Narrow Results

It is shown that a process running in a dissipative system described by continuum equations may be organized so that the system under compression and heating will pass through a sequence of states the same as those occur in this system under rarefaction and cooling, but in the reverse time order. In the model reconstructed for such a process, the boundary regimes on the piston have a special form where the adiabatic exponent γ in the “forward” and “backward” motion of the medium are connected by a certain relation.

We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number $\approx 1.86$, below which profiles are known by energy estimates to be stable, to the upper stability boundary of $\approx 1669$, above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.

The aim of the present study is to calculate the process of detonation combustion of gas mixtures in engines. Development and verification of 3D transient mathematical model of chemically reacting gas mixture flows incorporating hydrogen was performed. Development of a computational model based on the mathematical one for parallel computing on supercomputers incorporating CPU and GPU units was carried out. Investigation of the influence of computational grid size on simulation precision and computational speed was performed. Investigation of calculation runtime acceleration was carried out subject to variable number of parallel threads on different architectures and implying different strategies of parallel computation.

In this short note, we consider smooth solutions to certain hyperbolic systems of equations. We present a condition which will ensure that no shocks develop and that solutions decay in L². The condition is restrictive in general; however, when applied to the system of one-dimensional gas dynamics it is shown that if the condition is satisfied initially then it will be satisfied for all time and therefore one obtains smooth solutions which decay.

Consider 1D flow of a compressible, ideal, and polytropic gas on a bounded interval in Lagrangian variables. We study the Cauchy problem when the initial data consist of four constant states that yield two contact waves bounding an interval of lower density, together with an admissible shock between them. To render the solution tractable for direct calculations, we also impose absorbing boundary conditions, at fixed locations (in Lagrangian coordinates) to the left and to the right of the two contacts. By estimating the wave strengths in shock–contact interactions, we show that the resulting flow is defined for all times. In particular, the pressure, density, particle velocities, and shock speeds are all uniformly bounded in time. We also record a scaling invariance of the system and comment on its relevance to large data solutions of the Euler system.

We derive explicit formulae for spherically symmetric solutions to the system of multidimensional zero-pressure gas dynamics and its adhesion approximation. The asymptotic behavior of the explicit solutions of the adhesion approximation is studied here. We observe that the radial components of the velocity and density satisfy a simpler equation, which enables us to get explicit formulae for different types of domains and study its asymptotic behavior. A class of solutions to the inviscid system with conditions on the mass instead of conditions at origin is also analyzed here.

Astrophysical problems such as modelling of core-collapse supernovae, collapse of protostellar clouds as well as other processes, involving collapsing matter, deal with regions (e.g. protostars, protoneutron stars), where a speed of sound has much larger values, than in remaining parts of a computational domain. A time-step in explicit numerical schemes, thus, has to be bounded by acoustic Courant-Friedrichs-Lewy condition, due to high speed of sound in these compact regions. In some cases, this condition can be very restrictive, and (semi-) implicit numerical schemes may outperform the explicit ones. We propose a semi-implicit solver on a collocated mesh for self-gravitating gas dynamical flows, in which only acoustic waves are treated implicitly. We use an operator-difference approach to construct difference analogues of vector differential operators on unstructured meshes in two and three dimensions, which allows us to save the conjugacy properties of the operators. A Rusanov-type dissipation was used to get monotonic flow profiles and usual linear flux reconstruction to improve an order of spatial approximation. Results of test calculations are presented.