Numerical dissipation switch for two-dimensional central-upwind schemes

Abstract

We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x- and y-directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.

Local Characteristic Decomposition Based Central-Upwind Scheme

“…In this example, we consider the explosion problem studied in [8,16,18,27]. This is a circularly symmetric problem with an initial circular region of higher density and pressure with the following initial conditions, (ρ(x, y, 0), u(x, y, 0), v(x, y, 0), p(x, y, 0)) = (1, 0, 0, 1), x 2 + y 2 < 0.16, (0.125, 0, 0, 0.1), otherwise, prescribed in the computational domain [0, 1.5] × [0, 1.5].…”

“…In this example, we consider the implosion problem taken from [8,16,18,27]. The initial conditions, (ρ(x, y, 0), u(x, y, 0), v(x, y, 0), p(x, y, 0)) = (0.125, 0, 0, 0.14…”

“…This shows that the New CU scheme is capable of resolving the same details on a substantially coarser grid due to the smaller amount of numerical diffusion present in the scheme. In this example, we study the KH instability, which develops in the test problem taken from [5,6,8,16,32]. We take the following initial data:…”

“…In the last example taken from [8,16,36,41], we investigate the RT instability, which is a physical phenomenon occurring when a layer of heavier fluid is placed on top of a layer of lighter fluid. The model is governed by the 2-D Euler equations (1.1), (B.1)-(B.2) with added gravitational source terms.…”

“…Even though the CU schemes from [17,19] are quite accurate, efficient and robust tools for a wide variety of hyperbolic systems, higher resolution of the numerical solutions can be achieved by further reducing numerical dissipation. This can be done in a number of different ways, for example: (i) by introducing a more accurate evolution procedure, which leads to a "built-in" antidiffusion term [18]; (ii) by implementing a numerical dissipation switch to control the amount of numerical dissipation present in the CU schemes [16]; (iii) by obtaining more accurate estimates for the one-sided local speeds of propagation using the discrete Rankine-Hugoniot conditions [8].…”