Numerical solution of the compressible ideal magnetohydrodynamics system:
Orszag-Tang Vortex Test. The scheme preserves positivity of density, positivity
of internal energy, total energy, as well as involution constraints of the magnetic
field. This work was carried out in collaboration with Prof. Murtazo Nazarov and
Tuan Dao from Uppsala University (Division of Scientific Computing, Department of
Information Technology). Check our article pre-print for more details
https://arxiv.org/abs/2310.18467. Please set resolution of 2160p for full enjoyment.

Cold-plasma Diocotron instability using the Euler-Poisson
model with a 'given' magnetic field. What you see in this
movie is the evolution of the electron-density driven by the
self-consistent electric field. Reproducing the Diocotron instability
appears to be a mandatory milestone towards full Euler-Maxwell
solvers that do not need to time-resolve cyclotron motions.
This particular simulation uses time-step sizes 10,000 times larger
than the electrostatic plasma and/or cyclotron oscillation periods.

Same as above, but the camera is placed on top.
Youtube tends to default to low resolution: you might
want to change resolution to 720p and watch it full screen.

Daru-Tenaud shock viscous-layer interaction. What is a challenging test?
Is it a high Reynolds number problem? Well, not always. This is a seemingly
innocuous low-Reynolds number problem, which exhibits a rather strong
interaction of a shock wave with a viscous-layer. This is indeed a very
challenging problem that can put to shame more than one compressible
Navier-Stokes solver.

Fake triple point.
This particular example contains discontinuous initial data
with pressure contrast-ratio of one billion. It is not really possible to
compute a genuine triple-point problem with a single-species model. The setup
of the initial conditions is meant to "mimic", to some extent, the so-called
triple-point problem while maximizing aesthetic impact.

Schardin's test (shock wave diffraction).
Schlieren-like visualization of Schardin's test
(right-moving shock towards a triangular obstacle).
This simple test can be challenging for many schemes
as it tends to produce negative density and/or negative
internal energy near the vertices of the triangle,
breaking down the whole computation. The numerical methods used to solve
this problem make no special treatment of the vertices of the triangle: the
scheme can be mathematically proven to be unconditionally robust (no
mysterious hacks, no fixes, nor 'fudging factors' here).

Mach 3 step. This movie uses 640x1920 elements,
so it's reasonable
to enjoy it using full screen and 720p (HD). Here I am using a
Q1-continuous finite element method with lumped mass matrix for
the time derivative. Note the nice vortices originating in the
upper left point where the three shocks meet. This is one of my very
first computations of Euler's equations (early 2017). For more
recent results, of much better quality, you might want to check
Matthias Maier webpage.

Normal Field Instability with uniform magnetic field:
also called Rosensweig instability.
We are using using a two-phase diffuse-interface ferrofluid model.

Normal Field Instability with non-uniform magnetic field (a.k.a. the hedgehog).
We are using using a two-phase diffuse-interface ferrofluid model.