Analytic discrete transparent boundary conditions for high-order Padé parabolic equations

Exact discrete Nonlocal Boundary Conditions (NLBCs) for high-order Padé PEs were obtained in an earlier paper [D. Yu. Mikhin, Exact discrete nonlocal boundary conditions for high-order Padé parabolic equations, J. Acoust. Soc. Am., 116 (2004), 2864-2875] using the Z transformation technique. The Z-transformed NLBC coefficients were found analytically as a product of several matrices, while the coefficients in the coordinate domain were obtained numerically by inverse Z transformation. This paper presents the analytical inverse of the matrix composed of Padé partial fractions that leads to closed-form analytic expressions for the elements of the Z-transformed NLBC matrix. The NLBC in the coordinate space are then obtained by direct decomposition of the matrix elements into their Laurent series. The new analytic solution allows calculating the elements of the Laurent series up to significantly higher order and with superior accuracy compared to the previous numerical technique. The analytic solution is also faster within the common domain of applicability. The results were generalized to other high-order PE algorithms and also to the NLBC at an interface with a density jump.