A local high-order doubly asymptotic open boundary for diffusion in a semi-infinite layer

A high-order open boundary for transient diffusion in a semi-infinite homogeneous layer is developed. The method of separation of variables is used to derive a relationship between the modal function and the flux at the near field/far field boundary in the Fourier domain. The resulting equation in terms of the modal impedance coefficient is solved by expanding the latter into a doubly asymptotic series of continued fractions. As a result, the open boundary condition in the Fourier domain is represented by a system of algebraic equations in terms of iω. This corresponds to a system of fractional differential equations of degree α = 0.5 in the time-domain. This temporally global formulation is transformed into a local description by introducing internal variables. The resulting local high-order open boundary condition is highly accurate, as is demonstrated by a number of heat transfer examples. A significant gain in accuracy is obtained in comparison with existing singly-asymptotic formulations at no additional computational cost.