Theseus is a compressible flow solver developed at the Center for Hypersonics and Entry Systems Studies (CHESS) at the University of Illinois [1]. Its formulation and numerics are based on the work of Hasanli [2]. The code uses the MFEM finite element library [3], and targets modern and emerging HPC architectures for execution.

Theseus started out as a refactor of Prandtl [4], the code produced by Hasanli, to enable GPU execution. The implementation diverged substantially into a new codebase that is lighter and faster, while currently less feature-complete than Prandtl.

Governing Equations

Theseus solves the compressible Navier-Stokes equations, written in conservative form as

$\frac{\partial \mathbf{Q}}{\partial t} + \nabla \cdot \left(\mathbf{F}^I - \mathbf{F}^V\right) = \mathbf{S}$

For a single component fluid, the state vector $\mathbf{Q}$ , inviscid flux $\mathbf{F}^I$ , and viscous flux $\mathbf{F}^V$ are

$\begin{bmatrix} \rho\\ \rho E\\ \rho \vec{v} \end{bmatrix}, \quad \begin{bmatrix} \rho\vec{v}\\ (\rho E + p)\vec{v}\\ \rho(\vec{v} \otimes \vec{v}) + p\delta_{ij} \end{bmatrix}, \quad \begin{bmatrix} 0\\ (\boldsymbol{\tau} \cdot \vec{v} - \mathbf{q})\\ \boldsymbol{\tau} \end{bmatrix}.$

where $(\rho, \vec{v}, E)$ are the density, velocity, and total energy of the fluid, respectively. The viscous stress tensor $\boldsymbol{\tau}$ and heat flux $\mathbf{q}$ are

$\boldsymbol{\tau} = \mu\left[\left(\nabla\vec{v} + (\nabla\vec{v})^T\right) - \frac{2}{3}(\nabla \cdot \vec{v})\mathbf{I}\right], \quad \mathbf{q} = -\kappa\nabla T,$

where $T$ is the fluid temperature and $(\mu, \kappa)$ are viscosity and thermal conductivity, respectively. Source terms are supported through $\mathbf{S}$ .

Model and Discretization

A modular gas model in Theseus implements the transport model and equation of state (EOS), providing transport and thermal properties as a function of the conserved state $\mathbf{Q}$ . Theseus currently implements a single-component calorically perfect gas model and is being extended to local thermal equilibrium (LTE) mixtures.

Theseus employs a nodal DGSEM formulation on tensor-product elements for spatial discretization [5, 6]. The solution strategy follows an entropy-conservative BR1 form [7], and uses explicit $s$ -stage Runge-Kutta methods for time advancement.

References

[1] Center for Hypersonics & Entry Systems Studies (CHESS), University of Illinois Urbana-Champaign. https://chess.grainger.illinois.edu/

[2] Hasanli, F. (2025). Implementation of a High Order Discontinuous Galerkin Spectral Element Method for the Euler and Navier–Stokes Equations on Unstructured Grids. M.S. thesis, UIUC. https://hdl.handle.net/2142/129734

[3] Anderson, R. et al. (2021). MFEM: A modular finite element methods library. Computers & Mathematics with Applications, 81, 42–74. https://doi.org/10.1016/j.camwa.2020.06.009

[4] Prandtl repository: https://github.com/chess-uiuc/Prandtl

[5] Fisher, T. C., & Carpenter, M. H. (2013). High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics, 252, 518–557.

[6] Gassner, G. J., Winters, A. R., & Kopriva, D. A. (2016). Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. Journal of Computational Physics, 327, 39–66.

[7] Gassner, G. J., Winters, A. R., Hindenlang, F. J., & Kopriva, D. A. (2018). The BR1 scheme is stable for the compressible Navier–Stokes equations. Journal of Scientific Computing, 77(1), 154–200. https://doi.org/10.1007/s10915-018-0702-1