# The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

## Abstract

We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable-resolution grids using the shallow-water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance, implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolution-dependent coefficient. For the spectral element method with variable-resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that, for regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity. With the tensor hyperviscosity, the large-scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications in which long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.

- Authors:

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1198008

- Grant/Contract Number:
- 11-014996; 12-015334

- Resource Type:
- Published Article

- Journal Name:
- Geoscientific Model Development (Online)

- Additional Journal Information:
- Journal Name: Geoscientific Model Development (Online) Journal Volume: 7 Journal Issue: 6; Journal ID: ISSN 1991-9603

- Publisher:
- European Geosciences Union

- Country of Publication:
- Germany

- Language:
- English

### Citation Formats

```
Guba, O., Taylor, M. A., Ullrich, P. A., Overfelt, J. R., and Levy, M. N. The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity. Germany: N. p., 2014.
Web. doi:10.5194/gmd-7-2803-2014.
```

```
Guba, O., Taylor, M. A., Ullrich, P. A., Overfelt, J. R., & Levy, M. N. The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity. Germany. doi:10.5194/gmd-7-2803-2014.
```

```
Guba, O., Taylor, M. A., Ullrich, P. A., Overfelt, J. R., and Levy, M. N. Thu .
"The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity". Germany. doi:10.5194/gmd-7-2803-2014.
```

```
@article{osti_1198008,
```

title = {The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity},

author = {Guba, O. and Taylor, M. A. and Ullrich, P. A. and Overfelt, J. R. and Levy, M. N.},

abstractNote = {We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable-resolution grids using the shallow-water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance, implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolution-dependent coefficient. For the spectral element method with variable-resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that, for regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity. With the tensor hyperviscosity, the large-scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications in which long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.},

doi = {10.5194/gmd-7-2803-2014},

journal = {Geoscientific Model Development (Online)},

number = 6,

volume = 7,

place = {Germany},

year = {2014},

month = {11}

}

DOI: 10.5194/gmd-7-2803-2014

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