Absorbing boundary conditions for diffusion equations
In dealing numerically with problems for unbounded regions, such as radiation problems, the reduction to bounded regions with artificial boundary conditions has received much attention, particularly in the hyperbolic case. The present paper deals with such reductions for diffusion equations \(Lu = 0\) (the heat equation \(\partial_t u - \Delta u = 0\) being a typical example) in a region with a smooth boundary \(\mathbb{R}_t \times S\). The Neumann operator \(N\), which yields the exterior normal derivative \(\partial u/ \partial v = - Nu\) on the boundary in terms of the boundary data of \(u\) itself for a solution of the equation, is studied. Its properties as a pseudodifferential operator --- in particular the leading terms of its symbol --- are discussed. For the heat operator \(N = (\partial_t - \Delta_S)^{1/2}\), \(\Delta_S\) being the Laplacian on \(S\). In general the actual computation of \(Nu\) is a problem from the numerical point of view. Construction of simple approximants are suggested, using only one or two leading terms of the symbol and rational approximants \[ z^{1/2} \approx \alpha z + \beta + \sum^m_{k = 1} \alpha_k z/(z + d_k), \] where, for the sake of stability, \(\alpha \geq 0\), \(\beta \geq 0\), \(\alpha_k > 0\), \(d_k > 0\). One of the two alternatives which are studied is \[ (\partial_t - \Delta_S)^{1/2} \approx \alpha (\partial_t - \Delta_S) + \sum^m_{k = 1} \alpha_k (\partial_t - \Delta_S) (\partial_t - \Delta_S + d_k)^{-1}. \] The existence of rational approximations of the type given above are discussed. No convergence proof is given but it is shown that the approximation has a smoothing property.
scientific article; zbMATH DE number 802461