We develop a new approach to finding bifurcations of solutions of nonlinear equation systems, which is based on the detection of extreme values of a new type of variational functional associated with the considered problems. The variational functional is obtained constructively by the extended functional method which can be applied to a wide class of parametric problems including nonlinear partial differential equations. Sufficient and necessary conditions for the existence of a maximal turning point by the approach are proved. Based on these, an algorithm of the quasi-direction of steepest ascent is introduced. Simulation experiments are used to illustrate the behaviour of the method and to discuss its advantages and disadvantages in comparison with the alternatives.

Computation of maximal turning points to nonlinear equations by nonsmooth optimization

Self-similar problems of the compression of an ideal gas and its dispersion from a point☆

The solutions of initial and boundary value problems of the outflow of an ideal (inviscid and non-heat-conducting) gas from cylindrical and spherical sources into a vacuum are obtained. Time is measured from the moment, when the source is turned on; at this moment the source is surrounded by a vacuum. The entropy, flow rate, and the Mach number of the gas outflowing from the source are given, together with the source radius; the Mach number can be greater of or equal to unity. If the source radius is greater than zero, then the flow domain in the “radial coordinate–time” plane consists of the stationary source flow and adjoining non-self-similar centered expansion wave consisting of C−-characteristics. The stationary flow is described by the known formulas, while the expansion wave is calculated by the method of characteristics. The calculations by this method confirm the earlier obtained laws for large values of the radial coordinate. The interface between the vacuum and the expansion wave is the straight trajectory of particles and, at the same time, a unique rectilinear C−-characteristic. For the source of zero radius (“pointwise” source) the velocity, density, and speed of sound of the outflowing gas are infinite. The gas velocity remains infinite everywhere, while the density and speed of sound become zero for any non-zero values of the radial coordinate. For the pointwise source the problem of outflow into a vacuum is self-similar. In the plane of the “self-similar” velocity and speed of sound its solution is given by three singular points of a differential equation in these variables. At one of these points the self-similar velocity is infinite, the self-similar speed of sound is zero, and the self-similar independent variable varies from zero to infinity, with the exception of the extreme values.

Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum

Simplification of Numerical and Analytical Tools for Sonic Boom Description

The Collatz–Wielandt formula obtained by Lothar Collatz (1942) and Helmut Wielandt (1950) provides a simple variational characterization of the Perron–Frobenius eigenvalue of certain types of matri...

Finding Saddle-Node Bifurcations via a Nonlinear Generalized Collatz–Wielandt Formula

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio ? (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that ? on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such ?, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.

Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

The stability of one-dimensional steady flows of an ideal (inviscid and non-heat-conducting) gas in channels of variable cross section with combustion in a structurally stable detonation wave is studied. The detonation wave represents a discontinuity surface propagating normally to the axis of the channel. It was previously established that, in this formulation, steady flows with combustion in a Chapman–Jouguet detonation wave are always unstable, in contrast to combustion flows in an overcompressed detonation wave. The stability analysis of such flows is reduced to the numerical solution of an initial-boundary value problem describing the evolution of finite flow perturbations between the moving detonation wave and the minimal nozzle cross section (in the case of a sudden contraction) or the exit nozzle cross section. The problem is solved by applying a modified Godunov scheme of higher order accuracy. More specifically, a Riemann solver with switching (between the overcompressed and Chapman–Jouguet detonation waves) with an explicitly represented detonation wave is created. Examples of stable flows and flows collapsing due to large initial perturbations are given, and their dynamics with transitions to a Chapman–Jouguet detonation wave and back are computed.

Stability of One-Dimensional Steady Flows with Detonation Wave in a Channel of Variable Cross-Sectional Area

For an unsteady spherical rarefaction wave in an ideal (inviscid and non-heat-conducting) gas, the features of the flow near the reflection of the first characteristic from the center of symmetry are investigated. Computations performed by the method of characteristics on nearly uniform grids usually used in such problems reveal sawtooth irregularities in the parameter distributions near the reflection point, whereas similar irregularities in the cylindrical and plane cases are absent. The amplitudes of the irregularities and the sizes of the domains where they are observed remain nearly unchanged when the number of points of the characteristic grid is increased by many times. Away from the reflection point in both time and space, the numerical solution completely “forgets” about the irregularities. This finding explains why these irregularities were ignored or overlooked earlier, but the nature of this phenomenon remains an open question. The present study has established that the spherical rarefaction flow near the reflection point differs fundamentally in structure from its plane and cylindrical counterparts. In the spherical case, the rarefaction flow near the reflection point was found to be nearly conical (entirely conical in the linear approximation). Allowance for this feature in the method of characteristics led to continuous regular distributions of the parameters. The performed analysis and computations revealed that a spherical rarefaction wave is strengthened (cumulates) theoretically unlimitedly in a small neighborhood of the reflection point (center of symmetry) of the first characteristic. Moreover, the claim that a gradient catastrophe occurs in this neighborhood was found to be untenable.

Reflection of a Rarefaction Wave from the Center of Symmetry: Theoretical Analysis of the Flow Features and Calculation by the Method of Characteristics

Kh. F. Valiyev

Papers

Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

Stability of One-Dimensional Steady Flows with Detonation Wave in a Channel of Variable Cross-Sectional Area

Reflection of a Rarefaction Wave from the Center of Symmetry: Theoretical Analysis of the Flow Features and Calculation by the Method of Characteristics