Multi-level methods with high order in time and space for the numerical simulation of incompressible flows
Final Report Abstract
Numerical methods with a high order of convergence can achieve significantly greater accuracy for comparable cost than the prevailing low-order methods based on piecewise constant or linear approximations. In computational fluid dynamics, the focus of recent research has been on high-order methods for spatial discretization. Considerable progress has been made especially in the development of continuous and discontinuous Galerkin methods, which decompose the computational domain into elements with polynomial approximations of arbitrarily high degree, usually 4 to 16, and thus achieve a correspondingly high convergence rate. In contrast, lower order methods are predominantly used for discretization in the time domain. When applied to incompressible flows, methods of order 2 dominate and those with order 3 or 4 are already an exception. To attain higher accuracy, the step size must be reduced much faster in time than in space. As a result the time integration becomes a limiting factor that prevents better use of resources such as computational time and energy. To overcome this handicap, time integration methods for incompressible flows that attain an arbitrarily high order of convergence were developed in the present project. These methods allow to adapt the temporal discretization to the spatial one and thus to achieve a cost reduction. The starting point is the spectral deferred correction (SDC) method developed since 2000. Within the scope of the project, this method was extended to incompressible flows. An essential step was the generalization of projection methods as a basic building block for the correction procedure. The semi-implicit SDC method developed in the project is more efficient and robust than comparable methods and, unlike these, allows variable material properties to be taken into account. For laminar flows with variable viscosity and time-dependent boundary values, the method achieved a temporal order of convergence of 12 and outperformed lower order methods already for moderate accuracy requirements. In simulations of turbulent flow, Runge-Kutta methods of order 3 proved superior to popular second-order methods. However, due to the marginal spatial resolution and stability limitations of the semi-implicit projection method, no further advantage could be achieved with methods of higher order. This finding suggests that further improvements can be achieved with implicit methods, which should be investigated in future projects.
Publications
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Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids. Journal of Computational Physics, 327 (2016, 12), 317-336.
Stiller, Jörg
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A spectral deferred correction method for incompressible flow with variable viscosity. Journal of Computational Physics, 423 (2020, 12), 109840.
Stiller, Jörg
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Evaluation and development of Spectral Deferred Correction Methods for convection-diffusion equations. GAMM Jahrestagung, Kassel, 17.03.2021
Grotteschi M., Stiller J., Guesmi M.
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Assessment of high‐order IMEX methods for incompressible flow. International Journal for Numerical Methods in Fluids, 95(6), 954-978.
Guesmi, Montadhar; Grotteschi, Martina & Stiller, Jörg
