Balancing spatial and temporal accuracy in high fidelity simulations of incompressible flows

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Flow simulations are becoming increasingly complex and detailed, but also more expensive. In order to further push the limits of what is feasible, approaches are needed that save computing time and energy. Considerable progress has already been made through the development of efficient numerical methods with a high order of convergence. Nevertheless, it is important that the spatial and temporal resolution are tuned to each other and the required accuracy is maintained globally and locally. The goal of this project is the development of a balanced method for the simulation of incompressible flows to increase the accuracy while reducing the costs. Essential subgoals are the control of the error by adjusting the global spatial and temporal resolution on the basis of reliable error estimators and increasing the efficiency through a local adaptation of the temporal resolution. This is to be achieved with a multilevel method that combines the discontinuous Galerkin method in space with the spectral deferred correction method for time integration. Both methods use element-based polynomial approaches, which offer decisive advantages for achieving the goals: On the one hand, they enable extremely flexible control of the numerical resolution, since the element size as well as the polynomial degree can be adjusted. On the other hand, the aspired high polynomial orders are ideally suited for the use of spectral error estimators. These are potentially much more accurate than the prevailing residual-based estimators, which is of crucial for the success of balancing. The generalization, further development and validation of element-based spectral estimators for the temporal, spatial and total errors is therefore an important part of the planned work. Based on these estimators, strategies are devised that guide the decision of whether to adapt the temporal or the spatial resolution, when adjust the step size, and when to change the polynomial degree of the expansions. Additionally, an approach to local time adaptivity is developed, which allows to focus the resources on regions with high dynamics such as transitional processes or turbulence.