A Sparse Fully Spectral Method for Simulations in Spherical, Cylindrical and Cartesian Geometries

English Abstract:

Due to the complexity of core flow problems, they need to be tackled from all possible angles. Direct Numerical Simulations (DNS), asymptotically reduced models and lab experiments allow for the investigation of different aspects related to core dynamics. While a majority of DNS studies are performed in spherical geometries, reduced models and lab experiments often involve Cartesian and cylindrical geometries as well; thus requiring efficient numerical methods that are applicable to a variety of coordinate systems. Furthermore, for numerical stability, accuracy and performance, the linear terms of the governing equations need to be treated implicitly. For example, the explicit treatment of the Coriolis force leads to a stringent timestep constraint. We have developed a fully spectral coding framework that is capable of solving a variety of equation sets in many different geometries. Depending on the geometry, it uses a combination of Chebyshev series, Fourier series and spherical harmonics for the numerical discretization. By using the quasi-inverse method, our approach produces very sparse matrices that allow for the efficient solution of very large coupled systems of equations. We will present our coding framework as well as nonlinear results for different geometries.