A p-Multigrid Strategy with Anisotropic p-Adaptation Based on Truncation Errors for High-Order Discontinuous Galerkin Methods

Abstract

High-order DG methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom. In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [\textit{Rueda-Ramírez et al., Truncation Error Estimation in the p-Anisotropic DGSEM, J. of Scientific Computing}] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required. The proposed methods are tested for the compressible Navier-Stokes equations, where we investigate two cases: the 2D flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method; and the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations.

References in corpus (3)

• h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
• Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method
• On the role of numerical dissipation in stabilising under-resolved turbulent simulations using discontinuous Galerkin methods

Cited by in corpus (6)

• HORSES3D: a high-order discontinuous Galerkin solver for flow simulations and multi-physics applications
• Hybrid multigrid methods for high-order discontinuous Galerkin discretizations
• An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing
• Accelerating high order discontinuous Galerkin solvers using neural networks: 3D compressible Navier-Stokes equations
• Entropy-Stable Gauss Collocation Methods for Ideal Magneto-Hydrodynamics
• High--order discontinuous Galerkin approximation for a three--phase incompressible Navier--Stokes/Cahn--Hilliard model