Multi-Level Solution Strategies for Implicit Time Discontinuous Galerkin Discretizations

Mark W. Lohry, Luigi Martinelli

Research output: Contribution to journalArticlepeer-review


The discontinuous Galerkin method and related flavors of high-order spectral element methods provide many well-known benefits for the spatial discretization of partial differential equations such as the Navier–Stokes equations. However, practical problems of engineering relevance such as large-eddy simulation of turbulent flows over complex geometries are computationally intractable by standard explicit time integration methods, necessitating the use of implicit methods. The efficient solution of the nonlinear algebraic systems arising from implicit time integration methods applied to DG discretizations of nonlinear PDEs is challenging; standard linearization methods result in very stiff block-sparse systems with prohibitive computational and memory requirements. This paper presents a low-memory, computationally efficient, implicit solution method that combines a framework of nonlinear polynomial multigrid, adaptive explicit Runge–Kutta smoothers, implicit Jacobian-free coarse level smoothers, nonlinear Krylov subspace acceleration and adaptive time stepping using feedback control of the nonlinear solver convergence rate.

Original languageEnglish (US)
Pages (from-to)509-521
Number of pages13
JournalInternational Journal of Computational Fluid Dynamics
Issue number6
StatePublished - 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Aerospace Engineering
  • Condensed Matter Physics
  • Energy Engineering and Power Technology
  • Mechanics of Materials
  • Mechanical Engineering


  • discontinuous Galerkin
  • High-order methods
  • multigrid


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