| Popis: |
The global and local stability of an nth‐order time‐domain paraxial approximation (TDPAn) to an acoustic wave equation [M. D. Collins, J. Acoust. Soc. Am. 86, 1097–1102 (1989)] is resolved. An operator splitting technique and the Trotter product formula are used to determine the conditional (i.e., parameter dependent) stability of the complete paraxial operator from the stability of the individual diffractive, refractive, and dissipative components. The local stability of a finite difference implementation of the TDPAn is analyzed using Von Neumann and matrix methods. The Von Neumann analysis, which provides a necessary (but not sufficient) stability condition, suggests that the algorithm is unconditionally stable. The matrix method provides two requirements. A necessary condition for stability on the spectral radius of the amplification matrix W is unconditionally satisfied. However, a sufficient condition on the norm ‖W‖2 is only conditionally satisfied. Consequently, the parameter and step‐size values ... |
