Speaker: Dr. Vikram Singh, Max-Planck-Institute for Meteorology
Date & time: Friday, 21st May 2021, 10 am (CEST)
Meeting link: Please request via email from Michael Schlottke-Lakemper

Abstract:
Split form schemes for Euler and Navier-Stokes equations are useful for computation of turbulent flows due to their better robustness. This is because they satisfy additional conservation properties of the governing equations like kinetic energy preservation leading to a reduction in aliasing errors at high orders. Recently, linear stability issues have been pointed out for these schemes for a density wave problem and we investigate this behaviour for some standard split forms. By deriving linearized equations of split form schemes, we show that most existing schemes do not satisfy a perturbation energy equation that holds at the continuous level. A simple modification to the energy flux of some existing schemes is shown to yield a scheme that is consistent with the energy perturbation equation. Numerical tests are given using a discontinuous Galerkin method to demonstrate these results.

Note: This meeting will be recorded.

Speaker: Dr. Andrew Giuliani, New York University
Date & time: Monday, 10th May 2021, 4pm (CEST)/10am (EDT)
Meeting link: Please request via email from Michael Schlottke-Lakemper

Abstract:
In this talk, we present a state redistribution method for high order discontinuous Galerkin methods on curvilinear embedded boundary grids. State redistribution relaxes the overly restrictive CFL condition that results from arbitrarily small cut cells when explicit time steppers are used. Thus, the scheme can take time steps that are proportional to the size of cells in the background grid. The discontinuous Galerkin scheme is stabilized by postprocessing the numerical solution after each stage or step of an explicit time stepping method. The advantage of this approach is that it uses only basic mesh information that is already available in many cut cell codes and does not require complex geometric manipulations. We prove that state redistribution is conservative and p-exact. Finally, we solve a number of test problems that demonstrate the encouraging potential of this technique for applications on curvilinear embedded geometries. Numerical experiments reveal that our scheme converges with order $p+1$ in $L_1$ and between $p$ and $p+1$ in $L_\infty$.

Trixi.jl is a numerical simulation framework for hyperbolic conservation laws written in Julia. A key objective for the framework is to be useful to both scientists and students. Therefore, next to having an extensible design with a fast implementation, Trixi is focused on being easy to use for new or inexperienced users, including the installation and postprocessing procedures. Its features include:

• Hierarchical quadtree/octree grid with adaptive mesh refinement
• Native support for 2D and 3D simulations
• High-order accuracy in space in time
• Nodal discontinuous Galerkin spectral element methods
  • Kinetic energy-preserving and entropy-stable split forms
  • Entropy-stable shock capturing
• Explicit low-storage Runge-Kutta time integration
• Square/cubic domains with periodic and Dirichlet boundary conditions
• Multiple governing equations:
  • Compressible Euler equations
  • Magnetohydrodynamics equations
  • Hyperbolic diffusion equations for elliptic problems
  • Scalar advection
• Multi-physics simulations
  • Self-gravitating gas dynamics
• Shared-memory parallelization via multithreading
• Visualization of results with Julia-only tools (2D) or ParaView/VisIt (2D/3D)

Trixi.jl was initiated by Michael Schlottke-Lakemper and Gregor Gassner (both University of Cologne, Germany). Together with Hendrik Ranocha (KAUST, Saudi Arabia) and Andrew Winters (Linköping University, Sweden), they are the principal developers of Trixi.

In case of questions, please feel free to create an issue. We are looking forward to feedback and/or potential scientific collaboration.