Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science

In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented

A Hybrid Scheme for Gas-Dust Systems Stiffly Coupled via Viscous Drag

We present a stable and convergent method for studying a system of gas and dust, coupled through viscous drag in both non-stiff and stiff regimes. To account for the effects of dust drag in the update of the fluid quantities, we employ a fluid description of the dust component and study the modified gas-dust hyperbolic system following the approach in Miniati & Colella (2007). In addition to two entropy waves for the gas and dust components, respectively, the extended system includes three waves driven partially by gas pressure and partially by dust drift, which, in the limit of vanishing coupling, tend to the two original acoustic waves and the unhindered dust streaming. Based on this analysis we formulate a predictor step providing first order accurate reconstruction of the time-averaged state variables at cell interfaces, whence a second order accurate estimate of the conservative fluxes can be obtained through a suitable linearized Riemann solver. The final source term update is carried out using a one-step, second order accurate, L-stable, predictor corrector asymptotic method (the alpha-QSS method suggested by Mott et. al. 2000). This procedure completely defines a two-fluid method for gas-dust system. Using the updated fluid solution allows us to then advance the individual particle solutions, including self-consistently the time evolution of the gas velocity in the estimate of the drag force. This is done with a suitable particle scheme also based on the alpha-QSS method. A set of benchmark problems shows that our method is stable and convergent. When dust is modeled as a fluid (two-fluid) second order accuracy is achieved in both stiff and non-stiff regimes, whereas when dust is modeled with particles (hybrid) second order is achieved in the non-stiff regime and first order otherwise.Comment: 41 pages, 3 figures, 14 tables, accepted to J. Comp. Phys

High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation

We construct a high order discontinuous Galerkin method for solving general hyperbolic systems of conservation laws. The method is CFL-less, matrix-free, has the complexity of an explicit scheme and can be of arbitrary order in space and time. The construction is based on: (a) the representation of the system of conservation laws by a kinetic vectorial representation with a stiff relaxation term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport solver; and (c) a stiffly accurate composition method for time integration. The method is validated on several one-dimensional test cases. It is then applied on two-dimensional and three-dimensional test cases: flow past a cylinder, magnetohydrodynamics and multifluid sedimentation

Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier--Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.Comment: 43 page

Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well