A flux-form version of the conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid

A conservative semi-Lagrangian cell-integrated transport scheme (CSLAM) was recently introduced, which ensures global mass conservation and allows long timesteps, multi-tracer efficiency, and shape preservation through the use of reconstruction filtering. This method is fully two-dimensional so that it may be easily implemented on non-cartesian grids such as the cubed-sphere grid. We present a flux-form implementation, FF-CSLAM, which retains the advantages of CSLAM while also allowing the use of flux-limited monotonicity and positivity preservation and efficient tracer sub-cycling. The methods are equivalent in the absence of flux limiting or reconstruction filtering.FF-CSLAM was found to be third-order accurate when an appropriately smooth initial mass distribution and flow field (with at least a continuous second derivative) was used. This was true even when using highly deformational flows and when the distribution is advected over the singularities in the cubed sphere, the latter a consequence of the full two-dimensionality of the method. Flux-limited monotonicity preservation, which is only available in a flux-form method, was found to be both less diffusive and more efficient than the monotone reconstruction filtering available to CSLAM. Despite the additional overhead of computing fluxes compared to CSLAM’s cell integrations, the non-monotone FF-CSLAM was found to be at most only 40% slower than CSLAM for Courant numbers less than one, with greater overhead for successively larger Courant numbers.     

Finite-volume transport on various cubed-sphere grids

The performance of a multidimensional finite-volume transport scheme is evaluated on the cubed-sphere geometry. Advection tests with prescribed winds are used to evaluate a variety of cubed-sphere projections and grid modifications including the gnomonic and conformal mappings, as well as two numerically generated grids by an elliptic solver and spring dynamics. We explore the impact of grid non-orthogonality on advection tests over the corner singularities of the cubed-sphere grids, using some variations of the transport scheme, including the piecewise parabolic method with alternative monotonicity constraints. The advection tests revealed comparable or better accuracy to those of the original latitudinal–longitudinal grid implementation. It is found that slight deviations from orthogonality on the modified cubed-sphere (quasi-orthogonal) grids do not negatively impact the accuracy. In fact, the more uniform version of the quasi-orthogonal cubed-sphere grids provided better overall accuracy than the most orthogonal (and therefore, much less uniform) conformal grid. It is also shown that a simple non-orthogonal extension to the transport equation enables the use of the highly non-orthogonal and computationally more efficient gnomonic grid with acceptable accuracy.     

Vorticity–divergence mass-conserving semi-Lagrangian shallow-water model using the reduced grid on the sphere

The semi-Lagrangian semi-implicit shallow water model on the sphere using the reduced latitude–longitude grid is presented. The key feature of the model is the vorticity–divergence formulation on the unstaggered grid. The new algorithm for the reconstruction of wind components from vorticity and divergence is described. The mass-conservative version of the model is developed. The conservative cascade scheme (CCS) by Nair et al. is modified to provide a locally-conservative semi-Lagrangian advection algorithm for the reduced grid. Some numerical advection tests are carried out to demonstrate the accuracy of the CCS with the reduced grid. The CCS-based discretization for the continuity equation and finite-volume Helmholtz problem solver are introduced to guarantee the mass-conservation.The results for shallow water tests on the sphere are presented. The results for different versions of the model are compared. They are also compared with the results for the same tests available in literature. The impact of the reduced grid is analyzed. The mass-conservative version of the model using the reduced grid with up to 20% reduction of grid points number has approximately the same accuracy as its non-conservative counterpart implemented on the regular latitude–longitude grid.     

Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow

In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes [5], [25], which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems.     

A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations

A spectral element semi-Lagrangian (SESL) method for the shallow water equations on the sphere is presented. The sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals. The equations are written in Cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates. In a previous paper [Int. J. Numer. Methods Fluids 35 (2001) 869] we showed how to construct an explicit Eulerian spectral element (SE) model on the sphere; we now extend this work to a semi-Lagrangian formulation. The novelty of the Lagrangian formulation presented is that the high order SE basis functions are used as the interpolation functions for evaluating the values at the Lagrangian departure points. This makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers. The equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points. By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method [Comput. Methods Appl. Mech. Eng. 163 (1998) 193]. Results for eight test cases are presented to confirm the accuracy and stability of the method. These results show that SESL yields the same accuracy as an Eulerian spectral element semi-implicit (SESI) while allowing for time-steps 10 times as large and being up to 70% more efficient.     

