Multi-moment Constrained Finite Volume Research Articles

Fourth-Order Conservative Transport on Overset Grids Using Multi-Moment Constrained Finite Volume Scheme for Oceanic Modeling

With an increase in model resolution, compact high-order numerical advection scheme can improve its effectiveness and competitiveness in oceanic modeling due to its high accuracy and scalability on massive-processor computers. To provide high-quality numerical ocean simulation on overset grids, we tried a novel formulation of the fourth-order multi-moment constrained finite volume scheme to simulate continuous and discontinuous problems in the Cartesian coordinate. Utilizing some degrees of freedom over each cell and derivatives at the cell center, we obtained a two-dimensional (2D) cubic polynomial from which point values on the extended overlap can achieve fourth-order accuracy. However, this interpolation causes a lack of conservation because the flux between the regions are no longer equal; thus, a flux correction is implemented to ensure conservation. A couple of numerical experiments are presented to evaluate the numerical scheme, which confirms its approximately fourth-order accuracy in conservative transportation on overset grid. The test cases reveal that the scheme is effective to suppress numerical oscillation in discontinuous problems, which may be powerful for salinity advection computing with a sharp gradient.

A 3D Nonhydrostatic Compressible Atmospheric Dynamic Core by Multi-moment Constrained Finite Volume Method

A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrain-following grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.

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.

A multi-moment constrained finite volume method on arbitrary unstructured grids for incompressible flows

We proposed a multi-moment constrained finite volume method which can simulate incompressible flows of high Reynolds number in complex geometries. Following the underlying idea of the volume-average/point-value multi-moment (VPM) method (Xie et al. (2014) [71]), this formulation is developed on arbitrary unstructured hybrid grids by employing the point values (PV) at both cell vertex and barycenter as the prognostic variables. The cell center value is updated via an evolution equation derived from a constraint condition of finite volume form, which ensures the rigorous numerical conservativeness. Novel numerical formulations based on the local PVs over compact stencil are proposed to enhance the accuracy, robustness and efficiency of computations on unstructured meshes of hybrid and arbitrary elements. Numerical experiments demonstrate that the present numerical model has nearly 3-order convergence rate with numerical errors much smaller than the VPM method. The numerical dissipation has been significantly suppressed, which facilitates numerical simulations of high Reynolds number flows in complex geometries.

An MCV Nonhydrostatic Atmospheric Model with Height-Based Terrain following Coordinate: Tests of Waves over Steep Mountains

A nonhydrostatic atmospheric model was tested with the mountain waves over various bell-shaped mountains. The model is recently proposed by using the MCV (multimoment constrained finite volume) schemes with the height-based terrain following coordinate representing the topography. As discussed in our previous work, the model has some appealing features for atmospheric modeling and can be expected as a practical framework of the dynamic cores, which well balances the numerical accuracy and algorithmic complexity. The flows over the mountains of various half widths and heights were simulated with the model. The semianalytic solutions to the mountain waves through the linear theory are used to check the performance of the MCV model. It is revealed that the present model can accurately reproduce various mountain waves including those generated by the mountains with very steep inclination and is very promising for numerically simulating atmospheric flows over complex terrains.

A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

AbstractThis paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

A high‐order multi‐moment constrained finite‐volume global shallow‐water model on the Yin‐Yang grid

A high‐order global shallow‐water model on a Yin‐Yang grid has been developed by using the multi‐moment constrained finite‐volume (MCV) method. Unlike the traditional finite‐volume method, more degrees of freedom (DOFs)–which are the values at the solution points within each mesh element–are defined and updated in time. The time evolution equations for these point values (PVs) are derived from a set of constraint conditions in terms of the so‐called multi‐moment quantities, such as the PV, the volume‐integrated average (VIA) and derivative. Different moments use different forms of equations which are all consistent with the shallow‐water equations, among which the VIA moment is computed from a finite‐volume formulation of flux form that guarantees rigorous numerical conservation. A fourth‐order formulation is devised with the third‐order reconstruction built over each element using the DOFs locally available. A simple and orthogonal overset grid, the Yin‐Yang grid, is used to represent the spherical geometry with quasi‐uniform grid spacing. The resulting global shallow‐water model is attractive in algorithmic simplicity and computational efficiency. The model has been validated by widely used benchmark tests. The numerical results of the present model are competitive with most existing advanced models.

