Convergence of energy stable finite-difference schemes with interfaces

Magnus Svärd,Jan Nordström

doi:10.1016/j.jcp.2020.110020

Abstract

We extend the convergence results in Svärd and Nordström (2019) [7] for single-domain energy-stable high-order finite difference schemes, to include domains split into several grid blocks. The analysis also demonstrates that reflective boundary conditions enjoy the same convergence properties. Finally, we briefly indicate that these results (and the previous ones in [7]) also hold in multiple dimensions.

Highlights

The convergence rate of finite-difference schemes that are closed at the boundaries with stencils of lower order accuracy1 than the interior stencils is a long-standing problem that has been treated extensively in the literature in e.g. [3,4,1,5]
In the theory for boundary conditions, the semi-discretisation of the initial-boundary value problem is Laplace transformed. This allows a decomposition of the modes into those that decay into the domain from the boundary, and should be supplied with boundary conditions, and those that do not
The programme for analysing interface accuracy is to view the interface as an external boundary and, with the aid of the requirements above, the a priori estimates and previous theory, infer the optimal convergence rates

Summary

Background

The convergence rate of finite-difference schemes that are closed at the boundaries with stencils of lower order accuracy than the interior stencils is a long-standing problem that has been treated extensively in the literature in e.g. [3,4,1,5]. The boundedness of the boundary data and the energy stability of the scheme, enabled a proof of the convergence result This proof, does not immediately encompass interfaces, since they take data from the conjoining domain, which may or may not be sufficiently bounded. The previous theory did not consider reflective boundaries, i.e., those that feed out-going waves back into the domain via the in-going characteristics, since sufficient boundedness of the solution that is fed back has to be established We address both interfaces and reflective boundary conditions and we discuss possible generalisations to multiple dimensions. The programme for analysing interface accuracy is to view the interface as an external boundary and, with the aid of the requirements above, the a priori estimates and previous theory, infer the optimal convergence rates We will demonstrate this for a few examples. The generalisations are obvious and will only be discussed briefly

The hyperbolic case

Reflective boundary conditions

The parabolic case

Multi-dimensions