Locally Divergence-Free Path-Conservative Central-Upwind Schemes for Ideal and Shallow Water Magnetohydrodynamics
Alexander Kurganov (Prof. Dr.)

I will present semi-discrete path-conservative central-upwind (PCCU) schemes for ideal and
shallow water magnetohydrodynamics (MHD) equations. These schemes possess several
important properties: they locally preserve the divergence-free constraint, they do not rely on
any (approximate) Riemann problem solver, and they robustly producehigh-resolution and non-
oscillatory results. The derivation of the schemes is based on the Godunov-Powell
nonconservative modifications of the studied MHD systems. The local divergence-free property
is enforced by augmenting the modified systems with the evolution equations for the
corresponding derivatives of the magnetic field components. These derivatives are then used to
design a special piecewise linear reconstruction of the magnetic field, which guarantees a non-
oscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization
accounts for the jump of the nonconservative product terms across cell interfaces, thereby
ensuring stability.
I will also discuss the extension of the proposed schemes to magnetic rotating shallow
water equations. The new scheme is both well-balanced and exactly preserves the divergence-
free condition of the magnetic field. The well-balanced property is enforced by applying a
flux globalization approach within the PCCU scheme. As a result, both still- and moving-
water equilibria can be exactly preserved at the discrete level. The proposed PCCU schemes
are tested on several benchmarks. The obtained numerical results illustrate the performance of
the new schemes, their robustness, and their ability not only to achieve high resolution, but also
preserve the positivity of computed quantities such as density, pressure, and water depth. The
talk is based on joint works with Alina Chertock (North Carolina State University, USA), Michael
Redle (RWTH Aachen University, Germany),Kailiang Wu (Southern University of Science and
Technology, China) and Vladimir Zeitlin (Sorbonne University, France).