Chungang Chen

Shallow water model on cubed-sphere by multi-moment finite volume method

A global numerical model for shallow water flows on the cubed-sphere grid is proposed in this paper. The model is constructed by using the constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM). Two kinds of moments, i.e. the point value (PV) and the volume-integrated average (VIA) are defined and independently updated in the present model by different numerical formulations. The Lax-Friedrichs upwind splitting is used to update the PV moment in terms of a derivative Riemann problem, and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the VIA moment. The cubed-sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry. Highly localized reconstruction in CIP/MM FVM is well suited for the cubed-sphere grid, especially in dealing with the discontinuity in the coordinates between different patches. The mass conservation is completely achieved over the whole globe. The numerical model has been verified by Williamsons standard test set for shallow water equation model on sphere. The results reveal that the present model is competitive to most existing ones.

A Multimoment Constrained Finite-Volume Model for Nonhydrostatic Atmospheric Dynamics

AbstractThe two-dimensional nonhydrostatic compressible dynamical core for the atmosphere has been developed by using a new nodal-type high-order conservative method, the so-called multimoment constrained finite-volume (MCV) method. Different from the conventional finite-volume method, the predicted variables (unknowns) in an MCV scheme are the values at the solution points distributed within each mesh cell. The time evolution equations to update the unknown point values are derived from a set of constraint conditions based on the multimoment concept, where the constraint on the volume-integrated average (VIA) for each mesh cell is cast into a flux form and thus guarantees rigorously the numerical conservation. Two important features make the MCV method particularly attractive as an accurate and practical numerical framework for atmospheric and oceanic modeling. 1) The predicted variables are the nodal values at the solution points that can be flexibly located within a mesh cell (equidistant solution poin...

An Adaptive Multimoment Global Model on a Cubed Sphere

An adaptive global shallow-water model is proposed on cubed-sphere grid using the multimoment finite volume scheme and the Berger-Oliger adaptive mesh refinement (AMR) algorithm that was originally designed for a Cartesian grid. On each patch of the cubed-sphere grid, the curvilinear coordinates are constructed in a way that the Berger-Oliger algorithm can be applied directly. Moreover, an algorithm to transfer data across neighboring patches is proposed to establish a practical integrated framework for global AMR computation on the cubed-sphere grid. The multimoment finite volume scheme is adopted as the fluid solver and is essentially beneficial to the implementation of AMR on the cubed-sphere grid. The multimoment interpolation based on both volume-integrated average (VIA) and point value (PV) provides the compact reconstruction that makes the present scheme very attractive not only in dealing with the artificial boundaries between different patches but also in the coarse fine interpolations required in the AMR computations. The single-cell-based reconstruction avoids involving more than two nesting levels during interpolations. The reconstruction profile of constrained interpolation profile-conservative semi-Lagrangian scheme with third-order polynomial function (CIP-CSL3) is adopted where the slope parameter provides a flexible and convenient switching to get the desired numerical properties, such as high-order (fourth order) accuracy, monotonicity, and positive definiteness. Numerical experiments with typical benchmark tests for both advection equation and shallow-water equations are carried out to evaluate the proposed model. The numerical errors and the corresponding CPU times of numerical experiments on uniform and adaptive meshes verify the performance of the proposed model. Compared to the uniformly refined grid, the AMR technique is able to achieve the similar numerical accuracy with less computational cost.

Global shallow water models based on multi-moment constrained finite volume method and three quasi-uniform spherical grids

This is a review article to present several accurate and computationally efficient global models for shallow water equations recently developed under a general numerical framework, the multi-moment constrained finite volume (MCV) method. The multi-moment constrained finite volume method defines the unknowns (prognostic variables) as the point values at the solution points located over each mesh element. The time evolution equations to update these unknowns are derived through the constraint conditions on different moments, e.g. the point value (PV) and the volume-integrated average (VIA). Rigorous numerical conservation is guaranteed by the constraint on the VIA moment through a finite volume formulation of flux form. The resulted numerical schemes are very simple, efficient and easy to implement for both structured and unstructured grids. We have implemented the MCV method to three major spherical grids, Yin-Yang overset grid, cubed-sphere grid and geodesic icosahedral grid, which have overall quasi-uniform grid spacings and are highly popular in the community of global modeling for atmospheric and oceanic dynamics. In this paper, we present the global shallow water models based on these three spherical grids and the third-order MCV scheme. We evaluate and compare the models by widely used benchmark tests, which show the third-order convergence rate for all models, and the numerical results are competitive to other exiting models. Using MCV method as a numerical formulation is well-balanced between solution quality and computational simplicity, the proposed models provide accurate and practical bases for developing dynamic core of general circulation models on different spherical grids.

