Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations
Energy-Conserving Spectral Spatial Discretization on Unstructured Tetrahedral Meshes
Dr. Marcus Thorne1*
Prof. Astrid Lindholm2
1 ETH Zürich, Seminar for Applied Mathematics, Zürich, Switzerland (*Corresponding author: m.thorne@ethz.ch)
2 Stockholm University, Department of Mathematics, Stockholm, Sweden
DOI: 10.5555/ijpacm.2026.1.1.01 Published: March 10, 2026
Structured Abstract
Numerical simulations of high Reynolds number turbulent flows demand spatial discretizations that strictly conserve kinetic energy while preserving divergence-free velocity fields. We formulate an energy-stable, high-order Discontinuous Galerkin (DG) method equipped with interior penalty formulations and symmetric flux splitting. Simulations on canonical Taylor-Green vortex benchmarks demonstrate optimal spatial convergence up to polynomial degree p=7 without non-physical numerical dissipation. The proposed framework provides an exceptionally stable computational foundation for direct numerical simulation of complex turbulent boundary layers.
Numerical simulations of high Reynolds number turbulent flows demand spatial discretizations that strictly conserve kinetic energy while preserving divergence-free velocity fields. Standard low-order finite volume schemes introduce artificial dissipation that smears small-scale turbulent vortices.
2. Mathematical Discretization & Energy Stability
We develop an energy-conserving high-order Discontinuous Galerkin formulation based on symmetric flux decomposition. Using Gauss-Legendre quadrature points, we prove unconditional nonlinear stability under exact polynomial integration.
3. Numerical Convergence Verification
Convergence rates were verified on 3D Taylor-Green vortex flows up to polynomial order $p=7$. The kinetic energy dissipation rates matched analytical direct numerical simulation (DNS) spectra with zero unphysical numerical diffusion.
Funding & Support
Swiss National Science Foundation (SNSF Project 198744).
Competing Interests
The authors declare no competing interests.
Author Contributions (CRediT Taxonomy)
Dr. Marcus Thorne: Conceptualization, Methodology, Writing – original draft, Funding acquisition
Prof. Astrid Lindholm: Formal analysis, Writing – review & editing
Figure 1
Kinetic energy dissipation spectrum for Discontinuous Galerkin $p=7$ polynomial orders in Taylor-Green vortex simulation.
[1]
Hesthaven, J. S., & Warburton, T. (2007). Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer New York.
Cockburn, B., & Shu, C. W. (2001). Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16(3), 173-261.
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Thorne, D. M., & Lindholm, P. A. (2026). Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
APA 7th Edition
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Thorne, D. M. and Lindholm, P. A. 2026. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics. 1, 1 (2026), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
ACM Format
Thorne, D. M.; Lindholm, P. A. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. IJPACM 2026, 1 (1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
ACS Format
Thorne, D. M., & Lindholm, P. A. (2026). Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
APA Format
THORNE, D. M.; LINDHOLM, P. A. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, v. 1, n. 1, p. 1-28, 2026. Disponível em: <https://doi.org/10.5555/ijpacm.2026.1.1.01>.
ABNT Format
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics 1, no. 1 (2026): 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
Chicago Format
Thorne, D. M. and Lindholm, P. A., 2026. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), pp.1-28. Available at: <https://doi.org/10.5555/ijpacm.2026.1.1.01>.
Harvard Format
D. M. Thorne and P. A. Lindholm, "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations," IJPACM, vol. 1, no. 1, pp. 1-28, 2026, doi: 10.5555/ijpacm.2026.1.1.01.
IEEE Format
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics, vol. 1, no. 1, 2026, pp. 1-28, https://doi.org/10.5555/ijpacm.2026.1.1.01.
MLA Format
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics 1, no. 1 (2026): 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
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ACM Format
Thorne, D. M. and Lindholm, P. A. 2026. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics. 1, 1 (2026), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
ACS Format
Thorne, D. M.; Lindholm, P. A. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. IJPACM 2026, 1 (1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
APA (7th) Format
Thorne, D. M., & Lindholm, P. A. (2026). Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
ABNT Format
THORNE, D. M.; LINDHOLM, P. A. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, v. 1, n. 1, p. 1-28, 2026. Disponível em: <https://doi.org/10.5555/ijpacm.2026.1.1.01>.
Chicago Format
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics 1, no. 1 (2026): 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
Harvard Format
Thorne, D. M. and Lindholm, P. A., 2026. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), pp.1-28. Available at: <https://doi.org/10.5555/ijpacm.2026.1.1.01>.
IEEE Format
D. M. Thorne and P. A. Lindholm, "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations," IJPACM, vol. 1, no. 1, pp. 1-28, 2026, doi: 10.5555/ijpacm.2026.1.1.01.
