An Efficient Sparse Approach for Core Flow Problems

Traditionally fully spectral simulations for core flows based on Chebyshev series, Fourier series and spherical harmonics do not require the solution of very large linear systems of equations to advance in time. The explicit treatment of the Coriolis term does generally lead to a large number of decoupled equations of moderate size. It is possible in this context to work with dense matrices and dense solvers. On the other hand, an implicit treatment of the Coriolis term or certain sets of asymptotically reduced equations can not be treated in this way. The time marching of these equations requires solving few very large linear systems. Dense matrices and dense solvers become prohibitively expensive due to a very high memory footprint as well as a very slow execution time. We present a numerical approach converting theses dense systems into equivalent sparse systems that can be solved efficiently. We demonstrate our approach on a set of rapidly rotating flow problems in Cartesian, cylindrical and spherical geometry and compare it to a standard approach.