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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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volume formulation

mcd

dt

?1

ρc

?

=

?

Pnzrdl

∂Ωc

u · nrdl

mcd

dtuz

c= −

?

?

?

∂Ωc

mcd

dtur

c= −

∂Ωc

Pnrrdl +

??

Ωc

Pdrdz

mcd

dtEc= −

∂Ωc

Pu · nrdl (2.5)

where mc=??

time marching according to the first equation in (2.3). ρc, uc= (uz

Ωcρrdrdz denotes the mass in the cell Ωc, which keeps constant during the

c,ur

c) and Ecrepresent

the density, velocity and total energy of the cell Ωcwhich are defined as follows

ρc=1

Vc

??

??

??

??

Ωc

ρrdrdz,

uz

c=

1

mc

1

mc

1

mc

Ωc

ρuzrdrdz,

ur

c=

Ωc

ρurrdrdz,

Ec=

Ωc

ρErdrdz.

The coordinates and velocity of the point p are denoted as (zp,rp) and up = (uz

p,ur

p)

respectively. lpp− and lpp+ denote the lengths of the edges [p−,p] and [p,p+], and npp− and

npp+ are the corresponding unit outward normals, where p−,p+are the two neighboring

points of the point p (see Figure 2.1).

In the cell-centered control volume Lagrangian scheme presented in [14], the discrete

gradient operators over the cell Ωcare constructed by introducing two nodal pressures at

each node p of the cell Ωc. These pressures are denoted as πc

pand πc

p, see Figure 2.1. They

are related to the two edges sharing the node p. The half lengths and the unit outward

normals of the edges connected to the point p are denoted as follows

lc

p=1

2lpp−,lc

p=1

2lpp+,

nc

p= npp−,

nc

p= npp+.(2.6)

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u

p?

p

nc

p

p?

nc

p

c

p

c

pr

c

p

?

c

pr

c

p

?

c

?

( , ,

c

? u

)

ccp

lp

clp

Figure 2.1: Notations related to the cell Ωc.

The pseudo radii rc

pand rc

pare defined as

rc

p=1

3(2rp+ rp−),rc

p=1

3(2rp+ rp+).(2.7)

2.2.2Computation of nodal velocity and pressure

In the paper [14], the specific way in determining the nodal velocity and pressure guarantees

that the scheme satisfies the following sufficient condition for total energy conservation

?

c∈c(p)

(rc

plc

pπc

pnc,r

p + rc

plc

pπc

pnc,r

p) = 0,(2.8)

where c(p) is the set of the cells around the point p.

The specific formulas to calculate the nodal velocity upand nodal pressures πc

pand πc

p

are as follows

up= M−1

p

?

p(up− uc) · nc

c∈c(p)

(rc

plc

pnc

p+ rc

plc

pnc

p)Pc+ uc,

Pc− πc

Pc− πc

p= zc

p,

p= zc

p(up− uc) · nc

p, (2.9)

where Pcis the pressure of the cell Ωcdetermined by {ρc,uc,Ec}. The 2 × 2 matrices Mpc

and Mpare denoted as

Mpc= zc

prc

plc

p(nc

p⊗ nc

p) + zc

prc

plc

p(nc

p⊗ nc

p),Mp=

?

c∈c(p)

Mpc

(2.10)

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where zc

pand zc

pare the mass fluxes swept by the waves which can be determined in several

ways, such as the Dukowicz approach or the acoustic approach. We refer the reader to the

paper [14] for more details. In this paper, we will only use the acoustic approach, that is

zc

p= zc

p= ρcac.(2.11)

where acis the local isentropic speed of sound.

After we get the nodal velocity upat point p, the point moves with the following local

kinematic equation

d

dtxp= up,

xp(0) = x0

p, (2.12)

where xp= (zp,rp) defines the position of point p at t > 0 and x0

pdenotes its initial position.

2.2.3Spatial discretization

The semi-discrete finite volume scheme of the governing equations (2.5) is of the following

form

mcd

dt

?1

ρc

?

−

?

(rc

p∈p(c)

(rc

plc

pnc

p+ rc

plc

pnc

p) · up= 0,

mcd

dtuz

c+

?

?

?

p∈p(c)

plc

pπc

pnc,z

p + rc

plc

pπc

pnc,z

p) = 0,

mcd

dtur

c+

p∈p(c)

(rc

plc

pπc

pnc,r

p+ rc

plc

pπc

pnc,r

p) − AcPc= 0,

mcd

dtEc+

p∈p(c)

(rc

plc

pπc

pnc

p+ rc

plc

pπc

pnc

p) · up= 0(2.13)

where nc,z

p,nc,z

p

and nc,r

p,nc,r

p

are the components of nc

pand nc

palong the z and r directions

respectively.

