An Unsteady Two-Dimensional Complex Variable Boundary Element Method

Abstract

The Complex Variable Boundary Element Method (CVBEM) procedure is extended to modeling applications of the two-dimensional linear diffusion partial differential equation (PDE) on a rectangular domain. The methodology in this work is suitable for modeling diffusion problems with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The underpinning of the modeling approach is to decompose the global initial-boundary value problem into a steady-state component and a transient component. The steady-state component is governed by the Laplace PDE and is modeled using the Complex Variable Boundary Element Method. The transient component is governed by the linear diffusion PDE and is modeled by a linear combination of basis functions that are the products of a two-dimensional Fourier sine series and an exponential function. The global approximation function is the sum of the approximate solutions from the two components. The boundary conditions of the steady-state problem are specified to match the boundary conditions from the global problem so that the CVBEM approximation function satisfies the global boundary conditions. Consequently, the boundary conditions of the transient problem are specified to be continuously zero. The initial condition of the transient component is specified as the difference between the initial condition of the global initial-boundary value problem and the CVBEM approximation of the steady-state solution. Therefore, when the approximate solutions from the two components are summed, the resulting global approximation function approximately satisfies the global initial condition. In this work, it will be demonstrated that the coupled global approximation function satisfies the governing diffusion PDE. Lastly, a procedure for developing streamlines at arbitrary model time is discussed.