Elsevier

Journal of Computational Physics

Volume 227, Issue 2, 10 December 2007, Pages 1567-1596
Journal of Computational Physics

A high order ENO conservative Lagrangian type scheme for the compressible Euler equations

https://doi.org/10.1016/j.jcp.2007.09.017Get rights and content

Abstract

We develop a class of Lagrangian type schemes for solving the Euler equations of compressible gas dynamics both in the Cartesian and in the cylindrical coordinates. The schemes are based on high order essentially non-oscillatory (ENO) reconstruction. They are conservative for the density, momentum and total energy, can maintain formal high order accuracy both in space and time and can achieve at least uniformly second-order accuracy with moving and distorted Lagrangian meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. One and two-dimensional numerical examples in the Cartesian and cylindrical coordinates are presented to demonstrate the performance of the schemes in terms of accuracy, resolution for discontinuities, and non-oscillatory properties.

Introduction

In numerical simulations of multi-dimensional fluid flow, there are two typical choices: a Lagrangian framework, in which the mesh moves with the local fluid velocity, and an Eulerian framework, in which the fluid flows through a grid fixed in space. More generally, the motion of the grid can also be chosen arbitrarily, this method is called the Arbitrary Lagrangian–Eulerian method (ALE; cf. [14], [2], [21], [16], [25]). Most ALE algorithms consist of three phases, a Lagrangian phase in which the solution and the grid are updated, a rezoning phase in which the nodes of the computational grid are moved to a more optimal position and a remapping phase in which the Lagrangian solution is transferred to the new grid.

In this paper, we focus on computational hydrodynamic methods for the Euler equations where the mesh moves with the flow velocity. Such methods, which we refer to as Lagrangian type methods, imply the use of distorted or non-uniform meshes. Particular examples include the Lagrangian methods, or the ALE methods which contain a Lagrangian phase.

Pure Lagrangian methods, and certain ALE methods which can capture contact discontinuities sharply (see e.g. [19]), are widely used in many fields for multi-material flow simulations such as astrophysics and computational fluid dynamics (CFD). We will only consider single material in this paper, however the pure Lagrangian method and the ALE method based on the HLLC flux have the potential to be applied to multi-material flows. Comparing with Eulerian methods, Lagrangian type methods avoid or can reduce a source of numerical error due to the advection terms in the conservation equations. For this reason, Lagrangian type methods are frequently preferred in one-dimensional computations where mesh distortion plays no role. Even though the Euler equations are much simpler in the Lagrangian framework as they do not contain the advection terms, in two or more space dimensions they are actually more difficult to solve since the mesh moves with the fluid and can easily lose its quality. In the past years, many efforts have been made to develop Lagrangian type methods. Some algorithms are developed from the non-conservative form of the Euler equations, for example, those discussed in [23], [3], [4], [5], [18], [37]. The other class of Lagrangian type algorithms starts from the conservative form of the Euler equations which usually can guarantee exact conservation. See for example [2], [8], [9], [7], [17], [22], [34], [20] etc.

Most existing Lagrangian type schemes for the Euler equations have first or at most second-order accuracy. Among them many Lagrangian schemes of non-conservative form are only first-order accurate, because of a first-order error due to the non-conservative formulation of the momentum equation. On the other hand, some of the conservative Lagrangian type schemes apply the linear interpolation strategy to achieve second-order accuracy, meanwhile they usually use a flux limiter to control spurious oscillations which leads to a possible loss of this second-order accuracy at some special points such as smooth extrema and sonic points.

Essentially non-oscillatory (ENO) schemes, first introduced by Harten and Osher [13] and Harten et al. [12], can achieve uniformly high order accuracy with sharp, essentially non-oscillatory shock transitions. In the subsequent years, ENO schemes in the Eulerian formulation have accomplished successful applications in many fields especially with problems containing both shocks and complicated smooth flow structures, see for example [29]. Eulerian ENO schemes on unstructured meshes are developed in [1]. However, the application of the ENO methodology in the Lagrangian formulation does not seem to have been extensively explored.

