Spatially Non-Uniform Time-Step Adaptation for Functional Outputs in Unsteady Flow Problems

This paper presents a space-time finite-volume formulation for the Euler equations, which allows for the use of spatially non-uniform time-steps. The formulation also inherently accounts for the effect of dynamically deforming computational meshes. The space and time dimensions are treated in a unified manner so as to permit the variation of control volume sizes in both dimensions. The primary goal is to maintain solution accuracy while reducing the number of unknowns in the overall solution process and potentially lower computational expense. While the formulation presented here is capable of simultaneously handling non conformal meshes in both space and time, the scope of the paper is limited to conformal spatial meshes with non conformal temporal meshes. At any slice in time, the number of spatial elements remains the same, but across any slice in space, the number of time-steps is allowed to vary. In traditional terms, this translates to non-uniform temporal advancement of spatial elements in an unsteady problem. Two unsteady problems, one where an isentropic vortex is convected through a rectangular domain and one of a pitching NACA64A010 airfoil in transonic conditions are presented to demonstrate the algorithm. In the vortex convection problem, a local temporal error indicator is used to identify space-time elements which require higher resolution in the time dimension, thus marking them for temporal refinement. In the case of the transonic pitching airfoil, the adjoint-weighted residual method targeting the lift is used as the error indicator. The results indicate that significant reduction in the overall degrees-of-freedom required to solve an unsteady problem can be achieved using the proposed algorithm. Modest improvements in computational expense for specific problems are also observed.

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