5:15 PM - 6:30 PM
[MGI35-P07] Stability analysis of the RBF advection models with local node refinement
Keywords:geophysical fluid dynamics, advection scheme
A radial basis function (RBF) is a function only of the distance from a node. Because RBF do not depend on particular coordinates, it can be applicable to unstructured quasi-uniform nodes on a sphere and derivative operators are not singular at the poles. In addition, RBF-based methods are straightforward to implement and do not become complicated at higher dimensions. Flyer and Wright (2007) constructed an advection model on a sphere and showed a spectral convergence.
Local node refinement is used for efficiency and accuracy in a region with a large variation such as vortices. Flyer and Lehto (2010) conducted a vortex roll-up experiment to demonstrate improvement in accuracy of the refined over uniform nodes. In addition to the higher resolution in vortices, the reduction of condition number of the RBF interpolation matrix contributes to the improved results.
Ogasawara and Enomoto (2020) adopted a modified Schmidt transform for local refinement. The Schmidt transform does not require iterations and is suitable for a large number of nodes compared to the existing method using charged particles, whose convergence is not guaranteed. Moreover, to suppress the Runge phenomenon, the map factor that depends only on latitude can be used to scale the shape parameter in place of the distance between nodes. However, Ogasawara and Enomoto (2020) reported the reduction of error of refined over quasi-uniform nodes in short integrations and did not conduct stability analysis to show the stability in long integrations. Therefore, this study performs stability analysis in the vortex roll-up experiment.
The eigen-analysis for stability shows that local refinement stabilizes the integration owing to the reduction of the number of eigenvalues outside the stable domain of the fourth-order Runge-Kutta scheme used for time integration and the maximum amplification factor. The results of eigen-analysis are confirmed by the emergence and absence of instability with quasi-uniform and refined nodes, respectively.
References
Flyer, N. and E. Lehto, 2010: Rotational transport on a sphere: Local node refinement with radial basis functions. J. Comput. Phys. 229, 1954–1969.
Schmidt, F., B., 1977: Variable fine mesh in spectral global models, Phys. Atmos., 50, 211–217.
Flyer, N., and G. B. Wright, 2007: Transport schemes on a sphere using radial basis functions. J. Comput. Phys., 226, 1059–1084.
Courtier, P., and J.-F. Geleyn, 1988: A global numerical weather prediction model with variable resolution: Application to the shallow-water equations. Quart. J. Roy. Meteor. Soc., 114, 1321–1346.
Nair, R., J. Côté, and A. Staniforth, 1999: Cascade interpolation for semi-Lagrangian advection over the sphere. Quart. J. Roy. Meteor. Soc., 125.
Ogasawara and Enomoto, 2020: Reproducibility of Vortex Roll-up using Locally Refines Nodes. DPRI Annuals, 63 B. In Japanese.
Local node refinement is used for efficiency and accuracy in a region with a large variation such as vortices. Flyer and Lehto (2010) conducted a vortex roll-up experiment to demonstrate improvement in accuracy of the refined over uniform nodes. In addition to the higher resolution in vortices, the reduction of condition number of the RBF interpolation matrix contributes to the improved results.
Ogasawara and Enomoto (2020) adopted a modified Schmidt transform for local refinement. The Schmidt transform does not require iterations and is suitable for a large number of nodes compared to the existing method using charged particles, whose convergence is not guaranteed. Moreover, to suppress the Runge phenomenon, the map factor that depends only on latitude can be used to scale the shape parameter in place of the distance between nodes. However, Ogasawara and Enomoto (2020) reported the reduction of error of refined over quasi-uniform nodes in short integrations and did not conduct stability analysis to show the stability in long integrations. Therefore, this study performs stability analysis in the vortex roll-up experiment.
The eigen-analysis for stability shows that local refinement stabilizes the integration owing to the reduction of the number of eigenvalues outside the stable domain of the fourth-order Runge-Kutta scheme used for time integration and the maximum amplification factor. The results of eigen-analysis are confirmed by the emergence and absence of instability with quasi-uniform and refined nodes, respectively.
References
Flyer, N. and E. Lehto, 2010: Rotational transport on a sphere: Local node refinement with radial basis functions. J. Comput. Phys. 229, 1954–1969.
Schmidt, F., B., 1977: Variable fine mesh in spectral global models, Phys. Atmos., 50, 211–217.
Flyer, N., and G. B. Wright, 2007: Transport schemes on a sphere using radial basis functions. J. Comput. Phys., 226, 1059–1084.
Courtier, P., and J.-F. Geleyn, 1988: A global numerical weather prediction model with variable resolution: Application to the shallow-water equations. Quart. J. Roy. Meteor. Soc., 114, 1321–1346.
Nair, R., J. Côté, and A. Staniforth, 1999: Cascade interpolation for semi-Lagrangian advection over the sphere. Quart. J. Roy. Meteor. Soc., 125.
Ogasawara and Enomoto, 2020: Reproducibility of Vortex Roll-up using Locally Refines Nodes. DPRI Annuals, 63 B. In Japanese.