What is it about?

We investigated the usage of modern programming paradigms to translate mathematical expressions derived from partial differential models directly into programmable source code. One example of such a framework is the Unified Form Language (UFL) within the FEniCS framework. Since the extended gas models are a complex set of equations with tensorial structures, we used the capabilities of the UFL efficiently to simplify the implementation significantly. The result is a simulation framework that is both compact and easily maintainable. The one-to-one correspondence of actual mathematics and resulting implementation makes it much easier to modify the model's complexity and extend the solver for future applications.

Featured Image

Why is it important?

For most fluid simulations under standard conditions, the classical Navier-Stokes and Fourier models are sufficiently precise to resolve the critical physical phenomena. However, for simulations of non-standard, so-called non-equilibrium conditions, a variety of strange effects appears. Therefore, simulation engineers need to use advanced models to describe the behavior of, for example, dilute gases or liquids in micro applications. Accurately solving these models is a significant challenge among non-equilibrium scientists. One reason for this is our technical advancement nowadays that allows us to build devices smaller and smaller or to explore the universe, with its very diluted environment. The current solver considers the most straightforward case of a linear, time-independent, and two-dimensional gas flow.


The current compact implementation serves as the perfect starting point for future applications. We can now move on to successively include more submodels to ultimately solve the nonlinear, time-dependent equations in all spatial domains.

Lambert Theisen
Rheinisch Westfalische Technische Hochschule Aachen

Read the Original

This page is a summary of: fenicsR13, ACM Transactions on Mathematical Software, June 2021, ACM (Association for Computing Machinery), DOI: 10.1145/3442378.
You can read the full text:




The following have contributed to this page