Author Archives: Christof Czernik

Snapshot: Higher-order schemes for the MHD equations

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.

New paper published: Entropy-Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier–Stokes Equations

The entropy-conservative/stable, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernández et al. (2019) is extended from the compressible Euler equations to the compressible Navier–Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to construct the interface coupling terms, thereby localizing the discrete geometric conservation law constraints to each individual element. The resulting scheme is entropy conservative/stable, element-wise conservative, and freestream preserving. Viscous interface dissipation operators that retain the entropy stability of the base scheme are developed. The accuracy and stability of the resulting numerical scheme are shown to be comparable to those of the original conforming scheme in Carpenter et al. (2014) and Parsani et al. (2016), i.e., this scheme achieves ~p+1/2 convergence on geometrically high-order distorted element grids; this is demonstrated in the context of the viscous shock problem, the Taylor–Green vortex problem at a Reynolds number of Re = 1,600 and a subsonic turbulent flow past a sphere at Re = 2,000.

sciencedirect
arxiv

New article published in ECCOMAS Newsletter (p.16-20): Split Form Discontinuous Galerkin Methods For Implicit Large Eddy Simulation Of Compressible Turbulence

As a teaser we show a numerical demonstration of the capabilities of the split form DG approach. We consider the flow past a plunging SD7003 airfoil at Mach number and cord length based Reynolds number Rec = 40,000. We subdivide the computational domain into 58,490 unstructured curvilinear hexahedral elements and use a polynomial degree N = 7 resulting in a total of about 150 million degrees of freedom.

The Newsletter is available via: eccomas Newsletter.

Snapshot: 2D PyOpenGL real-time simulation of the forward-facing step test case

WENO methods refers to a class of nonlinear finite volume or finite difference methods which are well suited to approximate solutions of hyperbolic conservation laws with high order accuracy in smooth regions and essentially non-oscillatory transition for solution discontinuities. In the following video we used a 5th order accurate WENO-FD scheme with immersed boundary conditions.

Here we solve the forward-facing step test case in real-time using Python3/PyOpenGL/GLFW with an Intel Corporation HD Graphics 620 (rev 02). As one can see, it is also possible to add pressure peaks to the simulation by simply clicking with the mouse on the desired field.

Snapshot: Smoothed-Particle Hydrodynamics (SPH) simulation of the Numerical Simulation Research Group

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.

Snapshot: Blast wave simulation computed in FLUXO with an entropy-stable high-order Discontinuous Galerkin/Finite Volume hybrid scheme

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

Talk: Lea Miko Versbach, Lund University, Sweden, “Implicit DG solvers and multigrid preconditioners” [05.02.2020, 11am]

On Wednesday, 5 February 2020 at 11:00 Lea Miko Versbach will talk on the topic “Implicit DG solvers and multigrid preconditioners” 

Location: Gyrhofstraße 8a (Gebäude 158a), Room 1.105 (1st floor), 50931

Abstract: In this talk I will give an introduction to implicit DG solvers and the challenges which arise when solving the large nonlinear systems coming from the implicit temporal discretization. We will solve these system using a Jacobian-free Newton-Krylov method. The problem with this method is that the Krylov subspace method converges slowly for discretized PDEs. In order to speed up computations, we need to construct preconditioners. This can be done using iterative methods. Carefully constructed multigrid methods are cheap and effective preconditioners. I will explain more details about the construction of these preconditioners and show some numerical results.