A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows

The term ‘Convected Scheme’ (CS) refers to a family of algorithms, most usually applied to the solution of Boltzmann’s equation, which uses a method of characteristics in an integral form to project an initial cell forward to a group of final cells. As such the CS is a ‘forward-trajectory’ semi-Lagrangian scheme. For multi-dimensional simulations of neutral gas flows, the cell-centered version of this semi-Lagrangian (CCSL) scheme has advantages over other options due to its implementation simplicity, low memory requirements, and easier treatment of boundary conditions. The main drawback of the CCSL-CS to date has been its high numerical diffusion in physical space, because of the 2nd order remapping that takes place at the end of each time step. By means of a modified equation analysis, it is shown that a high order estimate of the remapping error can be obtained a priori, and a small correction to the final position of the cells can be applied upon remapping, in order to achieve full compensation of this error. The resulting scheme is 4th order accurate in space while retaining the desirable properties of the CS: it is conservative and positivity-preserving, and the overall algorithm complexity is not appreciably increased. Two monotone (i.e. non-oscillating) versions of the fourth order CCSL-CS are also presented: one uses a common flux-limiter approach; the other uses a non-polynomial reconstruction to evaluate the derivatives of the density function. The method is illustrated in simple one- and two-dimensional examples, and a fully 3D solution of the Boltzmann equation describing expansion of a gas into vacuum through a cylindrical tube.     

A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh

A conservative formulation of the Lorentz force is given here for magnetohydrodynamic (MHD) flows at a low magnetic Reynolds number with the current density calculated based on Ohm’s law and the electrical potential formula. This conservative formula shows that the total momentum contributed from the Lorentz force is conservative when the applied magnetic field is constant. For the case with a non-constant applied magnetic field, the Lorentz force has been divided into two parts: a strong globally conservative part and a weak locally conservative part.The conservative formula has been employed to develop a conservative scheme for the calculation of the Lorentz force on an unstructured collocated mesh. Only the current density fluxes on the cell faces, which are calculated using a consistent scheme with good conservation, are needed for the calculation of the Lorentz force. Meanwhile, a conservative interpolation technique is designed to get the current density at the cell center from the current density fluxes on the cell faces. This conservative interpolation can keep the current density at the cell center conservative, which can be used to calculate the Lorentz force at the cell center with good accuracy. The Lorentz force calculated from the conservative current at the cell center is equivalent to the Lorentz force from the conservative formula when the applied magnetic field is constant, which can conserve the total momentum. We will further prove that the simple interpolation scheme used in the Part I [M.-J. Ni, R. Munipalli, N.B. Morley, P.Y. Huang, M. Abdou, A current density conservative scheme for MHD flows at a low magnetic Reynolds number. Part I. On a rectangular collocated grid system, Journal of Computational Physics, in press, doi:10.1016/j.jcp.2007.07.025] of this series of papers is conservative on a rectangular grid and can keep the total momentum conservative in a rectangular grid.     

Vlasov-Poisson方程半拉格朗日守恒算法

利用三阶迎风插值多项式结合限制子方法,构造Vlasov-Poisson方程的半拉格朗日守恒型格式,可保持Vlasov-Poisson方程解的正性.计算朗道阻尼,双束不稳定性等典型问题,并与样条插值方法、UGKS方法进行比较,模拟结果表明半拉格朗日守恒性格式在Vlasov-Poisson方程求解中具有较高分辨率.     

A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: On a rectangular collocated grid system

A consistent, conservative and accurate scheme has been designed to calculate the current density and the Lorentz force by solving the electrical potential equation for magnetohydrodynamics (MHD) at low magnetic Reynolds numbers and high Hartmann numbers on a finite-volume structured collocated grid. In this collocated grid, velocity (u), pressure (p), and electrical potential (φ) are located in the grid center, while current fluxes are located on the cell faces. The calculation of current fluxes on the cell faces is conducted using a conservative scheme, which is consistent with the discretization scheme for the solution of electrical potential Poisson equation. A conservative interpolation is used to get the current density at the cell center, which is used to conduct the calculation of Lorentz force at the cell center for momentum equations. We will show that both “conservative” and “consistent” are important properties of the scheme to get an accurate result for high Hartmann number MHD flows with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers of Hunt’s conductive walls and Shercliff’s insulated walls. A general second-order projection method has been developed for the incompressible Navier–Stokes equations with the Lorentz force included. This projection method can accurately balance the pressure term and the Lorentz force for a fully developed core flow. This method can also simplify the pressure boundary conditions for MHD flows.     

Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers

We present an immersed-boundary algorithm for incompressible flows with complex boundaries, suitable for Cartesian or curvilinear grid system. The key stages of any immersed-boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary (a line in 2D, a surface in 3D), and the spreading of a force field from the immersed boundary to the neighboring mesh points, to enforce the desired boundary conditions on the immersed-boundary points. We propose a technique that uses the Reproducing Kernel Particle Method [W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Fluids 20(8) (1995) 1081–1106] for the interpolation and spreading. Unlike other methods presented in the literature, the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved, independent of the grid topology (uniform or non-uniform, Cartesian or curvilinear). The technique is easy to implement, and is able to maintain the order of the original underlying spatial discretization. Applications to two- and three-dimensional flows in Cartesian and non-Cartesian grid system, with uniform and non-uniform meshes are presented.     

NS方程的各向异性笛卡尔网格法研究

将近年发展起来的用于Euler方程求解的具有局部均匀网格总体非结构特性的笛卡尔网格法推广到NS方程的求解。为了与流场的各向异性相适应、减少网格点数量,提出了一种各向异性网格加密法。另外还研究了分级笛卡尔网格对内点格式稳定性的影响和插值固体边界条件的稳定性。数值结果表明各向异性笛卡尔网格法相对于传统的各向同性网格方法能大量节省网格点数量而且与后者具有同样的精度。     

An FDTD scheme on a face-centered-cubic (FCC) grid for the solution of the wave equation

A method is proposed to improve the numerical dispersion characteristics for simulations of the scalar wave equation in 3D using the FDTD method. The improvements are realized by choosing a face-centered-cubic (FCC) grid instead of the typical Cartesian (Yee) grid, which exhibits non-physical distortions of the wavefront due to the FD stencil. FCC grids are the logical extension of hexagonal grids in 2D, and have been shown previously to provide optimal sampling of space based on close packing of spheres (highest density). The difference equations are developed for the wave equation on this alternative grid, and the dispersion relationship and stability for grids of equal and non-equal aspect ratios are derived. A comparison is made between FCC and Cartesian formulations, based upon having an equal volume density of gridpoints in each method (i.e. the computational storage requirements of each method would be the same for the same simulated space). The comparison shows that the FCC grid exhibits a much more isotropic dispersion relation than the Cartesian grid of equivalent density. Furthermore, for an equivalent density, the FCC method has a more relaxed stability criterion by a factor of approximately 1.35, resulting in a further reduction in computational resources.     

Bi2Te3: implications of the rhombohedral k-space texture on the evaluation of the in-plane/out-of-plane conductivity anisotropy

Different computational scheme for calculating surface integrals in anisotropic Brillouin zones are compared. An example of the transport distribution function (plasma frequency) of the thermoelectric material Bi(2)Te(3) near the band edges is discussed. The layered structure of the material together with the rhombohedral symmetry causes a strong anisotropy of the transport distribution function for the directions in the basal plane (in-plane) and perpendicular to the basal plane (out-of-plane). It is shown that a thorough reciprocal space integration is necessary to reproduce the in-plane/out-of-plane anisotropy. A quantitative comparison can be made at the band edges, where the transport anisotropy is given in terms of the anisotropic mass tensor.     

Completely Conservative and Oscillationless Semi-Lagrangian Schemes for Advection Transportation

In this paper, we present a new type of semi-Lagrangian scheme for advection transportation equation. The interpolation function is based on a cubic polynomial and is constructed under the constraints of conservation of cell-integrated average and the slope modification. The cell-integrated average is defined via the spatial integration of the interpolation function over a single grid cell and is advanced using a flux form. Nonoscillatory interpolation is constructed by choosing proper approximation to the cell-center values of the first derivative of the interpolation function, which appears to be a free parameter in the present formulation. The resulting scheme is exactly conservative regarding the cell average of the advected quantity and does not produce any spurious oscillation. Oscillationless solutions to linear transportation problems were obtained. Incorporated with an entropy-enforcing numerical flux, the presented schemes can accurately compute shocks and sonic rarefaction waves when applied to nonlinear problems.     

The HyperCASL algorithm: A new approach to the numerical simulation of geophysical flows

We describe a major extension to the Contour-Advective Semi-Lagrangian (CASL) algorithm [D.G. Dritschel, M.H.P. Ambaum, A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields, Quart. J. Roy. Meteorol. Soc. 123 (1997) 1097–1130; D.G. Dritschel, M.H.P. Ambaum, The diabatic contour advective semi-Lagrangian algorithm, Mon. Weather Rev. 134 (9) (2006) 2503–2514]. The extension, called ‘HyperCASL’ (HCASL), uses Lagrangian advection of material potential vorticity contours like CASL, but a Vortex-In-Cell (VIC) method for the treatment of diabatic forcing or damping. In this way, HyperCASL is fully Lagrangian regarding advection. A grid is used as in CASL to deal with ‘inversion’ (computing the velocity field from the potential vorticity field).     

