The logic we are going to employ to survey the similarity structure of several kinds of PDEs is the following. The preliminary requirement is that the problem under consideration should be well-posed, or the uniqueness should be guaranteed at least. Let u(x, t) be a solution to such a problem and consider a rescaled function v given by u(x, t) = av(bx, ct), a, b, c > 0. (1.1) The main step is t...

The general d-dimensional Riemann problem raises naturally the question of resolving the interaction of d planar shocks merging at a point. In gas dynamics, we may consider only standing shocks. This problem has received a satisfactory answer in dimension d = 2 (see [2, 3]). We investigate the 3-dimensional case. We restrict to the irrotational case, in order to keep the complexity of the solut...

In this paper we answer some open questions concerning totally degenerate systems of conservation laws. We study the augmented Born-Infeld system, which is the Born-Infeld model augmented by two additional conservations laws. This system is a nice example of totally degenerate system of conservation laws and, global smooth solutions are conjectured to exist when the initial-data is smooth. We s...

Polygonal hybrid systems (SPDIs) are planar hybrid systems, whose dynamics are defined in terms of constant differential inclusions, one for each of a number of polygonal regions partitioning the plane. The reachability problem for SPDIs is known to be decidable, but depends on the goodness assumption — which states that the dynamics do not allow a trajectory to both enter and leave a region th...

We present a new expansion of the zeta-function of Riemann. It is given by the formula (1) below. The current formalism – which combines both the idea of interpolation with constraints and the concept of hypergeometric functions – can, in a natural way, be generalised within the theory of the zetafunction of Hawking offering thus a variety of applications in quantum field theory, quantum cosmol...

An approximate solution of the Riemann problem associated with a realisable and objective turbulent second-moment closure, which is valid for compressible flows, is examined. The main features of the continuous model are first recalled. An entropy inequality is exhibited, and the structure of waves associated with the non-conservative hyperbolic convective system is briefly described. Using a l...

In this work, two extensiosn of our previous work are proposed for the simulation of full-Mach-range cavitating flow with sharp interface model. These extensions are: a new interface-interaction solver, which includes a new HLLC Riemann solver suitable for both strong and weak interface interaction with general EOS, and a simple phase-change model which assumes that the phase change is in therm...

We present a theoretical solution for the Riemann problem for the five-equation two-phase non-conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non-conservative methods to solve the general initial-boundary value problem for the two-phase flow model in non-conservative form. The basic upwind scheme constructed is the non-conservative analogue...

In this paper we extend the fluctuation-splitting method by allowing the nonvanishing fluctuations to drive the vertex placements as well as the vertex values. For the generalised Cauchy-Riemann (PrandtlGlauert) equations, a simple least-squares minimisation carried out with respect to u, v, x, y as unknowns creates a grid responsive to the physics of the solution, if the objective function is ...

We discuss the procedure for the exact solution of the Riemann problem in special relativistic MHD. We consider both initial states leading to a set of only three waves analogous to the ones in relativistic hydrodynamics, as well as generic initial states leading to the full set of seven MHD waves. Because of its generality, the solution presented here could serve as an important if not indispe...