The Laplace transform boundary element method for diffusion-type problems
Abstract
Diffusion-type problems are described by parabolic partial differential
equations; they are defined on a domain involving both time and space. The
usual method of solution is to use a finite difference time-stepping process
which leads to an elliptic equation in the space variable. The major drawback
with the finite difference method in time is the possibility of severe
stability restrictions.
An alternative process is to use the Laplace transform. The transformed
problem can be solved using a suitable partial differential equation solver
and the solution is transformed back into the time domain using a suitable
inversion process. In all practical situations a numerical inversion is
required. For problems with discontinuous or periodic boundary conditions,
the numerical inversion is not straightforward and we show how to overcome
these difficulties.
The boundary element method is a well-established technique for solving
elliptic problems. One of the procedures required is the evaluation of
singular integrals which arise in the solution process and a new formulation
is developed to handle these integrals.
For the solution of non-homogeneous equations an additional technique
is required and the dual reciprocity method used in conjunction with the
boundary element method provides a way forward.
The Laplace transform is a linear operator and as such cannot handle
non-linear terms. We address this problem by a linearisation process
together with a suitable iterative scheme. We apply such a procedure to
a non-linear coupled electromagnetic heating problem with electrical and thermal properties exhibiting temperature dependencies.
Publication date
2005Published version
https://doi.org/10.18745/th.14240https://doi.org/10.18745/th.14240