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2.2.4Time discretization

The time discretization for the equation of nodal movement (2.12) is the Euler forward

method given as follows

zn+1

p

= zn

p+ ∆tnuz,n

p

rn+1

p

= rn

p+ ∆tnur,n

p

(2.14)

where uz,n

p,ur,n

p

are the z and r components of upat the nth time step.

As a first order scheme, the time marching for the semi-discrete scheme (2.5) can also

be accomplished by the Euler forward method. Thus the fully discretized scheme can be

written as follows

mc

1

ρn+1

c

−

− uz,n

− ur,n

− En

1

ρn

c

uz,n+1

c

ur,n+1

c

En+1

c

c

c

c

p∈p(c)(rc,n

p∈p(c)(rc,n

?

(2.15)

= ∆tn

−

−?

plc,n

plc,n

plc,n

plc,n

pnc,n

pπc,n

pπc,n

pπc,n

p + rc,n

pnc,z,n

p

pnc,r,n

p

pnc,n

plc,n

+ rc,n

+ rc,n

plc,n

pnc,n

plc,n

plc,n

pπc,n

p) · un

pπc,n

pπc,n

pnc,n

p

?

?

pnc,z,n

pnc,r,n

p) · un

p

)

)

p∈p(c)(rc,n

p∈p(c)(rc,n

p

p + rc,n

p

+

0

0

cPn

0

An

c

Here the variables with the superscripts ’n’ and ’n + 1’ represent the values of the cor-

responding variables at the nth and (n + 1)th time steps respectively. The scheme (2.15) is

consistent with the Euler equations (2.5) and has first order accuracy in space and time.

In [14], the time step ∆tnis controlled by both the CFL condition and the criterion on

the variation of volume, that is, the CFL condition is satisfied as follows

∆te= Cemin

c(ln

c/an

c)

where ln

cis the shortest edge length of the cell Ωc, and an

cis the sound speed within this

cell. Ceis the Courant number which is set to be 0.5 unless otherwise stated in the following

tests.

The criterion on the variation of volume is accomplished as

∆tv= Cv

?

Vn

c

|d

dtVc(tn)|

?

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where

d

dtVc(tn) =

Vn+1

c

−Vn

c

∆tn

. The parameter Cv= 0.1 is used in the numerical simulations.

Finally, the next time step ∆tn+1is given by

∆tn+1= min(∆te,∆tv,Cm∆tn)

where Cm= 1.01.

2.3The improvement of the scheme on the symmetry property

The control volume scheme (2.15) is theoretically proven to have many good properties

such as the conservation, GCL and entropy inequality, and is also verified to have good

performance in practical simulations. However, it has a limitation in not being able to

preserve the spherical symmetry. In this paper, we attempt to improve the scheme in this

aspect using the strategy of our recent work [8]. Somewhat differently from the approach

of the explicit local coordinate transform in [8], here we perform the improvement on the

scheme (2.13) in z-r coordinates directly and use local coordinate transform only in the

symmetry-preserving proof but not in the actual implementation of the algorithm. In fact,

the key ingredient to make the scheme satisfy the spherical symmetry property in a 2D

cylindrical geometry is the treatment of the source term. To be more specific, we replace Pc

in the source term of the r-momentum equation in (2.13) by Pawhich is defined as follows

Pa=1

4(πc

1+ πc

2+ πc

3+ πc

4)(2.16)

where πc

1, πc

2, πc

3and πc

4are the values of pressure related to the two radial edges of the cell

Ωc(see Figure 2.2). We can easily see that AcPaapproximates??

order as AcPc.

ΩcPdrdz with the same

Thus, the difference between the modified scheme and the original scheme lies only in

the expression of the r-momentum equation. In summary, the modified semi-discrete scheme

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z

r

r

?

?

?

4

?

1

2

3

4

l

?

c

?

t

?

b

?

??

000

,, p

?

u

111

,, p

? u

222

,, p

?

u

1

c

?

2

c

?

3

c

?

c

Figure 2.2: Equi-angular polar grid for the cylindrical geometry.

can be expressed as follows

mcd

dt

?1

ρc

?

−

?

(rc

p∈p(c)

(rc

plc

pnc

p+ rc

plc

pnc

p) · up= 0,

mcd

dtuz

c+

?

?

?

p∈p(c)

plc

pπc

pnc,z

p + rc

plc

pπc

pnc,z

p) = 0,

mcd

dtur

c+

p∈p(c)

(rc

plc

pπc

pnc,r

p+ rc

plc

pπc

pnc,r

p) − AcPa= 0,

mcd

dtEc+

p∈p(c)

(rc

plc

pπc

pnc

p+ rc

plc

pπc

pnc

p) · up= 0.(2.17)

The fully discretized modified scheme is of the following expression

mc

1

ρn+1

c

−

− uz,n

− ur,n

− En

1

ρn

c

uz,n+1

c

ur,n+1

c

En+1

c

c

c

c

p∈p(c)(rc,n

p∈p(c)(rc,n

?

