Joint Seminar with Department of Mathematics

Stability of Strong Viscous Shock Layers in an Ideal Gas

By a combination of asymptotic ODE estimates and numberical Evans function computations, we examine the spectral stability of shock-wave solutions of the compressible Navier--Stokes equations with ideal gas equation of state, for arbitrary strength waves.

Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave, an analytic function analogous to the characteristic polynomial whose zeros correspond to eigenvalues of L, (i) converges in the strong shock limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one endstate; and (ii) has no unstable (positive real part) zeros outside a uniform ball. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that may be conveniently studied numerically. An intensive numerical study then yields unconditional stability, independent of amplitude, for a range of parameter values including all common gases.

Besides its physical interest, we believe that this analysis has interest as an example where it is possible to carry out a rigorous globl stability analysis by numerical techniques, the obvious obstace being the need to treat an unbounded parameter range using finitely many operations.