New options for explicit all Mach number schemes by suitable choice of time integration methods

Many low-Mach or all-Mach number codes are based on space discretizations which in combination with the first order explicit Euler method as time integration would lead to an unstable scheme. In this paper, we investigate how the choice of a suitable explicit time integration method can stabilize these schemes. We restrict ourselves to some old prototypical examples in order to find directions for further research in this field

Hydrodynamic simulations of AGN jets:The impact of Riemann solvers and spatial reconstruction schemes on jet evolution

Numerical simulations play an essential role in helping us to understand the physical processes behind relativistic jets in active galactic nuclei. The large number of hydrodynamic codes available today enables a variety of different numerical algorithms to be utilized when conducting the simulations. Since many of the simulations presented in the literature use different combinations of algorithms it is important to quantify the differences in jet evolution that can arise due to the precise numerical schemes used. We conduct a series of simulations using the FLASH (magneto-)hydrodynamics code in which we vary the Riemann solver and spatial reconstruction schemes to determine their impact on the evolution and dynamics of the jets. For highly refined grids the variation in the simulation results introduced by the different combinations of spatial reconstruction scheme and Riemann solver is typically small. A high level of convergence is found for simulations using third-order spatial reconstruction with the Harten–Lax–Van-Leer with contact and Hybrid Riemann solvers

Contributions to the Development of Entropy-Stable Schemes for Compressible Flows

Entropy-Stable (ES) schemes have gathered considerable attention over the last decade, especially in the context of under-resolved simulations of compressible turbulent flows, where achieving both high-order accuracy and robustness is difficult. ES schemes provide stability in a nonlinear and integral sense: the total entropy of the discrete solution can be made non-decreasing, in agreement with the second principle of thermodynamics. Additionally, the amount of entropy produced by the scheme is known and can be modified, making room for analysis and improvements. This thesis delves into some of the challenges currently limiting their use in practice. The current state of the art solves the compressible Navier-Stokes equations for a single-component perfect gas in chemical and thermal equilibrium. This model is inappropriate in aerospace engineering applications such as hypersonics and combustion, which typically involve chemically reacting gas mixtures far from equilibrium. As a first step towards enabling their use for these applications, we formulated ES schemes for the multicomponent compressible Euler equations. Special care had to be taken as we found out that the theoretical foundations of ES schemes begin to crumble in the limit of vanishing partial densities. The realization that ES schemes can only go as far as their theory led us to review some of it. A fundamental result supporting the development of limiting strategies for high-order methods is the minimum entropy principle for the compressible Euler equations. It states that the specific entropy of the physically relevant weak solution does not decrease. We proved that the same result holds for the specific entropy of the gas mixture in the multicomponent case. While entropy-stability is a valuable property, it does not imply a well-behaved solution. One must recall that the second principle is a prescription on the correct behavior of a system at the global level only. To better understand how ES schemes may or may not improve the quality of the numerical solution, we revisited two classical problems encountered in the development of shock-capturing techniques. First, we studied the receding flow problem, which is a simple setup used to study the anomalous temperature rise, termed "overheating", typically observed in shock reflection and shock interaction calculations. Previous studies showed that the anomaly can be cured if conservation of entropy is enforced, but at the considerable price of total energy conservation. Entropy-Conservative (EC) schemes, a particular instance of ES schemes, can achieve both simultaneously and therefore appeared as a potential solution. We showed that while the overheating is correlated to entropy production, entropy conservation does not necessarily prevent it. Second, we studied the behavior of ES schemes in the low Mach number regime, where shock-capturing schemes are known to suffer from severe accuracy degradation issues. A classic remedy to this problem is the flux-preconditioning technique, which consists in modifying artificial dissipation terms to enforce consistent low Mach behavior. We showed that ES schemes suffer from the same issues and that the flux-preconditioning technique can improve their behavior without interfering with entropy-stability. Furthermore, we demonstrated analytically that these issues stem from an acoustic entropy production field which scales improperly with the Mach number, generating spatial fluctuations that are inconsistent with the equations. An important outgrowth of this effort is the discovery that skew-symmetric dissipation operators can alter the way entropy is produced or conserved locally.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155304/1/gouasmia_1.pd