Department of Information Technology

# Improved methods for global climate simulations

Global climate simulation entails numerical solution of partial differential equations (PDEs) with the whole earth (an approximate sphere) as the computational domain. This leads to very large computational problems and requires high performance computers combined with advanced numerical algorithms in order to be feasible.

The methods currently in use are typically spectral methods based on expansions in spherical harmonics, orthogonal polynomials, or Fourier series. With these methods, several different problems can occur that are not really related to the physics of the problem. For example

1. Due to the use of spherical coordinates, the poles of the earth become singular points, which of course is not physically correct.
2. If the simulation is designed to track a specific phenomenon, extra resolution is most likely needed in the area where the phenomenon is active. With the standard mesh-based methods it is difficult to add local resolution. Either the whole earth must be resolved more accurately (prohibitively expensive) or locations totally unrelated to the phenomenon such as the mirror location with respect to the center of the earth must be resolved simultaneously (unnecessarily expensive).
3. The time evolution of the simulation must be computed with much smaller time increments than those dictated by the physics of the PDE problem.

The idea of this project is to develop radial basis function (RBF) based methods for global climate modeling. RBF methods are also spectral with the right choice of basis functions, hence sharing the property of having a high order of accuracy with the traditional methods. However, these new methods do not suffer from the problems mentioned above.

1. RBF methods are mesh-free, that is, they do not depend on some specific choice of coordinates and accompanying grid. Therefore, the poles of the earth are not numerically different from other locations.
2. Extra resolution (adaptivity) can be employed locally anywhere without affecting the rest of the domain (since there is no mesh).
3. A first numerical study [FG07] shows that it is possible to use much larger steps in time with RBF methods than traditional methods, thus reducing the computational time significantly for a given problem. Furthermore, the overall number of data points (resolution) needed is smaller than for a traditional method for a given problem, which reduces both computational time and memory requirements.

Another very beneficial aspect of RBF methods is that they are generally much easier to implement than mesh-based methods, simply because there is less need for bookkeeping.

The impact of introducing RBF methods into the global climate modeling community would be significant.

• With reduced computational time and memory requirements it is possible to do more simulations and/or do more accurate simulations.
• With the possibility of adding resolution anywhere at a low cost it is possible to add a lot of local detail in a simulation without ending up with an intractable problem.
• With the large number of high quality simulations that would be performed, it might become easier to persuade the world leaders that global warming is an immediate problem.

## Current research topics

While RBFs naturally allow for adaptive node refinement, few methods have been developed for time-dependent problems. There are several issues to handle for such a method, for instance how to update the node set and how to implement the RBF method in a computationally feasible way. The first adaptive RBF method for a linear convective PDE on the sphere was implemented by Flyer and Lehto [FL10]. For this problem, the refinement was given by simulation of electrostatic repulsion where the charge of each point was dictated by features of the flow. The accuracy of the method far surpassed any previous result in the literature.

### Local RBF methods

When combining non-linear problems with dynamically adaptive nodes, the computational cost of the global RBF method is prohibitively high. If local RBF interpolants are used instead, the cost may be reduced to the same order as for classical finite difference methods. This does however reduce the accuracy and it introduces stability issues. The latter have been successfully dealt with by a novel hyper-viscosity filter introduced by Fornberg and Lehto [BL11]. With the aid of this stabilizing method, the local RBF-FD method was shown to outperform global RBFs in terms of efficiency for linear advection on the surface of the sphere.

An updated implementation using local RBFs for the shallow water equations, based on the previous implementation by Flyer and Wright [FG09], is currently under development. A manuscript featuring results for some of the Williamson test cases and the Galewsky test case is in preparation. A few numerical results are shown in the figures below.

The project is a collaboration with Dr. Natasha Flyer at the National Center for Atmospheric Research (NCAR) and is partly funded by the National Science Foundation. NCAR is the institution where the global climate models used in the US are being developed.

Collaborators:

[BL11] B. Fornberg and E. Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comp. Phys., 230, 2270-2285, 2011.

[FL10] N. Flyer and E. Lehto, Rotational transport on a sphere: Local node refinement with radial basis functions, J. Comp. Phys., 229(6), 1954-1969, 2010.

[FG09] N. Flyer and G. Wright, A radial basis function method for the shallow water equations on a sphere, Proc. R. Soc. A , 465(2106), 2009, 1949-1976.

[FG07] N. Flyer and G. Wright, Transport schemes on a sphere using radial basis functions, J. Comp. Phys., 226, 2007, 1059-1084.