Efficient $p$-multigrid method based on an exponential time discretization for compressible steady flows
This work addresses the need for faster convergence in high-order discontinuous Galerkin simulations of steady compressible flows, a domain-specific problem.
The authors developed an efficient p-multigrid method using exponential time integration for steady-state compressible flows, achieving rapid and p-independent convergence in 2D and 3D.
An efficient multigrid framework is developed for the time marching of steady-state compressible flows with a spatially high-order ($p$-order polynomial) modal discontinuous Galerkin method. The core algorithm that based on a global coupling, exponential time integration scheme provides strong damping effects to accelerate the convergence towards the steady state, while high-frequency, high-order spatial error modes are smoothed out with a $s$-stage preconditioned Runge-Kutta method. Numerical studies show that the exponential time integration substantially improves the damping and propagative efficiency of Runge-Kutta time-stepping for use with the $p$-multigrid method, yielding rapid and $p$-independent convergences to steady flows in both two and three dimensions.