Christian Klingenberg

Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior

Existence and uniqueness is proved, in the class of functions satisfying a wave entropy condition, of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable. The large time limit is then studied, and explicit formulas for this limit is given in the case where the initial data as well as the x dependency of the flux vary periodically. Throughout the paper, front tracking is used as a method of analysis. A numerical example which illustrates the results and method of proof is also presented.

Vertical Structure of a Supernova-driven Turbulent, Magnetized Interstellar Medium

Stellar feedback drives the circulation of matter from the disk to the halo of galaxies. We perform three-dimensional magnetohydrodynamic simulations of a vertical column of the interstellar medium with initial conditions typical of the solar circle in which supernovae drive turbulence and determine the vertical stratification of the medium. The simulations were run using a stable, positivity-preserving scheme for ideal MHD implemented in the FLASH code. We find that the majority (90%) of the mass is contained in thermally stable temperature regimes of cold molecular and atomic gas at T 3 kpc. The magnetic field in our models has no significant impact on the scale heights of gas in each temperature regime; the magnetic tension force is approximately equal to and opposite the magnetic pressure, so the addition of the field does not significantly affect the vertical support of the gas. The addition of a magnetic field does reduce the fraction of gas in the cold (<200 K) regime with a corresponding increase in the fraction of warm (~104 K) gas. However, our models lack rotational shear and thus have no large-scale dynamo, which reduces the role of the field in the models compared to reality. The supernovae drive oscillations in the vertical distribution of halo gas, with the period of the oscillations ranging from 30 Myr in the T < 200 K gas to ~100 Myr in the 106 K gas, in line with predictions by Walters & Cox.

Front tracking and two-dimensional Riemann problems

A substantial improvement in resolution has been achieved for the computation of jump discontinuities in gas dynamics using the method of front tracking. The essential feature of this method is that a lower dimensional grid is fitted to and follows the discontinuous waves. At the intersection points of these discontinuities, two-dimensional Riemann problems occur. In this paper we study such two-dimensional Riemann problems from both numerical and theoretical points of view. Specifically included is a numerical solution for the Mach reflection, a general classification scheme for two-dimensional elementary waves, and a discussion of problems and conjectures in this area.

A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests

The ideal MHD equations are a central model in astrophysics, and their solution relies upon stable numerical schemes. We present an implementation of a new method, which possesses excellent stability properties. Numerical tests demonstrate that the theoretical stability properties are valid in practice with negligible compromises to accuracy. The result is a highly robust scheme with state-of-the-art efficiency. The schemes robustness is due to entropy stability, positivity and properly discretised Powell terms. The implementation takes the form of a modification of the MHD module in the FLASH code, an adaptive mesh refinement code. We compare the new scheme with the standard FLASH implementation for MHD. Results show comparable accuracy to standard FLASH with the Roe solver, but highly improved efficiency and stability, particularly for high Mach number flows and low plasma @b. The tests include 1D shock tubes, 2D instabilities and highly supersonic, 3D turbulence. We consider turbulent flows with RMS sonic Mach numbers up to 10, typical of gas flows in the interstellar medium. We investigate both strong initial magnetic fields and magnetic field amplification by the turbulent dynamo from extremely high plasma @b. The energy spectra show a reasonable decrease in dissipation with grid refinement, and at a resolution of 512^3 grid cells we identify a narrow inertial range with the expected power law scaling. The turbulent dynamo exhibits exponential growth of magnetic pressure, with the growth rate higher from solenoidal forcing than from compressive forcing. Two versions of the new scheme are presented, using relaxation-based 3-wave and 5-wave approximate Riemann solvers, respectively. The 5-wave solver is more accurate in some cases, and its computational cost is close to the 3-wave solver.

A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework

We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisfy discrete entropy inequalities, and preserve positivity of density and internal energy. The subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate Riemann solver itself. The 7-wave version of the solver is able to resolve exactly all material and Alfven isolated contact discontinuities. Practical considerations and numerical results will be provided in another paper.

A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units

We study a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by the initial value problem for a one-dimensional scalar conservation law with a flux function that depends discontinuously on the spatial position. We construct a weak solution to the sedimentation model by proving the convergence of a front tracking method. The basic building block in this method is the solution of the Riemann problem, which is complicated by the fact that the flux function is discontinuous. A feature of the convergence analysis is the difficulty of bounding the total variation of the conserved variable. To overcome this obstacle, we rely on a certain non-linear Temple functional under which the total variation can be bounded. The total variation bound on the transformed variable also implies that the front tracking construction is well defined. Finally, via some numerical examples, we demonstrate that the front tracking method can be used as a highly efficient and accurate simulation tool for continuous sedimentation.

A relaxation scheme for conservation laws with a discontinuous coefficient

lWe study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k E BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat-Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 x 2 Riemann solver.

Numerical comparison of Riemann solvers for astrophysical hydrodynamics

The idea of this work is to compare a new positive and entropy stable approximate Riemann solver by Francois Bouchut with a state-of the-art algorithm for astrophysical fluid dynamics. We implemented the new Riemann solver into an astrophysical PPM-code, the Prometheus code, and also made a version with a different, more theoretically grounded higher order algorithm than PPM. We present shock tube tests, two-dimensional instability tests and forced turbulence simulations in three dimensions. We find subtle differences between the codes in the shock tube tests, and in the statistics of the turbulence simulations. The new Riemann solver increases the computational speed without significant loss of accuracy.

Astrophysical hydrodynamics with a high-order discontinuous Galerkin scheme and adaptive mesh refinement

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two- and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.

Entropy Stable Finite Volume Scheme for Ideal Compressible MHD on 2-D Cartesian Meshes

We present a finite volume scheme for ideal compressible magnetohydrodynamic (MHD) equations on two-dimensional Cartesian meshes. The semidiscrete scheme is constructed to be entropy stable by using the symmetrized version of the equations as introduced by Godunov. We first construct an entropy conservative scheme for which sufficient condition is given and we also derive a numerical flux satisfying this condition. Second, following a standard procedure, we make the scheme entropy stable by adding dissipative flux terms using jumps in entropy variables. A semi-discrete high resolution scheme is constructed that preserves the entropy stability of the first order scheme. We demonstrate the robustness of this new scheme on several standard MHD test cases.