Numerical representation of geostrophic modes on arbitrarily structured C-grids

A C-grid staggering, in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces (or edges in two dimensions) is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes. However, the discretization of the Coriolis terms is non-trivial. For constant Coriolis parameter, the linearized shallow water equations support geostrophic modes: stationary solutions in geostrophic balance. A naive discretization of the Coriolis terms can cause geostrophic modes to become non-stationary, causing unphysical behaviour of numerical solutions. Recent work has shown how to discretize the Coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the Coriolis terms remain energy conserving. In this paper this result is extended to arbitrarily structured C-grids. An explicit formula is given for constructing an appropriate discretization of the Coriolis terms. The general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal C-grid and the spherical longitude–latitude C-grid. Numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal–pentagonal and triangular geodesic C-grids on the sphere.     

水斗非定常自由水膜流三维贴体数值模拟

本研究采用水斗三维非正交贴体坐标系进行了非定常自由水膜流动的数值解析。对不规则水斗内表面采用三维非正交贴体坐标系下离散点进行拟合,推导了曲面离散点的法向矢量和曲面微元面高斯曲率、平均曲率等几何特征量的计算公式,进而导出流体粒子在运动方向上曲率计算式。在水斗三维贴体坐标系中,还推导了流体粒子在水斗曲面上的运动控制方程。最后对某水轮机水斗内表面非定常自由水膜流进行了数值模拟,得到其非定常水膜流态分布。     

Divergence formulation of source term

In this paper, we propose to write a source term in the divergence form. A conservation law with a source term can then be written as a single divergence form. We demonstrate that it enables to discretize both the conservation law and the source term in the same framework, and thus greatly simplifies the construction of numerical schemes. To illustrate the advantage of the divergence formulation, we apply the new formulation to construct a uniformly third-order accurate edge-based finite-volume scheme for conservation laws with a source term. Third-order accuracy is demonstrated for regular and irregular triangular grids for the linear advection and Burgers’ equations with a source term.     

A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal–toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.     

Effects of orientation and shape of holes on the band gaps in water waves over periodically drilled bottoms

The complete band gaps (CBGs) of shallow water waves propagating over bottoms with periodically drilled holes are investigated numerically by the plane wave expansion method. Four different patterns are considered, containing triangular, square, hexagonal and circular cross-sectioned holes arranged into triangular lattices. Results show that the width of CBGs can be changed by adjusting the orientation of noncircular holes and the effect of hole shape on the width of the maximal CBGs is discussed.     

不规则对接网格的分区求解计算

采用一种保持通量守恒的不规则对接网格分区求解中交界面耦合条件的计算方法, 结合有限体积法求解了Euler 方程, 无粘通量取用Van Leer 分裂格式, 构造了一种限制器以实现格式的二阶精度和TVD 性质, 并给出数值算例。     

Vector Arithmetic in the Triangular Grid

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.     

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid on slope in the shallow water approximation

Magnetohydrodynamic equations for a heavy fluid over an arbitrary surface are studied in the shallow water approximation. While solutions to the shallow water equations for a neutral fluid are well known, shallow water magnetohydrodynamic (SMHD) equations over a nonflat boundary have an additional dependence on the magnetic field, and the number of equations in the magnetic case exceeds that in the neutral case. As a consequence, the number of Riemann invariants defining SMHD equations is also greater. The classical simple wave solutions do not exist for hyperbolic SMHD equations over an arbitrary surface due to the appearance of a source term. In this paper, we suggest a more general definition of simple wave solutions that reduce to the classical ones in the case of zero source term. We show that simple wave solutions exist only for underlying surfaces that are slopes of constant inclination. All self-similar discontinuous and continuous solutions are found. Exact explicit solutions of the initial discontinuity decay problem over a slope are found. It is shown that the initial discontinuity decay solution is represented by one of four possible wave configurations. For each configuration, the necessary and sufficient conditions for its realization are found. The change of dependent and independent variables transforming the initial equations over a slope to those over a flat plane is found.     

Variational Equations and Symmetries in the Lagrangian Formalism; Arbitrary Vector Fields

We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the generalized Helmholtz equations (sometimes called the Anderson-Duchamp-Krupka equations). For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance.     

求解二维Euler方程有限单元边插值的降维重构算法

数值求解二维Euler方程的有限体积法（如k-exact，WENO重构、紧致重构等），无一例外地要进行耗时的网格单元上的二维重构.然而这些二维重构最后仅用于确定网格单元边界上高斯积分点处的解值，单元上二维重构似乎并非必需的.因此，文章提出用网格边上的一维重构来取代有限体积法中网格单元上的二维重构，分别在一致矩形网格和非结构三角形网格上发展了基于网格边重构的求解二维Euler方程的新方法，称为降维重构算法.数值算例表明该算法可以计算有强激波的无黏流动问题，且有较高的计算效率.      

一种非结构/结构多层混合网格方法及其应用

对于粘性绕流的数值模拟,在自适应直角网格基础上,结合三角形非结构网格和结构化网格,利用其各自的优势和特点,提出一种生成混合杂交网格的思路和方法.在物面附近生成适合粘性流计算的大长宽比结构化网格,在远场分布自适应直角网格,快速离散计算空间.对于复杂的多体问题,采用三角形网格来连接各体网格,并运用网格合并的方法,保证各网格之间的光滑过渡与连接,提高网格质量.针对一些二维、三维外形的绕流问题,在上述网格基础上,采用B-L代数湍流模型和中心有限体积法,完成Navier-Stokes和Euler方程数值模拟的对比计算,结果表明网格生成和流场计算是正确的.     

