Fluid mechanics has proven to be remarkably successful in describing a wide variety of substances. Of these, the quark-gluon plasma is of particular interest, as experiments are now sensitive to dissipative effects typically neglected in relativistic models. This fact, combined with indications that similar effects might also impact the dynamics of neutron star mergers, provides strong motivation for the development of relativistic fluid theories with dissipation.

In this talk, I will review the modern interpretation of fluid mechanics as a gradient expansion about equilibrium, and define two fluid theories which include dissipation: Muller-Israel-Stewart (MIS) theory, the standard in the field, and the promising alternative known as BDNK theory. The main difference between the two is that BDNK arises at lower order in the gradient expansion, rendering it simpler in many respects than MIS and allowing for easier control over the so-called hydrodynamic frame, a property crucial to ensuring the theory be free of pathologies. To capitalize on these advantages, one must solve a series of technical problems in designing a numerical scheme for the BDNK equations. I will review these challenges and their solutions, and evaluate our BDNK solver on a suite of standard tests. To conclude, I will provide a roadmap of steps to generalize our approach for applications in nuclear physics and astrophysics.