An efficient WENO limiter for discontinuous Galerkin transport scheme on the cubed sphere

Guo, Wei; Nair, Ramachandran D.; Zhong, Xinghui

doi:10.1002/fld.4171

Cited by 19 publications

(26 citation statements)

References 40 publications

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“…The errors quickly reach an extremum at about day 2 and remain nearly unchanged in later days. Similar results are observed in Guo et al s*().…”

Section: Numerical Tests and Resultssupporting

confidence: 92%

“…Three kinds of initial conditions are checked in this test (Guo et al , ), including the following. Twin slotted cylinders :

q^{s c} = \{\begin{matrix} 1 & if r_{i} \leq r_{0} and (||, λ - λ_{i}) \geq \frac{r_{0}}{6} for i = 1, 2, \\ 1 & if r_{1} \leq r_{0} and (||, λ - λ_{1}) < \frac{r_{0}}{6} \\ and θ - θ_{1} < - \frac{5 r_{0}}{12}, \\ 1 & if r_{2} \leq r_{0} and (||, λ - λ_{2}) < \frac{r_{0}}{6} \\ and θ - θ_{2} > \frac{5 r_{0}}{12}, \\ 0.1 & otherwise, \end{matrix}

where r 0 = 0.5 and r i ( i = 1,2) denote the great circle distant between the point of interest and the centers of two slotted cylinders initially located at

(λ_{1}, θ_{1}) = (\frac{5 π}{6}, 0) and (λ_{2}, θ_{2}) = (\frac{7 π}{6}, 0) .

Twin cosine bells :

q^{c b} = ({, \begin{matrix} array \\ array 0.1 + 0.9 h_{i} (λ, θ) & array if ... \end{matrix})

…”

Section: Numerical Tests and Resultsmentioning

confidence: 99%

“…At present, there is still a very active research field aimed at designing WENO‐type limiters in DG models for various types of practical application. In Guo et al s*(), a simple WENO limiter is adopted in the DG schemes to develop a non‐oscillatory transport model on a cubed sphere. In this model, an indicator is used to find the “troubled” cell and the corresponding spatial reconstructions are then revised by using a simple WENO method to avoid oscillations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

Tang

Chen

et al. 2018

Quart J Royal Meteoro Soc

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By using CSL3 multimoment interpolation, a piecewise cubic polynomial for spatial reconstruction can be obtained with four multimoment constraint conditions consisting of two point values at cell boundaries, one volume-integrated average and one slope parameter at the cell center. The resulting multimoment finite-volume scheme is of fourth-order accuracy. A non-oscillatory scheme can be derived by designing the proper formula to calculate the slope parameter at the cell center. A new strategy was recently proposed, using the Weighted Essentially Non-Oscillatory (WENO) concept to determine the slope parameter. Using a WENO-type limiter, the multimoment reconstruction can effectively remove nonphysical oscillations while keeping fourth-order accuracy in smooth regions. In this study, a WENO-type slope limiter is proposed and implemented in our multimoment finite-volume global transport model based on the cubed-sphere grid. The widely used benchmark tests, including both solid rotation and complicated deformational advection cases, are checked to verify the performance of the proposed global transport model. Numerical results reveal that a WENO-type slope limiter can greatly improve the accuracy of the multimoment finite-volume model compared with the former Total Variation Diminishing (TVD)-type limiter. Furthermore, the proposed limiter is constructed over a compact stencil of only three adjacent cells. Without any user-defined or problem-dependent parameters, the present model is very promising for practical applications. KEYWORDS cubed-sphere grid; global model; multimoment scheme; slope limiter; transport model; WENO INTRODUCTIONDue to polar problems on the latitude-longitude grid, global computational meshes with a quasi-uniform grid spacing and free of polar problems, such as the cubed-sphere grid (Sadourny, 1972), icosahedral grid (Sadourny et al., 1968;Williamson, 1968), and yin-yang grid (Kageyama and Sato, 2004), are becoming popular in constructing global models, especially for those aiming at very high-resolution simulations of atmospheric dynamics. One of the key issues in implementing research and operational models on these quasi-uniform grids is the relatively complex grid structure, which causes essential difficulties in constructing high-order models and introduces extra grid-imprinting errors. In the past decade, some accurate global models have been proposed on these grids through the application of so-called "local" schemes, such as the discontinuous Galerkin scheme (Nair et al., 2005a;2005b), the spectral element scheme (Dennis et al., 2012), the multimoment scheme (Chen et al., 2014) and so on. These schemes are accomplished based on compact stencils (usually within a single computational element) and are very convenient to implement in spherical geometry to develop high-order models. Furthermore, these schemes are computationally intensive, so they are highly scalable on massively parallel clusters. A comprehensive review on the applications of quasi-uniform grids can be found in the work of...

