Exact discrete nonlocal boundary conditions for high-order Padé parabolic equations

Abstract

The nonlocal boundary conditions (NLBCs) for high-order finite-difference parabolic equations (PEs) are obtained by Z transformation of the discrete PE in a homogeneous medium. The considered NLBCs include the free-space radiation condition, possibly with a density jump at the NLBC interface, the NLBC at an arbitrary impedance interface, and the NLBCs for sources and the starting field beyond the NLBC interface. The derivation is presented for the multiterm Padé PE model OWWE (one-way wave equation), but the developed technique is applicable to a broad class of finite-difference PEs. The obtained NLBCs are exact for the given finite-difference scheme. They are not limited by the order of Padé approximation or by the PE steps in range and depth. The NLBC convolution coefficients are calculated by the numerical inverse-Z transformation. The accuracy and performance of the algorithm are analyzed for several benchmark problems. The solution is robust for the range steps over 25 wavelengths and/or the approximations up to the tenth order. The NLBCs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution. Precomputation of the NLBC coefficients may be required in time-demanding applications. The results are compared with earlier proposed NLBCs for high-order PEs.