*Simone Chiocchetti, Michael Dumbser, Ilya Peshkov*

The aim of this research project is the construction of accurate and efficient numerical schemes for the solution of systems of Partial Differential Equations (PDE) describing compressible and weakly compressible multiphase flows. The topic is relevant in the study of combustion phenomena, jet atomisation, droplet transport and the methods developed in this work can be made available to a wider range of applications, for example geophysics [4] or non-newtonian fluids [6]. Such generality is aided by our modeling choice of employing a single first order hyperbolic PDE system [3, 7] that is able of describing the behaviour of both solid and fluid media, by simply changing the parameters characterising the material being studied. This allows to simulate a very wide variety of continua (elastic solids, viscous fluids, visco-elasto-plastic materials in general) within a unified framework.

To illustrate the generality of our framework, we report an anecdote regarding a numerical solver [1] that we engineered with the purpose of handling stiff Ordinary Differential Equations describing finite-rate pressure/velocity relaxation, which are necessary for modelling compressible two-phase flows with PDEs of the Baer–Nunziato type. The solver was later repurposed for an entirely different task in solid mechanics, even though it was initially designed as a single-purpose scheme. Pressure/velocity relaxation equations are reminiscent of ODEs used in chemistry for the description of reaction processes involving different time scales and strong nonlinearities such as thermal activation thresholds and they share this commonality with the damage kinetics for a diffuse-interface model of elasto-plastic solids with material failure, object of some recent works [4, 8]. Reliably and efficiently solving such equations in the context of PDE codes is a challenging task, which often requires the development of ad-hoc, equations-specific techniques. Indeed, our novel methodology did take advantage of the peculiar structure of the relaxation equations for which it had been developed, but we were able to readily extend it to handle the aforementioned damage kinetics, a different task altogether, with only relatively small modifications.

Up to now, particular attention has been devoted to the treatment of the so-called involution constraints that must be satisfied when numerically solving some PDE systems, like the first order hyperbolic reduction systems that we employ. Involutions are additional PDEs that are implied by the governing equations but due to accumulation of numerical errors, they may be not satisfied by the discrete solutions. A well known example is the divergence-free condition of the magnetic field in Maxwell’s equations of electromagnetism. We employed both GLM-type cleaning techniques [5], as well as structure preserving schemes that, by construction, satisfy the constraints exactly.

Future work will focus on the extension of the general unified model of continuum mechanics to multiphase flows with phase change, as well as to the diffuse interface treatment of fluid-solid interaction.

#### Bibliography

*[1] S. Chiocchetti and C. Müller. A Solver for Stiff Finite-Rate Relaxation in Baer-Nunziato Two-Phase Flow Models. Fluid Mechanics and its Applications, 2020.
[2] S. Chiocchetti, I. Peshkov, S. Gavrilyuk, and M. Dumbser. High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension. Journal of Computational Physics, 2020.
[3] M. Dumbser, I. Peshkov, E. Romenski, and O. Zanotti. High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids. Journal of Computational Physics, 2016.
[4] A. Gabriel, D.Li, S. Chiocchetti, M. Tavelli, I. Peshkov, E. Romenski, and M. Dumbser. A unified first order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones. Philosophical Transactions of the Royal Society A, 2021.
[5] C. Munz, P. Omnes, R. Schneider, E. Sonnendrücker, and U. Voss. Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model. Journal of Computational Physics, 2000.
[6] I. Peshkov, M. Dumbser, W. Boscheri, E. Romenski, S. Chiocchetti, and M. Ioriatti. Modeling solid-fluid transformations in non-Newtonian viscoplastic flows with a unified flow theory. Computers and fluids, 2021.
[7] I. Peshkov and E. Romenski. A hyperbolic model for viscous Newtonian flows. Continuum Mechanics and Thermodynamics, 2016.
[8] M. Tavelli, S. Chiocchetti, E. Romenski, A. Gabriel, and M. Dumbser. Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure. Journal of Computational Physics, 2020.*