Residual-Based Time-Step Control for High-Order Discretizations

This paper presents a study of temporal errors, as well as a time-step control strategy for their reduction, in high-order spatial discretizations of convection-dominated flow equations. The discontinuous Galerkin finite-element method serves as the spatial discretization, and it is combined with implicit, semi-discrete time-marching schemes. Efficiency of the schemes, measured in terms of the number of implicit system solutions for a given level of accuracy, is compared for different spatial orders and mesh sizes. For uniform time stepping, certain hybrid multi-step/stage schemes can be more efficient than popular diagonally-implicit Runge-Kutta methods. A residual-based adaptive time-step control strategy is also presented for balancing spatial and temporal errors in each time step, with a focus on performance on spatially-adapted meshes. The methods are tested on an unsteady manufactured solution for scalar advection-diffusion, and for the Navier-Stokes equations on two problems with moving geometry and mesh.