The Laplace transform boundary element methods for diffusion problems with periodic boundary conditions

摘要

The Laplace transform has been shown to be well-suited to the solution of diffusion problems and provides an alternative to the finite difference method. Such problems, parabolic in time, are transformed to elliptic problems in the space variables. Any suitable solver may be used in the space domain and a numerical inversion of" the transform is then performed. For parabolic problems the Stehfest numerical method has been shown to be accurate, robust and easy to implement. The boundary element method has been used by a variety of authors to solve the resulting elliptic problem. The initial conditions lead to a non-homogenous Helmholte-type problem which may be solved using the dual reciprocity method. Time-dependent boundary conditions are, in principle, easily implemented. However, problems can occur if the conditions are not monotonic in time. The authors have already considered problems with a discontinuity in the boundary condition and have shown that the Laplace transform can be used to find the solution up to the discontinuity and then, using the computed solution as a new initial condition, to proceed past the continuity. Similarly, for boundary conditions with period 2T, the Laplace transform is used in time intervals of length (1/2)T, where the boundary condition is monotonic, and the computed solution at time (1/2)T is used to Wove onto the next monotonic phase.