Positive trigonometric polynomials for strong stability of difference equations

We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5

Servo-stabilization of combustion in rocket motors

This paper shows that the combustion in the rocket motor can be stabilized against any value of time lag in combustion by a feedback servo link from a chamber pressure pickup, through an appropriately designed amplifier, to a control capacitance on the propellant feed line. The technique of stability analysis is based upon a combination of the Satche diagram and the Nyquist diagram. For simplicity of calculation, only low-frequency oscillations in monopropellant rocket motors are considered. However, the concept of servo-stabilization and method of analysis are believed to be generally applicable to other cases

A New Domain Decomposition Method for the Compressible Euler Equations

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The basis is the equivalence via the Smith factorization with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be preserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ....).Comment: Submitte

Stabilization in relation to wavenumber in HDG methods

Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex. Numerical methods of the Hybrid Discontinuous Galerkin (HDG) type are considered for simulating waves that satisfy the Helmholtz and Maxwell equations. It is shown that these methods, when wrongly used, give rise to singular systems for complex wavenumbers. A sufficient condition on the HDG stabilization parameter for guaranteeing unique solvability of the numerical HDG system, both for Helmholtz and Maxwell systems, is obtained for complex wavenumbers. For real wavenumbers, results from a dispersion analysis are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal that some choices of the stabilization parameter are better than others. To summarize the findings, there are values of the HDG stabilization parameter that will cause the HDG method to fail for complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. When the wavenumber is real, values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors are found on the imaginary axis. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method showed that its wavenumber errors are an order smaller than those of the HDG method