Weak solutions for partial differential equations with source terms: Application to the shallow water equations

Weak solutions of problems with m equations with source terms are proposed using an augmented Riemann solver defined by m + 1 states instead of increasing the number of involved equations. These weak solutions use propagating jump discontinuities connecting the m + 1 states to approximate the Riemann solution. The average of the propagated waves in the computational cell leads to a reinterpretation of the Roe’s approach and in the upwind treatment of the source term of Vázquez-Cendón. It is derived that the numerical scheme can not be formulated evaluating the physical flux function at the position of the initial discontinuities, as usually done in the homogeneous case. Positivity requirements over the values of the intermediate states are the only way to control the global stability of the method. Also it is shown that the definition of well-balanced equilibrium in trivial cases is not sufficient to provide correct results: it is necessary to provide discrete evaluations of the source term that ensure energy dissipating solutions when demanded. The one and two dimensional shallow water equations with source terms due to the bottom topography and friction are presented as case study. The stability region is shown to differ from the one defined for the case without source terms, and it can be derived that the appearance of negative values of the thickness of the water layer in the proximity of the wet/dry front is a particular case, of the wet/wet fronts. The consequence is a severe reduction in the magnitude of the allowable time step size if compared with the one obtained for the homogeneous case. Starting from this result, 1D and 2D numerical schemes are developed for both quadrilateral and triangular grids, enforcing conservation and positivity over the solution, allowing computationally efficient simulations by means of a reconstruction technique for the inner states of the weak solution that allows a recovery of the time step size.     

The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.     

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.     

A global shallow water model using high order multi-moment constrained finite volume method and icosahedral grid

A novel accurate numerical model for shallow water equations on sphere have been developed by implementing the high order multi-moment constrained finite volume (MCV) method on the icosahedral geodesic grid. High order reconstructions are conducted cell-wisely by making use of the point values as the unknowns distributed within each triangular cell element. The time evolution equations to update the unknowns are derived from a set of constrained conditions for two types of moments, i.e. the point values on the cell boundary edges and the cell-integrated average. The numerical conservation is rigorously guaranteed. In the present model, all unknowns or computational variables are point values and no numerical quadrature is involved, which particularly benefits the computational accuracy and efficiency in handling the spherical geometry, such as coordinate transformation and curved surface.Numerical formulations of third and fourth order accuracy are presented in detail. The proposed numerical model has been validated by widely used benchmark tests and competitive results are obtained. The present numerical framework provides a promising and practical base for further development of atmospheric and oceanic general circulation models.     

Computational study of some nonlinear shallow water equations

The shallow water equations have wide applications in ocean, atmospheric modeling and hydraulic engineering, also they can be used to model flows in rivers and coastal areas. In this article we obtained exact solutions of three equations of shallow water by using $\frac{{G'}} {G} $ -expansion method. Hyperbolic and triangular periodic solutions can be obtained from the $\frac{{G'}} {G} $ -expansion method.     

Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed-sphere

High resolution and scalable parallel algorithms for the shallow water equations on the sphere are very important for modeling the global climate. In this paper, we introduce and study some highly scalable multilevel domain decomposition methods for the fully implicit solution of the nonlinear shallow water equations discretized with a second-order well-balanced finite volume method on the cubed-sphere. With the fully implicit approach, the time step size is no longer limited by the stability condition, and with the multilevel preconditioners, good scalabilities are obtained on computers with a large number of processors. The investigation focuses on the use of semismooth inexact Newton method for the case with nonsmooth topography and the use of two- and three-level overlapping Schwarz methods with different order of discretizations for the preconditioning of the Jacobian systems. We test the proposed algorithm for several benchmark cases and show numerically that this approach converges well with smooth and nonsmooth bottom topography, and scales perfectly in terms of the strong scalability and reasonably well in terms of the weak scalability on machines with thousands and tens of thousands of processors.     

Numerical wave propagation on non-uniform one-dimensional staggered grids

The wave propagation behaviour of centered difference schemes on one-dimensional non-uniform staggered grids is investigated. Previous results for the linear advection equation are extended to the case of the shallow water equations on staggered grids. For waves of a given frequency, the wave field is decomposed into right- and left-propagating components, and a wave energy conservation law is derived in terms of these components. For slowly varying grids, separate evolution equations for the right- and left-propagating components are derived, leading to the result that there is asymptotically no reflection in the limit of a slowly varying grid, provided that waves of that frequency are resolvable. However, there will be reflection from any location at which the wave group velocity goes to zero. The possibility for wave energy to tunnel through a narrow region of the grid too coarse for propagation is noted. Grids with an abrupt jump in resolution are also investigated. It is possible to tailor the scheme at the jump to minimize spurious wave reflection over a range of frequencies provided the waves are resolvable on both sides of the jump. However, it does not appear possible to avoid complete reflection, except by introducing extra dissipation terms, if the waves are not resolvable on one side of the jump. An example is presented of a second-order accurate scheme that spontaneously radiates waves from the resolution jump.