MLA (9th) Format
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics, vol. 1, no. 1, 2026, pp. 1-28, https://doi.org/10.5555/ijpacm.2026.1.1.01.
Turabian Format
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics 1, no. 1 (2026): 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
TY - JOUR
TI - Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations
T2 - Energy-Conserving Spectral Spatial Discretization on Unstructured Tetrahedral Meshes
AU - Dr. Marcus Thorne
AU - Prof. Astrid Lindholm
JO - International Journal of Pure, Applied and Computational Mathematics
VL - 1
IS - 1
SP - 1
EP - 28
PY - 2026
DO - 10.5555/ijpacm.2026.1.1.01
UR - https://doi.org/10.5555/ijpacm.2026.1.1.01
PB - Academic Mathematical Publishing House
SN - 2348-0084
AB - Numerical simulations of high Reynolds number turbulent flows demand spatial discretizations that strictly conserve kinetic energy while preserving divergence-free velocity fields. We formulate an energy-stable, high-order Discontinuous Galerkin (DG) method equipped with interior penalty formulations and symmetric flux splitting. Simulations on canonical Taylor-Green vortex benchmarks demonstrate optimal spatial convergence up to polynomial degree p=7 without non-physical numerical dissipation. The proposed framework provides an exceptionally stable computational foundation for direct numerical simulation of complex turbulent boundary layers.
KW - Discontinuous Galerkin
KW - Navier-Stokes Equations
KW - High-Order Methods
KW - Computational Fluid Dynamics
KW - Energy Conservation
ER -
All 11 supported citation standards for this article are shown below. Click "Copy" next to any style to copy it directly to your clipboard.
ACM
Thorne, D. M. and Lindholm, P. A. 2026. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics. 1, 1 (2026), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
ACS
Thorne, D. M.; Lindholm, P. A. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. IJPACM 2026, 1 (1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
APA 7th
Thorne, D. M., & Lindholm, P. A. (2026). Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01
ABNT
THORNE, D. M.; LINDHOLM, P. A. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, v. 1, n. 1, p. 1-28, 2026. Disponível em: <https://doi.org/10.5555/ijpacm.2026.1.1.01>.
Chicago
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics 1, no. 1 (2026): 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
Harvard
Thorne, D. M. and Lindholm, P. A., 2026. Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations. International Journal of Pure, Applied and Computational Mathematics, 1(1), pp.1-28. Available at: <https://doi.org/10.5555/ijpacm.2026.1.1.01>.
IEEE
D. M. Thorne and P. A. Lindholm, "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations," IJPACM, vol. 1, no. 1, pp. 1-28, 2026, doi: 10.5555/ijpacm.2026.1.1.01.
MLA 9th
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics, vol. 1, no. 1, 2026, pp. 1-28, https://doi.org/10.5555/ijpacm.2026.1.1.01.
Turabian
Thorne, Dr. Marcus and Lindholm, Prof. Astrid. "Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations." International Journal of Pure, Applied and Computational Mathematics 1, no. 1 (2026): 1-28. https://doi.org/10.5555/ijpacm.2026.1.1.01.
International Journal of Pure, Applied and Computational Mathematics
Vol. 1, Issue 1, pp. 1-28 (2026)
DOI: 10.5555/ijpacm.2026.1.1.01
Open Access • CC BY 4.0
Higher-Order Discontinuous Galerkin Solvers for Incompressible Navier-Stokes Equations
Energy-Conserving Spectral Spatial Discretization on Unstructured Tetrahedral Meshes
Dr. Marcus Thorne*, Prof. Astrid Lindholm
ETH Zürich, Seminar for Applied Mathematics, Zürich, Switzerland
Stockholm University, Department of Mathematics, Stockholm, Sweden
Abstract
Numerical simulations of high Reynolds number turbulent flows demand spatial discretizations that strictly conserve kinetic energy while preserving divergence-free velocity fields. We formulate an energy-stable, high-order Discontinuous Galerkin (DG) method equipped with interior penalty formulations and symmetric flux splitting. Simulations on canonical Taylor-Green vortex benchmarks demonstrate optimal spatial convergence up to polynomial degree p=7 without non-physical numerical dissipation. The proposed framework provides an exceptionally stable computational foundation for direct numerical simulation of complex turbulent boundary layers.
Modern scientific workflows necessitate full algorithmic transparency and verifiable computation. In this investigation, we establish the foundational principles governing high-throughput data representations.
Continuous parameter spaces are mapped using adaptive loss functions with explicit bound convergence.
2. Experimental Framework
Empirical validation was performed on standardized benchmarking clusters. Communication profiles and latency bottlenecks were captured at sub-millisecond granularity.
All source algorithms are fully archived under open-source licenses for independent replication.
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