Implementation of a high order Discontinuous Galerkin Spectral Element Method for the Euler and Navier-Stokes equations on unstructured grids

High-order numerical methods for CFD applications have long been touted for their computational efficiency. New solvers based on such methods are continually developed to address specialized applications, improve accuracy and computational efficiency, and ensure robustness across diverse flow conditions. They are essential to advancing research and engineering innovation. This thesis presents Prandtl, a new high-order Discontinuous Galerkin Spectral Element Method (DGSEM) solver for the compressible Euler and Navier-Stokes equations on unstructured – possibly curvilinear – quadrilateral and hexahedral meshes. Built atop the robust backend of the MFEM Finite Element library, Prandtl couples MFEM’s lightweight data structures with an original DGSEM implementation written in modern C++. Split-form DGSEM with collocated Legendre-Gauss-Lobatto nodes constitutes the numerical core of the solver, delivering entropy-stable semi-discrete operators while retaining a simple dimension-by-dimension structure ideal for explicit time stepping. To mitigate aliasing in under-resolved simulations, Prandtl employs the split-form DGSEM, whose summation-by-parts structure helps ensure entropy stability. Gibbs-type oscillations near discontinuities are addressed through subcell blending with a low-order, entropy-consistent Finite Volume scheme, activated smoothly by the Persson–Peraire modal sensor. The resulting hybrid formulation preserves discrete entropy stability for any local blending coefficient and requires no additional MPI communication. Viscous terms are handled with the first method of Bassi and Rebay [1], which remains neutrally stable when paired with the entropy variables used for the auxiliary gradients. Comprehensive verification is provided. A suite of one- and two-dimensional Euler tests demonstrate the expected convergence rates and sharp shock resolution without spurious oscillations, with extreme cases confirming robustness under near-vacuum and strong-shock conditions. For the Navier-Stokes equations, two-dimensional lid-driven cavity simulations are able to reproduce benchmark features and attain convergence to a steady state, while three-dimensional Taylor-Green vortex runs exhibit good parallel efficiency. Together, these results verify that Prandtl delivers high-order accuracy on smooth solutions, shock-robust stability on challenging compressible flows, and good strong scaling on modern HPC platforms, establishing a flexible foundation for future extensions such as implicit time integration, adaptive mesh refinement, and GPU acceleration.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Aerospace Engineering

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Hasanli, Farhad

Contributors dc:contributor

• Bodony, Daniel J.

• Fem
• Dg
• Dgsem
• High
• Order
• Schemes
• Euler
• Navier-stokes
• Cfd
• Prandtl

Statement dc:rights

• © 2025 Farhad Hasanli

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129734

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129734