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.
Highlights
In the current work, the Complex Variable Boundary Element Method (CVBEM)is extended to modeling applications of the two-dimensional linear diffusion partial differential equation (PDE), uxx + uyy = ut
The steady-state component is governed by the Laplace PDE and is modeled using the Complex Variable Boundary Element Method
By resolving the global initial-boundary value problem into a steady-state component and a transient component, the CVBEM numerical technique can be applied to modeling applications of the two-dimensional linear diffusion PDE on a rectangular domain with Dirichlet boundary conditions
Summary
The methodology presented in this work is suitable for use in modeling problems in which the initial condition is equal on the boundary to the boundary conditions That is, this methodology is intended for modeling problems such that u ( x, y, 0) = f ( x, y) on Γ , where f ( x, y) represents the boundary conditions of the global BVP, and Γ is the boundary of the problem domain. This methodology is intended for modeling problems such that u ( x, y, 0) = f ( x, y) on Γ , where f ( x, y) represents the boundary conditions of the global BVP, and Γ is the boundary of the problem domain When this condition is not satisfied, this methodology can still be used, the global approximation function will not satisfy the initial condition on Γ
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