Efficient Solutions of the Euler Equations in a Time-Adaptive Space-Time Framework

Abstract

The inviscid Euler equations are solved using a space-time finite-volume discretization. The method inherently accounts for deforming computational meshes. Traditional implicit unsteady discretizations rely on solving the spatial problem with appropriate temporal terms added in an implicit sense at each time level while advancing forward in time one time-step at a time. The approach in this paper is to unify both time and space dimensions and operate on a single computational mesh that spans both the spatial and temporal domains of interest. In essence all unknowns at all spatial and temporal locations in the domain of interest are solved implicitly in one shot. The primary advantage of this approach is the spatially non-uniform variation of the time-step size. The goal is to reduce the total degrees-of-freedom by requiring that only certain spatial locations advance with high resolution in time. The adjoint weighted residual method is used to identify regions in the space-time domain that require higher temporal resolution, thus targeting only certain spatial locations that have to be advanced slowly in time while maintaining overall solution accuracy. Both the reduction of the total unknowns in the solution and the faster convergence due to implicit coupling in time combine to form a efficient unsteady solver. The actual non-linear problem is solved using the Newton-Krylov preconditioned GMRES method with ILU(0) as the preconditioner. While the method converges rapidly, the memory requirements are high even for relatively small problems. This problem is alleviated by splitting the time domain into a limited number of slabs and solving for all unknowns within a slab implicitly while advancing in time slab-by-slab.