# Professor Jan Nordström

"Provably Energy Stable Approximations of Linear and Nonlinear Hyperbolic Problems"

"Provably Energy Stable Approximations of Linear and Nonlinear Hyperbolic Problems"

Professor Jan Nordström is head of the Division of Computational Mathematics at the Department of Mathematics, Linköping University, Sweden. His main interest is Initial Boundary Value Problems (IBVPs), and in particular the fundamental effect of boundary and interface conditions on well-posedness and stability. Nordström has pioneered the use of summation-by-parts operators and weak boundary conditions for very high order accurate difference methods. He stresses the necessity to understand the IBVP during the development of numerical approximations.

He has published more than 160 journal papers, 90 conference reports, 14 book chapters and 1 book. He was main advisor for 18 PhD students (7 at Uppsala University and 11 at Linköping University, 2 with double degree at Stanford University) and currently supervise 3. He is a Senior Research Fellow at the Center for Turbulence Research, Stanford University, an Honorary Professor at the University of Cape Town and a Distinguished Visiting Professor at University of Johannesburg. He sits on the board of Linköping Institute of Technology and the National Supercomputer Centre. He is an associate editor of Journal of Computational Physics and BIT Numerical Mathematics.

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.

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). 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.

[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 & Andrew R. Winters Linear and nonlinear analysis of the shallow water equations arXiv:1907.10713, 2021

[3] J. Nordström. A new energy stable formulation of the compressible Euler equations. arXiv:2201.05423v2, 2022.

[4] M. Svärd & J. Nordström Review of Summation-By-Parts Schemes for Initial-Boundary-Value Problems. Journal of Computational Physics, Volume 268, pp. 17-38, 2014.