Discrete transparent boundary conditions for parabolic systems

In this work we construct and analyse transparent boundary conditions (TBCs) for general systems of parabolic equations. These TBCs are constructed for the fully discrete scheme (θ-method, finite differences), in order to maintain unconditional stability of the scheme and to avoid numerical reflections. The discrete transparent boundary conditions (DTBCs) are discrete convolutions in time and are constructed using the solution of the Z-transformed exterior problem. We will analyse the numerical error of these convolution coefficients caused by the inverse Z-transformation. Since the DTBCs are non-local in time and thus very costly to evaluate, we present approximate DTBCs of a sum-of-exponentials form that allow for a fast calculation of the boundary terms. Finally, we will use our approximate DTBCs for an example of a fluid stochastic Petri net and present numerical results.