A Global Multimoment Constrained Finite-Volume Scheme for Advection Transport on the Hexagonal Geodesic Grid

A third-order numerical model is developed for global advection transport computation. The multimoment constrained finite-volume scheme has been implemented to the hexagonal geodesic grid for spherical geometry. Two kinds of moments (i.e., point value and volume-integrated average) are used as the constraint conditions to derive the time evolution equations to update the computational variables, which are the values defined at the specified points over each mesh element in the present model. The numerical model has rigorous numerical conservation and third-order accuracy. One of the major merits of the present method is that it does not explicitly involve numerical quadrature, which leads to great convenience in accurately computing curved geometry and source terms. The present paper provides an accurate and practical formulation for advection calculation in the hexagonal-type geodesic grid.

An Accurate Multimoment Constrained Finite Volume Transport Model on Yin-Yang Grids

A global transport model is proposed in which a multimoment constrained finite volume (MCV) scheme is applied to a Yin-Yang overset grid. The MCV scheme defines 16 degrees of freedom (DOFs) within each element to build a 2D cubic reconstruction polynomial. The time evolution equations for DOFs are derived from constraint conditions on moments of line-integrated averages (LIA), point values (PV), and values of first-order derivatives (DV). The Yin-Yang grid eliminates polar singularities and results in a quasi-uniform mesh. A limiting projection is designed to remove nonphysical oscillations around discontinuities. Our model was tested against widely used benchmarks; the competitive results reveal that the model is accurate and promising for developing general circulation models.

Fourth order transport model on Yin-Yang grid by multi-moment constrained finite volume scheme

Abstract A fourth order transport model is proposed for global computation with the application of multi-moment constrained finite volume (MCV) scheme and Yin-Yang overset grid. Using multi-moment concept, local degrees of freedom (DOFs) are point-wisely defined within each mesh element to build a cubic spatial reconstruction. The updating formulations for local DOFs are derived by adopting multi moments as constraint conditions, including volume-integrated average (VIA), point value (PV) and first order derivative value (DV). Using Yin-Yang grid eliminates the polar singularities and results in a quasi-uniform mesh over the whole globe. Each component of Yin-Yang grid is a part of the LAT-LON grid, an orthogonal structured grid, where the MCV formulations designed for Cartesian grid can be applied straightforwardly to develop the high order numerical schemes. Proposed MCV model is checked by widely used benchmark tests. The numerical results show that the present model has fourth order accuracy and is competitive to most existing ones.

Large scale numerical simulations for multi-phase fluid dynamics with moving interfaces

This short communication presents our recent studies to implement numerical simulations for multi-phase flows on top-ranked supercomputer systems with distributed memory architecture. The numerical model is designed so as to make full use of the capacity of the hardware. Satisfactory scalability in terms of both the parallel speed-up rate and the size of the problem has been obtained on two high rank systems with massively parallel processors, the Earth Simulator (Earth simulator research center, Yokohama Kanagawa, Japan) and the TSUBAME (Tokyo Institute of Technology, Tokyo, Japan) supercomputers.

A non-oscillatory multi-moment finite volume scheme with boundary gradient switching

In this work we propose a new formulation for high-order multi-moment constrained finite volume (MCV) method. In the one-dimensional building-block scheme, three local degrees of freedom (DOFs) are equidistantly defined within a grid cell. Two candidate polynomials for spatial reconstruction of third-order are built by adopting one additional constraint condition from the adjacent cells, i.e. the DOF at middle point of left or right neighbour. A boundary gradient switching (BGS) algorithm based on the variation-minimization principle is devised to determine the spatial reconstruction from the two candidates, so as to remove the spurious oscillations around the discontinuities. The resulted non-oscillatory MCV3-BGS scheme is of fourth-order accuracy and completely free of case-dependent ad hoc parameters. The widely used benchmark tests of one- and two-dimensional scalar and Euler hyperbolic conservation laws are solved to verify the performance of the proposed scheme in this paper. The MCV3-BGS scheme is very promising for the practical applications due to its accuracy, non-oscillatory feature and algorithmic simplicity.

An Efficient RIGID Algorithm and Its Application to the Simulation of Particle Transport in Porous Medium

RIGID algorithm was recently proposed to identify the contact state between spherical particles and arbitrary-shaped walls, demonstrating significantly improved robustness, accuracy and efficiency compared to existing methods. It is an important module when coupling computational fluid dynamics with discrete element model to simulate particle transport in porous media. The procedure to identify particle and surface contact state is usually time-consuming and takes a large part of the CPU time for discrete element simulations of dense particle flow in complex geometries, especially in cases with a large number of particle–wall collisions (e.g. particle transport in porous media). This paper presents a new version of RIGID algorithm, namely ERIGID, which further improves the efficiency of the original algorithm through a number of new strategies including the recursive algorithm for particle-face pair selection, angle-testing algorithm for determining particle-face relations and the smallest index filter for fast rejection and storage of time invariant. Several specially designed numerical experiments have been carried out to test the performance of ERIGID and verify the effectiveness of these strategies. Finally, the improved algorithm is used to simulate particle transport in a rock treated as a porous medium. Our numerical results reveal several important flow phenomena and the primary reason for particle trapping inside the rock.