Arbitrarily High-Order-Accurate, Hermite WENO Limited, Boundary-Averaged Multi-Moment Constrained Finite-Volume (BA-MCV) Schemes for 1-D Transport

Abstract This study introduces the Boundary Averaged Multi-moment Constrained finite-Volume (BA- MCV) scheme for 1-D transport with Hermite Weighted Essentially Non-Oscillatory (HWENO) limiting using the ADER Differential Transform (ADER-DT) time discretization. The BA- MCV scheme evolves a cell average using a Finite-Volume (FV) scheme, and it adds further constraints as point wise derivatives of the state at cell boundaries, which are evolved in strong form using PDE derivatives. The resulting scheme maintains a Maximum Stable CFL (MSCFL) value of one no matter how high-order the scheme is. Also, parallel communication requirements are very low and will be described. h-refinement experiments demonstrate proper convergence order, and p-refinement experiments demonstrate expected exponential convergence. The over- all ADER-DT + BA-MCV + HWENO scheme is a scalable and larger time step alternative to Galerkin methods for multi-moment fluid simulation in climate and weather applications.

Developing A Large Time Step, Robust, and Low Communication Multi-Moment PDE Integration Scheme for Exascale Applications

The Boundary Averaged Multi-moment Constrained finite-Volume (BA-MCV) method is de- rived, explained, and evaluated for 1-D transport to assess accuracy, maximum stable time step (MSTS), oscillations for discontinuous data, and parallel communication burden. The BA-MCV scheme is altered from the original MCV scheme to compute the updates of point wise cell boundary derivatives entirely locally. Then it is altered such that boundary moments are replaced with the interface upwind value. The scheme is stable at a maximum stable CFL (MSCFL) value of one no matter how high-order the scheme is, giving significantly larger time steps than Galerkin methods, for which the MSCFL decreases nearly quadratically with in- creasing order. The BA-MCV method is compared against a SE method at varying order, both using the ADER-DT time discretization. BA-MCV error for a sine wave was comparable to the same order of accuracy for a SE method. The resulting large time step, multi-moment, low communication scheme is of great interest for exascale architectures.

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.

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.

A global shallow‐water model on an icosahedral–hexagonal grid by a multi‐moment constrained finite‐volume scheme

AbstractA third‐order global shallow‐water model was developed on an icosahedral–hexagonal grid with the application of the multi‐moment constrained finite‐volume (MCV) method. Using the hexagonal grid, the sphere is divided into quasi‐uniform elements and free of polar problems. The MCV model defines seven degrees of freedom (DOFs) as the prognostic variables which are the point values at the six vertices and the centre for each hexagonal element to construct a third‐order scheme. The time evolution equations to update the DOFs are derived through the constraint conditions on different moments, i.e. the point value and the volume‐integrated average (VIA) moments. Rigorous conservation is guaranteed by the constraint on the VIA through a flux form. The MCV formulation is very simple and easy to implement. We evaluated the model with benchmark tests, and the competitive results reveal the proposed model to be an accurate and practical framework for developing general circulation models.

A Multimoment Constrained Finite-Volume Model for Nonhydrostatic Atmospheric Dynamics

Abstract The 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 points are used in the present model). It is computationally efficient and provides great convenience in dealing with complex geometry and source terms. 2) High-order and physically consistent formulations can be built by choosing proper constraints in view of not only numerical accuracy and efficiency but also underlying physics. In this paper the authors present a dynamical core that uses the third- and the fourth-order MCV schemes. They have verified the numerical outputs of both schemes by widely used standard benchmark tests and obtained competitive results. The present numerical core provides a promising and practical framework for further development of nonhydrostatic compressible atmospheric models.

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

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.

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

Abstract 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.

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.

High order multi-moment constrained finite volume method. Part I: Basic formulation

A new high-order finite volume method based on local reconstruction is presented in this paper. The method, so-called the multi-moment constrained finite volume (MCV) method, uses the point values defined within single cell at equally spaced points as the model variables (or unknowns). The time evolution equations used to update the unknowns are derived from a set of constraint conditions imposed on multi kinds of moments, i.e. the cell-averaged value and the point-wise value of the state variable and its derivatives. The finite volume constraint on the cell-average guarantees the numerical conservativeness of the method. Most constraint conditions are imposed on the cell boundaries, where the numerical flux and its derivatives are solved as general Riemann problems. A multi-moment constrained Lagrange interpolation reconstruction for the demanded order of accuracy is constructed over single cell and converts the evolution equations of the moments to those of the unknowns. The presented method provides a general framework to construct efficient schemes of high orders. The basic formulations for hyperbolic conservation laws in 1- and 2D structured grids are detailed with the numerical results of widely used benchmark tests.

612 非圧縮性流体に対する高精度マルチモーメント制約型有限体積法(OS2.CIP法とその関連手法(3),オーガナイズドセッション)

This paper presents a new high order numerical method based on multi-moment idea developed by the CIP (Constrained Interpolation Profile) method, that is named the multi-moment constrained finite volume (MCV) method. In the MCV method, the locally defined point-values as the unknowns are updated respectively. Then, each of the time evolution equations in terms of the point-values are obtained from some kinds of moments such as the cell-averaged value, point-value and derivative value. Therefore, the MCV method is able to satisfy the mass conservation completely. In this paper, the method is extended to the incompressible fluid equations.