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Improvement on spherical symmetry in two-dimensional

cylindrical coordinates for a class of control volume Lagrangian

schemes

Juan Cheng1and Chi-Wang Shu2

Abstract

In [14], Maire developed a class of cell-centered Lagrangian schemes for solving Euler

equations of compressible gas dynamics in cylindrical coordinates. These schemes use a

node-based discretization of the numerical fluxes. The control volume version has several

distinguished properties, including the conservation of mass, momentum and total energy

and compatibility with the geometric conservation law (GCL). However it also has a lim-

itation in that it cannot preserve spherical symmetry for one-dimensional spherical flow.

An alternative is also given to use the first order area-weighted approach which can ensure

spherical symmetry, at the price of sacrificing conservation of momentum. In this paper,

we apply the methodology proposed in our recent work [8] to the first order control volume

scheme of Maire in [14] to obtain the spherical symmetry property. The modified scheme

can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry

when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original

good properties such as conservation and GCL. Several two-dimensional numerical examples

in cylindrical coordinates are presented to demonstrate the good performance of the scheme

in terms of symmetry, non-oscillation and robustness properties.

Keywords: control volume Lagrangian scheme; spherical symmetry preservation; con-

servative; cell-centered; compressible flow; cylindrical coordinates

1Institute of Applied Physics and Computational Mathematics, Beijing 100088, China.

cheng juan@iapcm.ac.cn. Research is supported in part by NSFC grants 10972043 and 10931004. Addi-

tional support is provided by the National Basic Research Program of China under grant 2005CB321702.

2DivisionofAppliedMathematics, Brown

shu@dam.brown.edu.Research is supported in part by ARO grant W911NF-08-1-0520 and NSF

grant DMS-0809086.

E-mail:

University, Providence,RI02912. E-mail:

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1Introduction

The Lagrangian method is one of the main numerical methods for simulating multidimen-

sional fluid flow, in which the mesh moves with the local fluid velocity. It is widely used

in many fields for multi-material flow simulations such as astrophysics, inertial confinement

fusion (ICF) and computational fluid dynamics (CFD), due to its distinguished advantage in

capturing material interfaces automatically and sharply. There are two kinds of Lagrangian

methods. One is built on a staggered discretization in which velocity (momentum) is stored

at vertices, while density and internal energy are stored at cell centers. The density / inter-

nal energy and velocity are solved on two different control volumes, see, e.g. [18, 1, 3]. This

kind of Lagrangian schemes usually uses an artificial viscosity term, for example [18, 4, 5],

to ensure the dissipation of kinetic energy into internal energy through shock waves. The

other is based on the cell-centered discretization in which density, momentum and energy are

all centered within cells and evolved on the same control volume, e.g. [10, 17, 15, 6, 7, 13].

This kind of schemes does not require the addition of an explicit artificial viscosity for shock

capturing. Numerical diffusion is implicitly contained in the Riemann solvers.

It is a critical issue for a Lagrangian scheme to keep certain symmetry in a coordinate

system different from that symmetry. For example, in the simulation of implosions, since

the small deviation from spherical symmetry due to numerical errors may be amplified by

Rayleigh-Taylor or other instability which may lead to unexpected large errors, it is very

important for the scheme to keep the spherical symmetry. In the past several decades,

many research works have been performed concerning the spherical symmetry preservation

in two-dimensional cylindrical coordinates. The most widely used method that keeps spher-

ical symmetry exactly on an equal-angle-zoned grid in cylindrical coordinates is the area-

weighted method [23, 2, 20, 22, 3, 14]. In this approach one uses a Cartesian form of the

momentum equation in the cylindrical coordinate system, hence integration is performed

on area rather than on the true volume in cylindrical coordinates. However, these area-

weighted schemes have a flaw in that they may violate momentum conservation. Margolin

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and Shashkov used a curvilinear grid to construct symmetry-preserving discretizations for

Lagrangian gas dynamics [16]. In our recent work [8], we have developed a new cell-centered

control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics

in two-dimensional cylindrical coordinates. Based on the strategy of local coordinate trans-

form and a careful treatment of the source term in the momentum equation, the scheme is

designed to be able to preserve one-dimensional spherical symmetry in a two-dimensional

cylindrical geometry when computed on an equal-angle-zoned initial grid. A distinguished

feature of our scheme is that it can keep both the symmetry and conservation properties on

the straight-line grid. However, our scheme in [8] does not satisfy the geometric conservation

law (GCL).

