A robust and easy way of simulating a hyperbolic test case with discontinuities is to use a 1st order finite volume scheme. Using such a method for magnetohydrodynamics (MHD) problems like the Orszag-Tang vortex leads to following results:

For this and the following examples we used a 4th order time integration scheme and 256 degrees of freedom in each spatial direction.

A way to generate more accurate results is to increase the order of the scheme, which has to be treated with caution near discontinuities because oscillations may occur. To overcome this issue one could use higher order schemes in smooth regions and lower order schemes in regions with discontinuities. An example for such an approach is our so called DGFV scheme, which blends e.g. a 4th order Discontinuous Galerkin scheme with a 1st order Finite Volume scheme.

Another way of doing it, is to use a suitable 4th order finite-volume scheme for MHD with a fitting limiter.

Variant of the classical Boss-Bodenheimer Test: We simulate the collapse and fragmentation of a rotating cloud core using a Discontinous Galerkin / Finite Volume hybrid high-order code. The self-gravity of the gas is calculated via the Fast-Multipole-Method.

Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and is being increasingly used to model fluid motion as well.
This method is well-suited for problems dominated by complex boundary dynamics, since SPH is a mesh-free method, as well as for mass conservation problems since the particles themselves represent mass.

We used a 2D Python/OpenCL SPH code that solves the incompressible Navier-Stokes equations in real time. In this simulation, 52,700 particles were used to smash the people of the Numerical Simulation Research Group against each other.

Blast wave simulation with periodic boundaries computed in FLUXO with an
entropy-stable high-order Discontinuous Galerkin/Finite Volume scheme.

The scheme is formulated in Hennemann and Gassner. “A provably entropy stable subcell shock capturing approach for high order split form DG.” Journal of Computational Physics (submitted). The scheme computes the spatial operator as F = (1-a) F_{DG} + a F_{FV}, where a is the blending function

Andrés M. Rueda-Ramírez completed his undergraduate studies in Mechanical Engineering at the National University of Colombia (Universidad Nacional de Colombia, UNAL). After graduating, he worked as a research and teaching assistant at the Research Group on biomechanics at UNAL, where he collaborated with a highly interdisciplinary team in the development of a Finite Element software to simulate bone growing processes.

AMRR completed his Ph.D. studies in Aerospace Engineering at the Polytechnic University of Madrid (Universidad Politécnica de Madrid). During his Ph.D., AMRR studied and developed p-adaptation algorithms, implicit time-integration schemes, and multigrid solvers for high-order Discontinuous Galerkin Spectral Element Methods (DGSEM).

AMRR is now a member of the Numerical Simulation Research Group at the University of Cologne, where he joined the development team of the DGSEM code FLUXO. AMRR is currently working on sub-cell shock-capturing schemes and time integrators for the Navier-Stokes and the MHD equations.

Supersonic 3D Blob Test computed with a ‘High-order Discontinuous Galerkin/Finite-Volume Hybrid Scheme’ on a non-conforming cartesian grid.