A Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods Applied to Atmospheric Flows
Birken, Philipp LU ; Dedner, Andreas and Klöfkorn, Robert LU
(2025)
In GEM - International Journal on Geomathematics
17(1).
- Abstract
Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in z-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. This motivates our research to... (More)
Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in z-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. This motivates our research to construct a Jacobian-free preconditioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.
(Less)
- author
- Birken, Philipp
LU
; Dedner, Andreas
and Klöfkorn, Robert
LU
- organization
- publishing date
- 2025-12-27
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- DG, DUNE, DUNE-FEM, FV, Implicit, Jacobian-free, Multigrid, Numerical weather prediction, Preconditioner
- in
- GEM - International Journal on Geomathematics
- volume
- 17
- issue
- 1
- article number
- 2
- publisher
- Springer
- external identifiers
-
- scopus:105026251459
- ISSN
- 1869-2672
- DOI
- 10.1007/s13137-025-00279-y
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © The Author(s) 2025.
- id
- ab290390-d42d-4c49-935e-5b9ed4c98bd9
- date added to LUP
- 2026-03-09 17:01:06
- date last changed
- 2026-03-09 17:02:08
@article{ab290390-d42d-4c49-935e-5b9ed4c98bd9,
abstract = {{<p>Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in z-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. This motivates our research to construct a Jacobian-free preconditioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.</p>}},
author = {{Birken, Philipp and Dedner, Andreas and Klöfkorn, Robert}},
issn = {{1869-2672}},
keywords = {{DG; DUNE; DUNE-FEM; FV; Implicit; Jacobian-free; Multigrid; Numerical weather prediction; Preconditioner}},
language = {{eng}},
month = {{12}},
number = {{1}},
publisher = {{Springer}},
series = {{GEM - International Journal on Geomathematics}},
title = {{A Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods Applied to Atmospheric Flows}},
url = {{http://dx.doi.org/10.1007/s13137-025-00279-y}},
doi = {{10.1007/s13137-025-00279-y}},
volume = {{17}},
year = {{2025}},
}