fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform

TL;DR

This paper introduces fenicsR13, a tensorial mixed finite element solver for the linear R13 equations in non-equilibrium gas dynamics, implemented in Python using FEniCS, demonstrating a new tensorial approach and validating its accuracy.

Contribution

The paper presents a novel tensorial approach for implementing the linear R13 equations in FEniCS, with a detailed Python implementation and validation, filling a gap in existing documentation and literature.

Findings

Solver accurately reproduces exact solutions

Supports applications involving rarefaction effects

Provides a publicly available, validated implementation

Abstract

We present a mixed finite element solver for the linearized R13 equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS's Unified Form Language (UFL) to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the UFL allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.