Stability analysis of higher-order time-domain paraxial equations

The global and local stability of a one-way, nth-order, time-domain paraxial equation (TDPEn) to an acoustic wave equation [M. D. Collins, J. Acoust. Soc. Am. 86, 1097–1102 (1989)] is resolved. An operator splitting technique is used to determine the conditional (i.e., parameter-dependent) stability of the complete paraxial operator from the stability of individual diffractive, refractive, and dissipative components. The local stability of a finite difference implementation for this TDPEn is analyzed using Von Neumann and matrix methods. It is significant that the former method provides only a necessary, but not sufficient, stability condition for the initial-boundary value problem considered here. The matrix method provides two requirements. A necessary condition for stability on the spectral radius of the amplification matrix Wn is unconditionally satisfied. However, a sufficient condition on the norm ∥Wn∥2 is only conditionally satisfied. Consequently, the parameter and step-size values required for use of the algorithm are determined. These results are illustrated with numerical examples, and implications for use of the TDPEn are described. The stability of an alternate time-marching TDPEn is also examined.