Well-Posedness, Stability and Conservation for a Discontinuous Interface Problem: An Initial Investigation
DOI: 10.1007/978-3-319-19800-2_11
- Cite this paper as:
- La Cognata C., Nordström J. (2015) Well-Posedness, Stability and Conservation for a Discontinuous Interface Problem: An Initial Investigation. In: Kirby R., Berzins M., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham
Abstract
A robust interface treatment for the discontinuous coefficient advection equation satisfying time-independent jump conditions is presented. The aim of the investigation is to show how the different concepts like well-posedness, conservation and stability are related. The equations are discretized using high order finite difference methods on Summation By Parts (SBP) form. The interface conditions are weakly imposed using the Simultaneous Approximation Term (SAT) procedure. Spectral analysis and numerical simulations corroborate the theoretical findings.
1 Introduction
In this paper we study fundamental properties such as well-posedness, stability and conservation for an advection equation, which changes wave-speed at the interface separating two spatial domains. The solution satisfies a time-independent jump-condition, which makes it discontinuous. The first goal is to show that for any piecewise constant advection velocity and interface jump condition the continuous problem is always well-posed. We provide a straightforward condition for checking conservation despite the presence of discontinuities. Applications where this is of interest include acoustic electromagnetism, seismology and fluid dynamics, [6, 9, 11, 12].
Stability and conservation at interfaces have also been studied in [2, 3, 4] for the case of identical velocities in the two domains. We extend this investigation by showing how well-posedness, conservation and stability are related in a more general setting. SBP-SAT schemes, [1, 13, 14], up to fifth order of accuracy are used to exemplify that the interface treatment is stable and accurate for all theoretically meaningful cases.
2 The Discontinuous Interface Problem
2.1 Well-Posedness and Conservation
Proposition 1
The interface problem defined by the coupled equations (1) is well-posed for any positive a,b and any constant\(c \in \mathbb{R}\).
Proof
Proposition 2
Proof
3 The Semi-discrete Approximation
3.1 Stability and Conservation Properties of the Semi-discrete Approximation
Proposition 3
One can also prove, see [8], that
Proposition 4
The conditions in (9) imply that\(P^{-1}\tilde{Q}\)has eigenvalues with negative semi-definite real parts.
As in the continuous case we rewrite (5) in a weak formulation to derive the conservation condition. We obtain the following discrete conservation criteria
Proposition 5
The conservative approximation requires a conservative continuous problem.
4 The Relation Between Stability and Conservation
In this section we present explicit stability condition for the penalty coefficients \(\sigma _{L,R}\) for different type of continuous problems and approximations. All the conditions are algebraically derived from (9).
Proposition 6
Proposition 7
with \(\theta = b/(a\alpha _{d}) \geq 1\).
Note that conservation and stability are two independent properties of the approximation (5). We have a stable and non-conservative semi-discretization if the assumptions of Proposition 6 are satisfied. Note also that for one norm, the stability requirements in Proposition 6 also lead to conservation. That norm is given by \(\alpha _{d} = b/a\).
5 Numerical Results
In order to show the effect of the interface treatment we must restrict ourselves to a finite spatial domain, we choose [−1, 1].
5.1 Order of Accuracy
L^{2} | SBP21 | SBP42 | SBP63 | SBP84 | |||||
---|---|---|---|---|---|---|---|---|---|
N | u_{l} | u_{r} | u_{l} | u_{r} | u_{l} | u_{r} | u_{l} | u_{r} | |
80 | 2.0124 | 2.0267 | 3.0397 | 3.0096 | 3.6801 | 3.8770 | 4.5847 | 5.0897 | |
160 | 2.0086 | 2.0102 | 3.0713 | 3.0083 | 3.8480 | 4.0149 | 4.7510 | 5.0624 | |
320 | 2.0059 | 2.0044 | 3.0359 | 3.0068 | 3.9590 | 4.0052 | 4.9033 | 5.0176 |
5.2 The Spectrum
6 Conclusions
We have presented a complete analysis of the discontinuous coefficient interface problem. We have shown that a such problem is always well-posed and we have investigated when it is conservative. We have derived a stable SBP-SAT scheme which can be made conservative or non-conservative depending on our choice. We have also shown that the approximation can be made strictly stable by adding artificial dissipation without reducing the accuracy.