Numerical testing of the stability of viscous shock waves

Abstract

Abstract.A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability. Accuracy of the method is demonstrated through comparison against exact solutions, a convergence study, and evaluation of approximate error equations. Robustness is demonstrated by applying the method to waves for which no current analytic results apply (highly nonlinear waves from the cubic model and strong shocks from gas dynamics). An interesting aspect of the analysis is the need to incorporate features from the analytic Evans function theory for purposes of numerical stability. For example, we find it necessary, for numerical accuracy, to solve ODEs on the space of wedge products.

Galloping instability of viscous shock waves

“…rectangular) crosssection; "spinning" instabilities for a circular cross-section, with one or more "hot spots", or "combustion heads", moving spirally along the duct. Stability of shocks and detonations may be studied within a unified mathematical framework; see, e.g., [Er1,Er2,Ko1,Ko2,D,BT,FD,LS,T,K,M1,M2,M3,FM,Me,CJLW] in the inviscid case (1.1), (1.5), and [S,Go1,Go2,KM,KMN,MN,L1,L3,GX,SX,GZ,ZH,Br1,Br2,BrZ,BDG,KK,Z1,Z2,Z3,Z4,MaZ2,MaZ3,MaZ4,GMWZ1,GWMZ2,GMWZ3,HZ,BL,LyZ1,LyZ2,JLW,…”

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

“…D has precisely one zero in Re ≥ 0 necessarily at = 0 and D a 0 = 0 While condition is generally quite difficult to verify analytically (see, for example, Jones, 1984 in the context of the FitzHugh-Nagumo equations, and Dodd, 1996;Freistuhler and Szmolyan, 2002;Goodman, 1986;Humpherys and Zumbrun, 2002;Kawashima and Matsumura, 1985;Kawashima et al, 1986;Nishihara, 1985, andPlaza and in the case of conservation laws), it can be checked numerically (see Brin, 2001;Oh and Zumbrun, 2003). A condition that lends itself more readily to exact study is the stability index, typically defined as…”

Grassmannian spectral shooting

“…They solved the flow for the corresponding linear vector field in the higher-dimensional Plücker embedding space. Details of this Plücker coordinate or compound matrix approach can be found in, for example, Alexander and Sachs [3], Brin [16,17] and Allen and Bridges [5]. Unfortunately the number of Plücker coordinates typically grows exponentially with the order of the original system, and so this approach cannot be used for medium to large order systems.…”