(2.18)

= ∆tn

−

−?

plc,n

plc,n

plc,n

plc,n

pnc,n

pπc,n

pπc,n

pπc,n

p + rc,n

pnc,z,n

p

pnc,r,n

p

pnc,n

plc,n

+ rc,n

+ rc,n

plc,n

pnc,n

plc,n

plc,n

pπc,n

p) · un

pπc,n

pπc,n

pnc,n

p

?

?

pnc,z,n

pnc,r,n

p) · un

p

)

)

p∈p(c)(rc,n

p∈p(c)(rc,n

p

p + rc,n

p

+

0

0

cPn

0

An

a

2.3.1The issue of conservation, GCL and entropy inequality

Compared with the scheme (2.15) of Maire [14], we only use an alternative approach (2.16)

to the pressure appearing in the source term of the r-momentum equation to guarantee

the scheme’s spherical symmetry preservation property that will be shown in the following

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section. Since we do not make any change on any other terms in the original scheme (2.15),

the modified scheme (2.18) can also satisfy the conservation for the mass, z-momentum

and total energy just as the original scheme. As to the conservation of the r-momentum,

summing the r-momentum equation in (2.17) over all the cells, we have

d

dt

??

c

mcur

c

?

= −

?

c

?

p∈p(c)

(rc

plc

pπc

pnc,r

p + rc

plc

pπc

pnc,r

p) − AcPa

(2.19)

Switching the summation over cells and the summation over nodes at the right-hand side of

Equation (2.19), it can be rewritten as

d

dt

??

c

mcur

c

?

= −

?

p

?

c∈c(p)

(rc

plc

pπc

pnc,r

p+ rc

plc

pπc

pnc,r

p) +

?

c

AcPa

(2.20)

Using the sufficient condition for total energy conservation (2.8), we have,

d

dt

??

c

mcur

c

?

=

?

c

AcPa

(2.21)

The equation (2.21) refers to the conservation of the r-momentum, which, together with the

conservation of the mass, z-momentum and total energy, guarantees the modified scheme

(2.18) satisfies the Lax-Wendroff theorem.

Since the scheme (2.18) employs the same compatible discretization of the geometry

conservation law (GCL) as that in the original scheme (2.15), it satisfies GCL naturally.

As to the entropy inequality, since we use the formulation of (2.16) to determine the

pressure in the source term rather than Pc, we can not write a similar entropy inequality for

the scheme (2.17) as that for the original scheme (2.13).

2.3.2The issue of symmetry preservation

In this section, we will prove the modified scheme (2.18) can keep the spherical symmetry

property computed on an equal-angle-zoned initial grid.

Theorem: The modified scheme (2.18) can keep the one-dimensional spherical symmetry

property computed on an equal-angle zoned initial grid. That is, if the solution has one-

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dimensional spherical symmetry at the initial time, then the computational solution will keep

this symmetry with the time marching.

Proof: Without loss of the generality, we only need to prove the solution of the modified

scheme (2.18) can keep the spherical symmetry at the (n + 1)th time step, if the solution is

known to be of spherical symmetry at the nth time step. Notice that, for the Lagrangian

solution, symmetry preserving refers to the evolution of both the conserved variables and

the grid.

To facilitate the proof, we first simplify the vertex indices of the cell Ω(c) as p = 1,2,3,4

(shown in Figure 2.2) and rewrite the momentum equations in (2.17) along the cell’s local

ξ-θ coordinates. The rewritten semi-discrete scheme is of the following form

mcd

dt

?1

ρc

?

=

?

p=1,4

(rc

plc

pnc

p+ rc

plc

pnc

p) · up,

mcd

dtuξ

c= −

?

?

?

p=1,4

(rc

plc

pπc

pnc,ξ

p + rc

plc

pπc

pnc,ξ

p) + AcPasinθc,

mcd

dtuθ

c= −

p=1,4

(rc

plc

pπc

pnc,θ

p + rc

plc

pπc

pnc,θ

p) + AcPacosθc,

mcd

dtEc= −

p=1,4

(rc

plc

pπc

pnc

p+ rc

plc

pπc

pnc

p) · up

(2.22)

where ξ is the radial direction passing through the cell center and the origin. For an equal-

angle-zoned grid, the cell shown in Figure 2.2 is an equal-sided trapezoid, it has the property

that the angles between ξ and the two equal sides of the cell are the same. θ is the angular

direction which is orthogonal to ξ, see Figure 2.2. uξ

cand uθ

care the component values of

velocity in the local ξ and θ directions respectively, nc,ξ

p,nc,ξ

p and nc,θ

p,nc,θ

p are the components

of nc

pand nc

palong ξ and θ directions respectively. θcis set to be the angle between the local

ξ direction and the z coordinate.