In this paper, we develop a class of Lagrangian type schemes for solving the Euler equations which are based on the high order ENO reconstruction both in the Cartesian and in the cylindrical coordinates. The schemes are conservative for the density, momentum and total energy, can maintain formal high order accuracy both in space and time and can achieve at least uniformly second-order accuracy on moving and distorted Lagrangian meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. They should also be generalizable to higher than second-order accuracy by using curved meshes, but this generalization is not carried out in this paper. Several one and two-dimensional numerical examples in the Cartesian and cylindrical coordinates are presented which demonstrate the good performance of the schemes both in purely Lagrangian and in ALE calculations.

An outline of the rest of this paper is as follows. In Section 2, we describe the individual steps of the ENO Lagrangian type scheme in one space dimension. In Section 3, we present one-dimensional numerical results. In Section 4, we extend the scheme to two space dimensions both in the Cartesian and in the cylindrical coordinates, while in Section 5 two-dimensional numerical examples are given to verify the performance of the ENO Lagrangian type method. In Section 6 we give concluding remarks.

Section snippets

The compressible Euler equations in Lagrangian formulation

The Euler equations for unsteady compressible flow in the reference frame of a moving control volume can be expressed in integral form in the Cartesian coordinates asddtΩ(t)UdΩ+Γ(t)FdΓ=0,where Ω(t) is the moving control volume enclosed by its boundary Γ(t). The vector of the conserved variables U and the flux vector F are given byU=ρME,F=(u-x˙)·nρ(u-x˙)·nM+p·n(u-x˙)·nE+pu·n,where ρ is the density, u is the velocity, M = ρu is the momentum, E is the total energy and p is the pressure, x˙ is the

Numerical results in one space dimension

In this section, we perform some numerical experiments in one space dimension. Purely Lagrangian computation and the ideal gas with γ = 1.4 are used to do the following tests unless otherwise stated. We mainly show the results obtained with the Dukowicz flux but we also show the results with the other fluxes (the Godunov flux, the L–F flux and the HLLC flux) for some test cases for comparison.

The scheme in the Cartesian coordinates

The 2D spatial domain Ω is discretized into M × N computational cells. Ii+1/2,j+1/2 is a quadrilateral cell constructed by the four vertices {(xi,j, yi,j), (xi+1,j, yi+1,j), (xi+1,j+1, yi+1,j+1), (xi,j+1, yi,j+1)}. Si+1/2,j+1/2 is denoted to be the area of the cell Ii+1/2,j+1/2 with i = 1,  , M, j = 1,  , N. For a given cell Ii+1/2,j+1/2, the location of the cell center is denoted by (xi+1/2,j+1/2, yi+1/2,j+1/2). The fluid velocity (ui,j, vi,j) is defined at the vertex of the mesh. On the non-staggered mesh,

Numerical results in two space dimensions

It is much more difficult to simulate a 2D problem than to simulate a 1D one in the Lagrangian framework, mainly because of the mesh distortion in multi-dimensions. In this section, although we have run most examples using the first, second and third-order schemes with the Godunov flux, the Dukowicz flux, the HLLC flux and the L–F flux, respectively, we will only show the results performed by the Dukowicz flux as representatives unless the results of the different fluxes are obviously different.

Concluding remarks

In this paper we have described a class of Lagrangian type schemes for solving Euler equations which are based on high order essentially non-oscillatory (ENO) reconstruction both in the Cartesian coordinates and in the cylindrical coordinates. The schemes are conservative for density, momentum and total energy, maintain formal high order accuracy both in space and time and can achieve at least uniformly second-order accuracy with moving and distorted Lagrangian meshes, are essentially

Acknowledgements

The research of the first author is supported in part by NSFC grant 10572028, with additional support provided by the National Basic Research Program of China under grant 2005CB321702, by the Foundation of National Key Laboratory of Computational Physics under grant 9140C6902010603 and by the National Hi-Tech Inertial Confinement Fusion Committee of China. The research of the second author is supported in part by NSFC grant 10671190 and by the Chinese Academy of Sciences during his visit to the

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