On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind scheme in 1D with numerical boundary conditions. The underlying approximated continuous problem is a hyperbolic partial differential equation. Our approach is based on the Uniform Kreiss-Lopatinskii Condition, using linear algebra and complex analysis to count the number of zeros of the associated determinant. The study is illustrated with the Beam-Warming scheme together with the simplified inverse Lax-Wendroff procedure at the boundary

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252-270. https://doi.org110.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells

On central-difference and upwind schemes

A class of numerical dissipation models for central-difference schemes constructed with second- and fourth-difference terms is considered. The notion of matrix dissipation associated with upwind schemes is used to establish improved shock capturing capability for these models. In addition, conditions are given that guarantee that such dissipation models produce a Total Variation Diminishing (TVD) scheme. Appropriate switches for this type of model to ensure satisfaction of the TVD property are presented. Significant improvements in the accuracy of a central-difference scheme are demonstrated by computing both inviscid and viscous transonic airfoil flows

A class of high resolution explicit and implicit shock-capturing methods

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems

One- and multi-dimensional CWENOZ reconstructions for implementing boundary conditions without ghost cells

We address the issue of point value reconstructions from cell averages in the context of third order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques known in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in boundary cells. In (Naumann, Kolb, Semplice, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests we compare the novel reconstruction with the standard approach using ghost cells

Discontinuous Galerkin methods: exploiting superconvergence for improved time-stepping

The discontinuous Galerkin (DG) methods are one of the most extensively researched classes of numerical methods for solving partial dfferential equations that display convective or diffusive qualities and have been popularly adopted by the scientific and engineering communities as a method capable of achieving arbitrary orders of accuracy in space. The choice of numerical flux function plays a pivotal role in the successful construction of DG methods and has an intrinsic effect on the superconvergence properties. As an inherent property of the spatial discretisation, superconvergence can only be retained in the solution through a sensitive pairing with a time integrator. The results of the literature and of this work suggest that an improved pairing between the spatial and temporal discretisations is both desirable and possible. We perform analysis of three different but related manifestations of superconvergence: the local, super-accurate points themselves; the subsequent global extraction via the Smoothness-Increasing Accuracy-Conserving (SIAC) filters; and the spectral properties that quantify, in terms of dispersion and dissipation errors, how accurately waves are convected. In order to explore the effect of the numerical flux function on superconvergence, we consider a generalisation of the “natural" upwind choice for a Method of Lines solution to the linear advection equation: the upwind-biased flux. We prove that the method is locally superconvergent at roots of a linear combination of the left- and right-Radau polynomials dependent on the value of a flux parameter and that the use of SIAC filters is still able to draw out the superconvergence information and create a globally smooth and superconvergent solution. In exploring the coupling of DG with a time integrator, we introduce a new scheme to a class of multi-stage multi-derivative methods, following recent incorporation of local DG technologies to recover superconvergence and achieve improved wave propagation properties

Upwind and symmetric shock-capturing schemes

The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis is on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the extact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows is discussed. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas dynamics problems