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. We discretize 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. We show how the spectrum of the single domain operator is contained in the multi domain operator spectrum when artificial interfaces are introduced. We study the impact on the spectrum and discretization errors depending on the interface treatment and verify that the results are carried over to higher order accurate schemes.