equation. Copyright © 2002 John Wiley & Sons, Ltd.

Analysis of a class of quasi-monotone and conservative semi-Lagrangian advection schemes

We analyze in the $L^\infty-$ norm a class of semi-Lagrangian advective schemes introduced by the author and A. Staniforth in 1992 to improve the solution produced by numerical models for weather prediction and climate studies that use semi-Lagrangian advective schemes. The new quasi-monotone and conservative semi-Lagrangian schemes are $L^\infty-$ stable and converge optimally when the solution is sufficiently smooth.

A two‐dimensional zonally averaged transport model including convective motions and a new strategy for the numerical solution

A two‐dimensional zonally averaged transport model of the atmosphere up to 10 mbar (∼33 km) has been developed. The transport is described by an effective stream function and a diffusion tensor derived from a three‐dimensional general circulation model. The numerical solution of the transport equations has been given special attention, and a positive definite mass conservative advection scheme which produces small numerical diffusion has been modified for use in the model. In addition to the vertical transport described by the transport equations, a parameterization of the vertical mixing due to convective clouds and frontal circulation is included in the model. The model has been tested with different tracers which indicated a realistic representation of the atmospheric transport.

The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection

Modeling of highly advective transport is embarrassingly difficult, even in the superficially simple case of one-dimensional constant-velocity flow. In this paper, a fresh approach is taken, based on an explicit conservative control-volume formulation that makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ‘ULTIMATE’ conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three representative test profiles: a discontinuous step, an isolated sine-squared wave, and a semi-ellipse. The goal is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservative advection schemes of any order of accuracy. Second-order methods are shown to be unsatisfactory because of steepening and clipping typical of currently popular so-called ‘high resolution’ shock-capturing or TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. By using adaptive stencil expansion to introduce (in principle arbitrarily) higher order resolution locally in isolated regions of high curvature or high gradient, extremely accurate coarse-grid results can be obtained with very little additional cost above that of the base (third-order) scheme.

Stability, Accuracy, and Improvement of Crowley Advection Schemes

The Crowley conservative advection scheme is shown to give difficulties even for constant initial concentrations in complex flows and to have a spectral radius greater than 1 for at least one such flow. Modifications are proposed that eliminate these problems without destroying conservation.