Recent developments in shock-capturing schemes

The development of the shock capturing methodology is reviewed, paying special attention to the increasing nonlinearity in its design and its relation to interpolation. It is well-known that higher-order approximations to a discontinuous function generate spurious oscillations near the discontinuity (Gibbs phenomenon). Unlike standard finite-difference methods which use a fixed stencil, modern shock capturing schemes use an adaptive stencil which is selected according to the local smoothness of the solution. Near discontinuities this technique automatically switches to one-sided approximations, thus avoiding the use of discontinuous data which brings about spurious oscillations

Hybrid Spectral Difference/Embedded Finite Volume Method for Conservation Laws

A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field. The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.Comment: 27 pages, 17 figures, 2 tables, Accepted for publication in the Journal of Computational Physics, April 201

Uniqueness in MHD in divergence form: right nullvectors and well-posedness

Magnetohydrodynamics in divergence form describes a hyperbolic system of covariant and constraint-free equations. It comprises a linear combination of an algebraic constraint and Faraday's equations. Here, we study the problem of well-posedness, and identify a preferred linear combination in this divergence formulation. The limit of weak magnetic fields shows the slow magnetosonic and Alfven waves to bifurcate from the contact discontinuity (entropy waves), while the fast magnetosonic wave is a regular perturbation of the hydrodynamical sound speed. These results are further reported as a starting point for characteristic based shock capturing schemes for simulations with ultra-relativistic shocks in magnetized relativistic fluids.Comment: To appear in J Math Phy

Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods

In this work, a novel artificial viscosity method is proposed using smooth and compactly supported viscosities. These are derived by revisiting the widely used piecewise constant artificial viscosity method of Persson and Peraire as well as the piecewise linear refinement of Kl\"ockner et al. with respect to the fundamental design criteria of conservation and entropy stability. Further investigating the method of modal filtering in the process, it is demonstrated that this strategy has inherent shortcomings, which are related to problems of Legendre viscosities to handle shocks near element boundaries. This problem is overcome by introducing certain functions from the fields of robust reprojection and mollififers as viscosity distributions. To the best of our knowledge, this is proposed for the first time in this work. The resulting $C_0^\infty$ artificial viscosity method is demonstrated to provide sharper profiles, steeper gradients and a higher resolution of small-scale features while still maintaining stability of the method.Comment: 21 pages, accepted for publication in Journal of Scientific Computin

An Entropy Stable Central Solver for Euler Equations

An exact discontinuity capturing central solver developed recently, named MOVERS (Method of Optimal Viscosity for Enhanced Resolution of Shocks, J Computat Phys 2009;228:770-798), is analyzed and improved further to make it entropy stable. MOVERS, which is designed to capture steady shocks and contact discontinuities exactly by enforcing the Rankine-Hugoniot jump condition directly in the discretization process, is a low diffusive algorithm in a simple central discretization framework, free of complicated Riemann solvers and flux splittings. However, this algorithm needs an entropy fix to avoid nonsmoothness in the expansion regions. The entropy conservation equation is used as a guideline to introduce an optimal numerical diffusion in the smooth regions and a limiter based switchover is introduced for numerical diffusion based on jump conditions at the large gradients. The resulting new scheme is entropy stable, accurate and captures steady discontinuities exactly while avoiding an entropy fix.Comment: 17 pages, 51 figures, Journal articl