Stellenbosch University
New provably energy stable formulations for hyperbolic
problems: application to the Euler and shallow water
equations
Jan Nordström
Department of Mathematics, Applied Mathematics, Linköping
University, Linköping, Sweden
Department of Mathematics and Applied Mathematics, University of
Johannesburg, South Africa
SAMS Subject Classification Number: 3, 23
We present the general stability theory for hyperbolic IBVPs developed in [1]. It extends the use of the energy method from linear to nonlinear problems, is easy to understand and leads to \(L_2\) estimates. The only requirements for an energy bound is that a specific skew-symmetric form of the equations exist and that proper boundary conditions are available. We will discuss the key steps to such a formulation and exemplify with the compressible Euler equations.
The new formulation also makes it possible to understand some confusing results obtained from linearisation, where in some case an energy bound exist for the nonlinear problem, but not for the linearised one (or vice versa) [1]. A nonlinear and linear analysis may also lead to different boundary conditions required for a bound [2]. The new formulation shed light on this confusing fact.
The new skew-symmetric formulation was shown to hold for the shallow water equations as well as for the incompressible and compressible Euler equations [1–3]. We will discuss how to determine nonlinear boundary conditions and relate that to a boundary condition analysis for linear problems.
Finally, by discretising using summation-by-parts (SBP) operators [4] which mimic integration-by-parts, we show that nonlinear stability follows automatically if boundary conditions leading to a continuous energy estimate are available.
References
[1] J. Nordström. Nonlinear and linearised primal and dual initial boundary value problems: When are they bounded? How are they connected? Journal of Computational Physics, vol 455, No 111001, 2022.
[2] J. Nordström and Andrew R. Winters Linear and nonlinear analysis of the shallow water equations, Journal of Computational Physics, Vol 463, 111254, 2022.
[3] J. Nordström, A skew-symmetric energy and entropy stable formulation of the compressible Euler equations, Journal of Computational Physics, Vol 470, 111573, 2022.
[4] M. Svärd and J. Nordström Review of Summation-By-Parts Schemes for Initial-Boundary-Value Problems. Journal of Computational Physics, Volume 268, pp. 17-38, 2014.
