Building blocks for a leading edge high‐order flow solver. Issue 1 (15th March 2018) - Huismann, Immo; Stiller, Jörg; Fröhlich, Jochen
Abstract: An important trend in Computational Fluid Dynamics is towards high‐order methods, as they offer a substantially lower discretization error for the same number of degrees of freedom (DOF). Examples are the Spectral‐Element Methods (SEM) and Discontinuous Galerkin (DG) methods. Unfortunately, with most implementations the work load of such solvers increases drastically with the number of DOF, for example, when increasing the polynomial degree of the approximation. This issue gets particular pressing for elliptic solvers which are a vital building block in the time‐stepping of the incompressible Navier‐Stokes equations, resulting from pressure projection methods or implicit treatment of viscous terms. So far, this drastic increase of resources has hampered the use of SEM for higher polynomial degrees, such as 16 or more. The present contribution is located at this particular "Frontier of CFD" and proposes an SEM with linear scaling in the number of degrees of freedom. It is achieved independent of the polynomial degree, independent of the aspect ratio of elements, and involves constant iteration count when increasing the number of elements. The method is based on combining static condensation with block‐Jacobi preconditioning and iterative substructuring. The latter two lead to a constant iteration count, while the former warrants linear operator complexity. In the presentation, the scheme is described in detail and applied to the construction of a Helmholtz solver. This solver is extremely fast and able to solve the Helmholtz equation with 10 10 unknowns on 240 cores in acceptable time. It thus enables competitive high‐order SEM simulations even on small clusters and in this way expands the frontier of CFD towards highly accurate results requiring comparatively modest resources. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)