In [14], Maire developed a class of high order cell-centered Lagrangian schemes for solv-

ing Euler equations of compressible gas dynamics in cylindrical coordinates. A node-based

discretization of the numerical fluxes is given which makes the finite volume scheme compat-

ible with the geometric conservation law. Both the control volume and area-weighted dis-

cretizations of the momentum equations are presented in [14]. The control volume scheme is

conservative for mass, momentum and total energy, and satisfies a local entropy inequality in

its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the

other hand, the first order area-weighted scheme is conservative for mass and total energy

and preserves spherical symmetry for one-dimensional spherical flow on equal-angle polar

grid, but it cannot preserve the momentum conservation and does not satisfy the entropy

inequality. Numerical tests are given in [14] which verify the robustness of the schemes.

In this paper, we attempt to apply the strategy proposed in [8] on Maire’s first order con-

trol volume Lagrangian scheme [14] to improve its property in symmetry preservation while

keeping its main original good properties including GCL, conservation of mass, momentum

and total energy.

An outline of the rest of this paper is as follows. In Section 2, we describe the modified

scheme and discuss some critical issues such as GCL, conservation and spherical symmetry

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preservation about the scheme. In Section 3, numerical examples are given to demonstrate

the performance of the new modified cell-centered Lagrangian scheme. In Section 4 we will

give concluding remarks.

2The improvement on the cell-centered control vol-

ume Lagrangian scheme of Maire in cylindrical co-

ordinates

2.1The compressible Euler equations in a Lagrangian formulation

in cylindrical coordinates

The compressible inviscid flow is governed by the Euler equations which have the following

integral form in the Lagrangian formulation

d

dt

d

dt

d

dt

??

??

Ω(t)ρdV

Ω(t)ρudV

Ω(t)ρEdV

= 0

= −?

??

Γ(t)Pnds

Γ(t)Pu · nds

= −?

(2.1)

where ρ is the density, P is the pressure, u is the vector of velocity, E is the specific total

energy, and n is the unit outward normal to the boundary Γ(t).

The geometric conservation law refers to the fact that the rate of change of a Lagrangian

volume should be computed consistently with the node motion, which can be formulated as

d

dt

??

Ω(t)

dV =

?

Γ(t)

u · nds.(2.2)

In this paper, we seek to study the axisymmetric compressible Euler system. Its specific

form in the cylindrical coordinates is as follows

d

dt

d

dt

d

dt

d

dt

d

dt

??

??

??

Ω(t)ρrdrdz

Ω(t)rdrdz

Ω(t)ρuzrdrdz = −?

Ω(t)ρErdrdz

=

=

0

??

??

?

Γ(t)u · nrdl

Γ(t)Pnzrdl

Γ(t)Pnrrdl +??

Ω(t)ρurrdrdz = −?

Ω(t)Pdrdz

= −?

Γ(t)Pu · nrdl

(2.3)

where z and r are the axial and radial directions respectively. u = (uz,ur), where uz, urare

the velocity components in the z and r directions respectively, and n = (nz,nr) is the unit

outward normal to the boundary Γ(t) in the z-r coordinates.

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The set of equations is completed by the addition of an equation of state (EOS) with the

following general form

P = P(ρ,e) (2.4)

where e = E −1

then the equation of state has a simpler form

2|u|2is the specific internal energy. Especially, if we consider the ideal gas,

P = (γ − 1)ρe

where γ is a constant representing the ratio of specific heat capacities of the fluid.

In the next subsection, we will first summarize the control volume scheme of Maire in

[14].

2.2The control volume scheme of Maire in cylindrical coordinates

2.2.1Notations and assumptions

We will mostly use the notations in [14].The 2D spatial domain Ω is discretized into

quadrangular computational cells. Each quadrangular cell is assigned a unique index c, and

is denoted by Ωc(t). The boundary of the cell Ωcis denoted as ∂Ωc. Each vertex of the mesh

is assigned a unique index p and we denote the counterclockwise ordered list of the vertices

of the cell Ωcby p(c). The cell Ωcis surrounded by four cells denoted as Ωb, Ωr, Ωt, Ωlwhich

correspond to the bottom, right, top and left positions respectively. Acdenotes the area of

the cell Ωc. Vcis the volume of the cell, that is, the volume of the circular ring obtained by

rotating this cell around the azimuthal z-axis (without the 2π factor).

Using these notations, the set of equations (2.3) can be rewritten in the following control

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