Meshless numerical methods for geophysical modeling

Global electric inputs
The electric potential (kV) at 1.5 km (top figure) and 6 km (bottom figure) above sea level. In the top figure, the regions in white are the intersection of the 1.5 km constant-height surface with the Earth’s topography.

While computer architectures have advanced rapidly in recent years, numerical schemes currently used for geoscience modeling have not kept pace with these technological developments. Radial basis functions (RBFs) offer a novel numerical approach for solving partial differential equations to high accuracy. Being a meshless method, RBFs excel in solving problems that require geometric flexibility, local refinement for small features, and with little increase in programming complexity when extended to higher dimensional spaces. In particular, the RBF-generated finite differences (RBF-FD) approach has allowed the RBF method to become computationally cost-effective in terms of scalability, memory, and runtime for solving systems of PDEs. The localized and accurate nature of the RBF-FD method:

  • Leads to matrices that are over 99% empty.

  • Allows it to scale as O(N) per time step, with N being with the total number of nodes.

  • Makes it highly suitable for parallelization on accelerator-based computer architectures.

A key advantage of RBFs for geophysical modeling is that they do not depend on any grid, mesh, or coordinate system, but only the Euclidean distance between node locations in any dimensional space. This makes it particularly easy for modelers to incorporate the Earth’s topography into physical models. This geometric flexibility is vital, since topography can play a crucial role in applications such as studying global electric currents (GEC) from thunderstorms in the Earth’s atmosphere. In fact, this electrical system can be considered the ultimate link between solar, galactic, ionospheric, and magnetospheric processes and processes in the lower atmosphere, cloud system dynamics, and climate evolution.

Topography discretizations
Examples of the discretization of the Earth’s topography: (a) 150 km resolution at sea level, (b) 400 km resolution at sea level.

The figure at right shows two sets of discretizations over the Earth’s surface that are used for solving the 3D GEC model with RBF-FD. Note that over the topographical features, the nodes are more irregularly spaced and twice as dense as over the oceans, with no meshes or grids involved. The illustration at top shows the solution for the atmospheric current in kiloVolts (kV) of the GEC RBF-FD model at 1.5 km and 6 km above sea level. Note the topography’s strong effect as the land areas that modify electrical potential are clearly defined.

The development of the 3D GEC RBF-FD model in FY2015 was made possible by the continual research advancements in meshless numerical methods over the last several fiscal years. In those previous years, modeling was restricted to 2D in regular geometries, such as a rectangle or a sphere. Building on those accomplishments, FY2015 saw the first-ever 3D RBF-FD approach relevant to atmospheric modeling. These advancements were made through the successful collaborations of the IMAGe Computational Mathematics Group working together with HAO and the University of Colorado at Boulder to continue research in the promising area of RBFs for geoscience modeling.

This work advances CISL’s scientific efforts to develop scalable algorithms for atmospheric modeling on massively parallel and accelerator-based computer architectures. This development effort at NCAR is supported by NSF grant DMS-0934317.