Parallel Semi-Implicit Spectral Element Methods for Atmospheric General Circulation Models

Spectral elements combine the accuracy and exponential convergence of conventional spectral methods with the geometric flexibility of finite elements. Additionally, there are several apparent computational advantages to using spectral element methods on microprocessors. In particular, the computations are naturally cache-blocked and derivatives may be computed using nearest neighbor communications. Thus, an explicit spectral element atmospheric model has demonstrated close to linear scaling on a variety of distributed memory computers including the IBM SP and Linux Clusters. Explicit formulations of PDE's arising in geophysical fluid dynamics, such as the primitive equations on the sphere, are time-step limited by the phase speed of gravity waves. Semi-implicit time integration schemes remove the stability restriction but require the solution of an elliptic BVP. By employing a weak formulation of the governing equations, it is possible to obtain a symmetric Helmholtz operator that permits the solution of the implicit problem using conjugate gradients. We find that a block-Jacobi preconditioned conjugate gradient solver accelerates the simulation rate of the semi-implicit relative to the explicit formulation for practical climate resolutions by about a factor of three.