News

  • New paper submitted: Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes September 29, 2020

    Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e. the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. The source code to reproduce all numerical experiments presented in this article is available online (DOI: 10.5281/zenodo.4054366).

    Preprint available at arXiv:2009.13139. Numerical results were obtained with Trixi.jl.

  • Snapshot: Purely hyperbolic gravity simulations with Trixi.jl September 28, 2020

    One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic compressible Euler equations, the gravity field is described by an elliptic Poisson equation. In [1], we present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime using the same explicit high-order discontinuous Galerkin method we use for the flow solution.

    We start with the Poisson equation for the Newtonian gravitational potential $\phi$,

    $$-\vec{\nabla}^2\phi = -4\pi G \rho\label{grav}$$

    where G is the universal gravitational constant and $\rho$ is the mass density. Following Nishikawa’s work, we convert the Poisson equation for the gravitational potential into the hyperbolic gravity equations,

    $$\frac{\partial}{\partial t}\begin{bmatrix}
    \phi\\[0.1cm]
    q_1\\[0.1cm]
    q_2\\[0.1cm]
    \end{bmatrix}
    +
    \frac{\partial}{\partial x}\begin{bmatrix}
    -q_1\\[0.1cm]
    -\phi/T_r\\[0.1cm]
    0\\[0.1cm]
    \end{bmatrix}
    +
    \frac{\partial}{\partial y}\begin{bmatrix}
    -q_2\\[0.1cm]
    0\\[0.1cm]
    -\phi/T_r\\[0.1cm]
    \end{bmatrix}
    =
    \begin{bmatrix}
    -4\pi G \rho\\[0.1cm]
    -q_1/T_r\\[0.1cm]
    -q_2/T_r\\[0.1cm]
    \end{bmatrix},\label{hyp}$$

    where the auxiliary variables $(q_1,q_2)^\intercal\approx\vec{\nabla}\phi$ and $T_r$ is the relaxation time. The steady-state solution of the hyperbolic system is, in fact, the desired solution of the original Poisson problem.

    The flow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupled in each Runge-Kutta stage via the source terms. A key benefit of our strategy is that it allows the reuse of existing explicit hyperbolic solvers without modifications, while retaining their advanced features such as non-conforming and solution-adaptive grids.

    We implemented this purely hyperbolic approach for self-gravitating gas dynamics in Trixi.jl, a tree-based numerical simulation framework for hyperbolic PDEs, which is written in Julia [2]. It was first validated by computing the Jeans gravitational instability, which demonstrates excellent agreement of the numerical results with the analytical solution:

    Evolution of the computed kinetic, internal, and potential energies for the Jeans
    instability using polynomial order N = 3 on a uniform 16 × 16 mesh. The analytical energies are included for reference.

    In a second example, we consider a modification of the Sedov blast wave problem that incorporates the effects of gravitational acceleration and which involves strong shocks and complex fluid interactions. We include it to demonstrate the shock capturing and AMR capabilities of Trixi.jl to resolve the cylindrical Sedov blast wave:

    Self-gravitating Sedov explosion approximated with polynomial order $N = 3$ on an AMR grid with four levels of possible refinement at $T = 0.5$ (left) and $T = 1.0$ (right). (first and third row) Two dimensional plots of the density and gravitational potential at two times. The white overlay of squares in the density plots shows the AMR grids used by both solvers. (second and last row) One dimensional slices of the solutions along a line from the origin to the edge of the domain in the positive x-direction.

    Finally, we show the evolution of the solution together with the adaptive mesh, which is dynamically refined and coarsened to maintain high accuracy while reducing the overall computation time:

    https://raw.githubusercontent.com/trixi-framework/paper-self-gravitating-gas-dynamics/master/assets/sedov-rho-phi-mesh.gif
    Simulation of self-gravitating Sedov blast wave with adaptive mesh refinement, where adaptation performed after every time step of the compressible Euler solver. The total run time compared to a uniformly refined mesh is reduced by a factor of 19, while the solutions are virtually unchanged.

    All results in the paper can be easily reproduced by using the resources provided in [3], including step-by-step instructions for how to obtain and install Trixi.jl.

    References

    [1] Schlottke-Lakemper, Michael and Winters, Andrew R and Ranocha, Hendrik and Gassner, Gregor J, A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics, submitted to J Comp Phys, 2020. arXiv:2008.10593

    [2] Schlottke-Lakemper, Michael and Gassner, Gregor J and Ranocha, Hendrik and Winters, Andrew R, Trixi.jl: A tree-based numerical simulation framework for hyperbolic PDEs written in Julia, 2020. doi:10.5281/zenodo.3996439

    [3] Schlottke-Lakemper, Michael and Winters, Andrew R and Ranocha, Hendrik and Gassner, Gregor J, Self-gravitating gas dynamics simulations with Trixi.jl, 2020. doi:10.5281/zenodo.3996575

  • Trixi.jl: A tree-based numerical simulation framework for hyperbolic PDEs written in Julia September 1, 2020

    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
    • 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.

  • New paper submitted: A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics August 26, 2020

    One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime using the same explicit high-order discontinuous Galerkin method we use for the flow solution. The flow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupled via the source terms. A key benefit of our approach is that it allows the reuse of existing explicit hyperbolic solvers without modifications, while retaining their advanced features such as non-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kutta stage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physics simulations. After verifying the expected order of convergence for single-physics and multi-physics setups, we validate our approach by a simulation of the Jeans gravitational instability. Furthermore, we demonstrate the full capabilities of our numerical framework by computing a self-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinement for the entire coupled system.

    Available at arXiv:2008.10593

  • Snapshot: A new approach for approximating conservation laws August 24, 2020

    A new approach for approximating conservation laws has been tested: instead of monitoring the changes of the means of a quantity within a certain volume over time like a finite volume procedure, this method mimics the behavior of simple solutions.

    First the current state is split into waves w, each aligned along one of the eigenvectors of the Fluxjacobian and living on its own grid. Then the corresponding grids are being moved with their eigenvelocity and finally all waves w are being overlayed and normalized yielding the solution after one timestep.

    The plots show the solution of isothermal Euler equations at Time T=0.0000 (initial condition) and T=0.0025 with N=40 gridpoints for each mesh.

  • Snapshot: Higher-order schemes for the MHD equations July 31, 2020

    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 July 16, 2020

    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 July 16, 2020

    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: Variant of the Boss-Bodenheimer test with a self-gravitating high order hydro code June 26, 2020

    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.

  • Snapshot: 2D PyOpenGL real-time simulation of the forward-facing step test case May 27, 2020

    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.