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“…The errors quickly reach an extremum at about day 2 and remain nearly unchanged in later days. Similar results are observed in Guo et al s*().…”

Section: Numerical Tests and Resultssupporting

confidence: 92%

“…Three kinds of initial conditions are checked in this test (Guo et al , ), including the following. Twin slotted cylinders :

q^{s c} = \{\begin{matrix} 1 & if r_{i} \leq r_{0} and (||, λ - λ_{i}) \geq \frac{r_{0}}{6} for i = 1, 2, \\ 1 & if r_{1} \leq r_{0} and (||, λ - λ_{1}) < \frac{r_{0}}{6} \\ and θ - θ_{1} < - \frac{5 r_{0}}{12}, \\ 1 & if r_{2} \leq r_{0} and (||, λ - λ_{2}) < \frac{r_{0}}{6} \\ and θ - θ_{2} > \frac{5 r_{0}}{12}, \\ 0.1 & otherwise, \end{matrix}

where r 0 = 0.5 and r i ( i = 1,2) denote the great circle distant between the point of interest and the centers of two slotted cylinders initially located at

(λ_{1}, θ_{1}) = (\frac{5 π}{6}, 0) and (λ_{2}, θ_{2}) = (\frac{7 π}{6}, 0) .

Twin cosine bells :

q^{c b} = ({, \begin{matrix} array \\ array 0.1 + 0.9 h_{i} (λ, θ) & array if ... \end{matrix})

…”

Section: Numerical Tests and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

Tang

Chen

et al. 2018

Quart J Royal Meteoro Soc

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show abstract

“…The three measurements of errors are computed after one period (t D 2:0) on a mesh of 100 cells. (2) Advection of a triangle wave (case 2): We have introduced the initial structure of this test in (43). The numerical errors are computed up to t D 0:7.…”

Section: Comparison Between Eulerian and Semi-lagrangian Schemesmentioning

confidence: 99%

“…A more general procedure that converts different moments to a set of mean values and constructs the WENO projection under the conventional FVM framework is reported in [40]. A WENO reconstruction, the so-called simple WENO limiter [41][42][43], was derived for the DG method, which uses the local DOFs over a compact stencil of only three cells.We have recently proposed a new scheme, called multi-moment constrained finite volume with WENO limiter of fourth-order (MCV-WENO4) method [44], by incorporating the WENO limiter to the multi-moment constrained FVM of third-order (MCV3) scheme. The new WENO limiter in MCV-WENO4 scheme distinguishes itself from the existing ones by taking fully use of the local nodal values (PV) in the WENO interpolation, which leads to a compact computational stencil involving merely the target cell and its adjacent cells.…”

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confidence: 99%

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Sun

Xiao

2016

Numerical Methods in Fluids

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SUMMARYNumerical oscillation has been an open problem for high order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods, and thus are not able to make full use of the sub-cell information available in the local high order reconstructions. This paper presents a novel algorithm which introduces a nodal-value based weighted essentially non-oscillatory limiter for constrained interpolation profile/multimoment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222(2007), 849-871 ) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP-CSL-WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the WENO reconstruction which makes use of the subcell information available from the local DOFs and is built from the point values (PVs) at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the 4th order accuracy and oscillation-suppressing property of the proposed scheme.

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A nonnegative and shape‐preserving global transport model on cubed sphere using high‐order conservative collocation scheme

Huang

Chen

et al. 2021

Numerical Methods in Fluids

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A numerical model for global tracer transport is implemented by using Gauss–Legendre‐point conservative collocation (GLPCC) scheme on cubed sphere. Three‐point GLPCC scheme can achieve fifth‐order accuracy in the spherical geometry. To remove the spurious oscillations in the regions with strong gradients or discontinuities, the hierarchical reconstruction (HR) is applied to the “troubled cells” identified by a smoothness indicator based on the WENO concept. To assure the positivity‐preserving property, a flux correction operation is adopted to impose the restriction on the mass reduction due to the fluxes leaving the cell edges. The proposed numerical scheme aims to remove the nonphysical oscillations while preserving the accuracy in the smooth regions and maintain the compact spatial stencil as far as possible by making full use of degrees of freedom. The proposed schemes are evaluated by simulating the widely used benchmark tests for advection equation in one dimension and multidimensions of spherical geometry. Numerical results show that the fifth‐order accuracy is well preserved for smooth problems, while numerical oscillations can be effectively removed and the nonnegativity of the solutions is strictly guaranteed. It is very promising to develop the practical numerical model for tracer transport computation in atmospheric general circulation models using the proposed numerical framework.

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An efficient WENO limiter for discontinuous Galerkin transport scheme on the cubed sphere

Cited by 19 publications

References 40 publications

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

A nonnegative and shape‐preserving global transport model on cubed sphere using high‐order conservative collocation scheme

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