A Quasi-Conservative Discontinuous Galerkin Method for Solving Five Equation Model of Compressible Two-Medium Flows
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A Quasi-Conservative Discontinuous Galerkin Method for Solving Five Equation Model of Compressible Two-Medium Flows Jian Cheng1
· Fan Zhang2 · Tiegang Liu3
Received: 10 February 2020 / Revised: 14 August 2020 / Accepted: 21 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, we develop a quasi-conservative discontinuous Galerkin method for the simulation of compressible gas-gas and gas-water two-medium flows by solving the five-equation transport model. This spatial discretization is a direct extension of the quasi-conservative finite volume discretization to the discontinuous Galerkin framework, thus, preserves uniform velocity and pressure fields at an isolated material interface. Furthermore, for discontinuities with a large pressure ratio, low density, and a dramatic change of material property where nonphysical values may occur, a strategy for imposing the bound-preserving limiting for volume fraction and a positivity-preserving limiting for density of each fluid and internal energy is developed and analyzed based on the quasi-conservative DG( p1 ) discretization. Typical test cases for both one- and two-dimensional problems are provided to demonstrate the performance of the proposed method. Keywords Discontinuous Galerkin method · Five-equation model · Bound-preserving · Positivity-preserving · Compressible two-medium flows
1 Introduction The simulation of different compressible fluids with immiscible interfaces is of great interest in many fields of science and engineering. Under the Eulerian framework, the numerical schemes can be generally classified into two major categories: sharp interface methods (SIM) [1–9] and diffuse interface methods (DIM) [10–18]. Each type of methods has its strength and weakness, in the following of this work, we will focus on the diffuse interface methods. Diffuse interface methods allow the material interface to numerically diffuse over a small but finite region. Nevertheless, several issues need to be considered in using diffuse interface
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Jian Cheng [email protected]
1
Institute of Applied Physics and Computational Mathematics, Beijing, People’s Republic of China
2
BIC-ESAT, College of Engineering, Peking University, Beijing, People’s Republic of China
3
School of Mathematical Sciences, Beihang University, Beijing, People’s Republic of China 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 85:12
methods. Firstly, the models are commonly described by a non-conservative system, which leads to potential difficulties in numerical discretization [19]. Within the traditional finite volume framework, two major strategies are developed: one is the quasi-conservative scheme [14,16,20], the other is the path-conservative scheme [19,21,22]. The quasi-conservative approach was first introduced in [14], in which pressure and velocity equilibrium condition (Abgrall condition) at an isolated material interface is applied as a basic principle to derive the spatial discretization.
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