An all-scale anelastic model for geophysical flows: dynamic grid deformation

We have developed an adaptive grid-refinement approach for simulating geophysical flows on scales from micro to planetary. Our model is nonoscillatory forward-in-time (NFT), nonhydrostatic, and anelastic. The major focus in this effort to date has been the design of a generalized mathematical framework for the implementation of deformable coordinates and its efficient numerical coding in a generic Eulerian/semi-Lagrangian NFT format. The key prerequisite of the adaptive grid is a time-dependent coordinate transformation, implemented rigorously throughout the governing equations of the model. The transformation enables mesh refinement indirectly via dynamic change of the metric coefficients, while retaining advantages of Cartesian mesh calculations (speed, low memory requirements, and accuracy) conducted fully in the computational domain. Diverse test results presented in this paper – simulations of a traveling stratospheric inertio-gravity-wave packet (with numerically advected dense-mesh region) and an idealized climate of the Earth (with analytically prescribed adaptive transformations) – demonstrate the potential and the efficacy of the new deformable grid model for tracing targeted flow features and dynamically adjusting to prescribed undulations of model boundaries.     

High-order finite-volume methods for the shallow-water equations on the sphere

This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM⁺-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge–Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM⁺-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.     

High resolution dynamic cardiac MRI using partial separability of spatiotemporal signals with a novel sampling scheme

The partial separability (PS) of spatiotemporal signals has been exploited to accelerate dynamic cardiac MRI by sampling two datasets (training and imaging datasets) without breath-holding or ECG triggering. According to the theory of partially separable functions, the wider the range of spatial frequency components covered by the training dataset, the more accurate the temporal constraint imposed by the PS model. Therefore, it is necessary to develop a new sampling scheme for the PS model in order to cover a wider range of spatial frequency components. In this paper, we propose the use of radial sampling trajectories for collecting the training dataset and Cartesian sampling trajectories for collecting the imaging dataset. In vivo high resolution cardiac MRI experiments demonstrate that the proposed data sampling scheme can significantly improve the image quality. The image quality using the PS model with the proposed sampling scheme is comparable to that of a commercial method using retrospective cardiac gating and breath-holding. The success of this study demonstrates great potential for high-quality, high resolution dynamic cardiac MRI without ECG gating or breath-holding through use of the PS model and the novel data sampling scheme.     

A conservative high order semi-Lagrangian WENO method for the Vlasov equation

In this paper, we propose a novel Vlasov solver based on a semi-Lagrangian method which combines Strang splitting in time with high order WENO (weighted essentially non-oscillatory) reconstruction in space. A key insight in this work is that the spatial interpolation matrices, used in the reconstruction process of a semi-Lagrangian approach to linear hyperbolic equations, can be factored into right and left flux matrices. It is the factoring of the interpolation matrices which makes it possible to apply the WENO methodology in the reconstruction used in the semi-Lagrangian update. The spatial WENO reconstruction developed for this method is conservative and updates point values of the solution. While the third, fifth, seventh and ninth order reconstructions are presented in this paper, the scheme can be extended to arbitrarily high order. WENO reconstruction is able to achieve high order accuracy in smooth parts of the solution while being able to capture sharp interfaces without introducing oscillations. Moreover, the CFL time step restriction of a regular finite difference or finite volume WENO scheme is removed in a semi-Lagrangian framework, allowing for a cheaper and more flexible numerical realization. The quality of the proposed method is demonstrated by applying the approach to basic test problems, such as linear advection and rigid body rotation, and to classical plasma problems, such as Landau damping and the two-stream instability. Even though the method is only second order accurate in time, our numerical results suggest the use of high order reconstruction is advantageous when considering the Vlasov–Poisson system.     

A class of Cartesian grid embedded boundary algorithms for incompressible flow with time-varying complex geometries

We present a class of numerical algorithms for simulating viscous fluid problems of incompressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing immersed boundary methods: the proposed algorithms calculate and apply the force density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid-body surface. The principle of the embedded boundary algorithm allows us to solve the fluid equations on a Cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure. The proposed algorithms use the MAC (marker and cell) algorithm to solve the incompressible Navier-Stokes equations. Unlike projection methods, the MAC scheme incorporates the gradient of the force density in solving the pressure Poisson equation, so that the dipole force, due to the jump of pressure across the solid-fluid interface, is directly balanced by the gradient of the force density. We validate the proposed algorithms via the classical benchmark problem of flow past a cylinder. Our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid-fluid interface. Finally, we consider the problem of a cylinder moving between two parallel plane walls. Numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions. We show that the two solutions are in good agreement.