Toward Adaptive Space-time Discontinuous Galerkin Discretizations of Unsteady Aerodynamics
Massachusetts Institute of Technology
Many challenging problems in aerodynamics involve long-time (i.e., unsteady) propagation of wave or wave-like phenomena. When these waves are highly anisotropic (i.e., the propagation distance is much larger than the wave's spatial "diameter"), current techniques can be inefficient when counting degrees of freedom (DOFs) required to obtain a particular error level. Most existing methods apply the method of lines or the Rothe method. This leads to a hierarchical or tensor-product type mesh structure in space-time, which is unable to capture arbitrary solution anisotropy. Additionally, rigorous error estimation and adaptation based on that information is nontrivial. An adaptive, finite element space-time discretization on unstructured, simplex meshes (in space-time) has the potential to resolve many of these issues. By working with time as an additional "spatial" dimension, the well-known Dual Weighted Residual framework (Becker 2001) for error estimation applies directly, without additional consideration. A new mesh adaptation framework (Yano 2012) is capable of producing meshes that optimally allocate DOFs to minimize the cost of obtaining the desired level of accuracy. In this work we will provide some analysis and preliminary results indicating that an adaptive, space-time technique can obtain super-convergence in error. In particular, proper adaptation can lead to an effective dimension reduction; i.e., the number of elements required for the space-time problem in N dimensions is only a constant factor more than the number of elements required for an "equivalent" steady problem in N-1 dimensions. We will also discuss challenges that remain in making adaptive space-time finite element methods practical and future research directions, since this work is still in its early stages.