Solving 3D relativistic hydrodynamical problems with WENO discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods coupled to WENO algorithms allow high order convergence for smooth problems and for the simulation of discontinuities and shocks. In this work, we investigate WENO-DG algorithms in the context of numerical general relativity, in particular for general relativistic hydrodynamics. We implement the standard WENO method at different orders, a compact (simple) WENO scheme, as well as an alternative subcell evolution algorithm. To evaluate the performance of the different numerical schemes, we study non-relativistic, special relativistic, and general relativistic testbeds. We present the first three-dimensional simulations of general relativistic hydrodynamics, albeit for a fixed spacetime background, within the framework of WENO-DG methods. The most important testbed is a single TOV-star in three dimensions, showing that long term stable simulations of single isolated neutron stars can be obtained with WENO-DG methods.Comment: 21 pages, 10 figure

Conservative and non-conservative methods based on hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations

We introduce a WENO reconstruction based on Hermite interpolation both for semi-Lagrangian and finite difference methods. This WENO reconstruction technique allows to control spurious oscillations. We develop third and fifth order methods and apply them to non-conservative semi-Lagrangian schemes and conservative finite difference methods. Our numerical results will be compared to the usual semi-Lagrangian method with cubic spline reconstruction and the classical fifth order WENO finite difference scheme. These reconstructions are observed to be less dissipative than the usual weighted essentially non- oscillatory procedure. We apply these methods to transport equations in the context of plasma physics and the numerical simulation of turbulence phenomena

A modified fifth-order WENO scheme for hyperbolic conservation laws

This paper deals with a new fifth-order weighted essentially non-oscillatory (WENO) scheme improving the WENO-NS and WENO-P methods which are introduced in Ha et al. J. Comput. Phys. (2013) and Kim et al., J. Sci. Comput. (2016) respectively. These two schemes provide the fifth-order accuracy at the critical points where the first derivatives vanish but the second derivatives are non-zero. In this paper, we have presented a scheme by defining a new global-smoothness indicator which shows an improved behavior over the solution to the WENO-NS and WENO-P schemes and the proposed scheme attains optimal approximation order, even at the critical points where the first and second derivatives vanish but the third derivatives are non-zero.Comment: 23 pages, 14 figure

Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations

In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high order weighted essentially non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under the constrained transport (CT) framework, can achieve high order accuracy, a discrete divergence-free condition and positivity of the numerical solution simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate the performance of the proposed method.Comment: 21 pages, 28 figure

Combination of WENO and Explicit Runge–Kutta Methods for Wind Transport in the Meso-NH Model

This paper investigates the use of the weighted essentially nonoscillatory (WENO) space discretization methods of third and fifth order for momentum transport in the Meso-NH meteorological model, and their association with explicit Runge–Kutta (ERK) methods, with the specific purpose of finding an optimal combination in terms of wall-clock time to solution. A linear stability analysis using von Neumann theory is first conducted that considers six different ERK time integration methods. A new graphical representation of linear stability is proposed, which allows a first discrimination between the ERK methods. The theoretical analysis is then completed by tests on numerical problems of increasing complexity (linear advection of high wind gradient, orographic waves, density current, large eddy simulation of fog, and windstorm simulation), using a fourth-order-centered scheme as a reference basis. The five-stage third-order and fourth-order ERK combinations appear as the time integration methods of choice for coupling with WENO schemes in terms of stability. An explicit time-splitting method added to the ERK temporal scheme for WENO improves the stability properties slightly more. When the spatial discretizations are compared, WENO schemes present the main advantage of maintaining stable, nonoscillatory transitions with sharp discontinuities, but WENO third order is excessively damping, while WENO fifth order provides better accuracy. Finally, WENO fifth order combined with the ERK method makes the whole physics of the model 3 times faster compared to the classical fourth-order centered scheme associated with the leapfrog temporal scheme

A comparative study of the efficiency of jet schemes

We present two versions of third order accurate jet schemes, which achieve high order accuracy by tracking derivative information of the solution along characteristic curves. For a benchmark linear advection problem, the efficiency of jet schemes is compared with WENO and Discontinuous Galerkin methods of the same order. Moreover, the performance of various schemes in tracking solution contours is investigated. It is demonstrated that jet schemes possess the simplicity and speed of WENO schemes, while showing several of the advantages as well as the accuracy of DG methods.Comment: 12 pages, 6 figures, presented at the conference Mathematical Modeling and Applications to Industrial Problems 201