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

Hasanli, Farhad

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

Description

Title

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

Author(s)

Hasanli, Farhad

Issue Date

2025-05-08

Director of Research (if dissertation) or Advisor (if thesis)

Bodony, Daniel J

Department of Study

Aerospace Engineering

Discipline

Aerospace Engineering

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

• FEM
• DG
• DGSEM
• high
• order
• schemes
• Euler
• Navier-Stokes
• CFD
• Prandtl

Abstract

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.

Graduation Semester

2025-05

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

© 2025 Farhad Hasanli

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