Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains

In this paper we study the heat and advection equation in single and multiple domains. The equations are discretized using a second order accurate finite difference method on Summation-By-Parts form with weak boundary and interface conditions. We derive analytic expressions for the spectrum of the continuous problem and for their corresponding discretization matrices. It is shown how the spectrum of the single domain operator is contained in the multi domain operator spectrum when artificial interfaces are introduced. The interface treatments are posed as a function of one parameter, and the impact on the spectrum and discretization error is investigated as a function of this parameter. Finally we briefly discuss the generalization to higher order accurate schemes.