For the convenience of notation, we adopt the convention that variables without the

superscript ’n + 1’ are those at the nth time step. Assume that at the nth time step the

grid is a polar grid with equal angles (see Figure 2.2) and the cell averages of the conserved

variables including density, momentum and total energy are symmetrical on this grid, namely

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these variables in the cells with the same radial position are identical. Consider the cell Ωc

and its neighboring cells, in the local ξ-θ coordinates of the cell Ωc, we have

ρc= ρt= ρb= ρ1,

Pc= Pt= Pb= P1,

ξ1= ξ4= ξl,

uc= (u1,0),

ρl= ρ0,

Pl= P0,

ρr= ρ2,

Pr= P2,

ξ2= ξ3= ξr,

ul= (u0,0),

ur= (u2,0)

(2.23)

where {ξk,k = 1,4} are the distance of the four vertices of the cell Ωc from the origin

respectively, and {uk,k = 0,2} are the magnitude of the cell velocity in the relevant cells.

Next we explain some notations concerning the grid geometry. For the cell Ωc, the lengths

of the cell edges are denoted as {l12,l23,l34,l41}. Since the grid is symmetrical, we can define

them as

l12= l34= lm,l14= ll,l23= lr. (2.24)

For the convenience of proof, in the following, we will project all the variables relative

to the determination of the cell’s and nodal velocity to the cell’s local ξ-θ coordinates. For

example, the outward normal direction of the cell’s four edges in the local ξ-θ coordinates

are as follows

n12= (nξ

n23= (nξ

n34= (nξ

n41= (nξ

12,nθ

23,nθ

34,nθ

41,nθ

12) = (−sin∆θ

23) = (1,0),

34) = (−sin∆θ

41) = (−1,0).

2,−cos∆θ

2),

2,cos∆θ

2),

As to the symmetry of the grid, in the case of a one-dimensional spherical flow computed

on an equal angle polar grid, Maire in the paper [14] has already given the proof that

the nodal velocity is radial and independent of the angular cell position which means the

symmetry preservation of the grid. From the results of the vertex velocity shown in [14], we

can deduce that the velocities at each vertex for the cell Ωcin its local ξ-θ coordinates are

14

Page 15

of the following form

uξ

1= uξ

4=z0u0+ z1u1− (P1− P0)

z0+ z1

4= −z0u0+ z1u1− (P1− P0)

3=z1u1+ z2u2− (P2− P1)

z1+ z2

3= −z1u1+ z2u2− (P2− P1)

,

uθ

1= −uθ

z0+ z1

tan∆θ

2,

uξ

2= uξ

,

uθ

2= −uθ

z1+ z2

tan∆θ

2

(2.25)

where z0= ρ0a0, z1= ρ1a1and z2= ρ2a2. {a0,a1,a2} are the speeds of sound in the cells

{Ωl,Ωc,Ωr} respectively.

Next we will prove the symmetry preservation of the evolved variables such as density,

cell velocity and total energy.

We will first write each variable appearing at the right-hand side of (2.22) in details. By

the following simple manipulation, the nodal pressures can be obtained

πc

1= P1− z1(u1− uc) · n41=z1P0+ z0P1− z0z1(u1− u0)

1= P1− z1(u1− uc) · n12= P1− z1u1sin∆θ

z0+ z1

,

πc

2,(2.26)

where uc= (u1,0), u1= (uξ

1,uθ

1).

Similarly

πc

2= πc

3= πc

4= πc

1= P1− z1u1sin∆θ

2,

πc

4= πc

1=z1P0+ z0P1− z0z1(u1− u0)

z0+ z1

3=z2P1+ z1P2− z1z2(u2− u1)

z1+ z2

,

πc

2= πc

.(2.27)

15

Page 16

For the simplicity of description, we denote

Pm= P1− z1u1sin∆θ

Pl=z1P0+ z0P1− z0z1(u1− u0)

z0+ z1

Pr=z2P1+ z1P2− z1z2(u2− u1)

z1+ z2

ul=z0u0+ z1u1− (P1− P0)

z0+ z1

ur=z1u1+ z2u2− (P2− P1)

z1+ z2

2,

,

,

,

.

By the formula (2.16), we have

Pa= Pm= P1− z1u1sin∆θ

2

(2.28)

The other corresponding variables appearing at the right hand side of (2.22) can be

described in the following details

lc

1= lc

4=1

2ll,

lc

1= lc

2= lc

3= lc

4=1

2lm,

lc

2= lc

3=1

2lr,

(nc,ξ

1,nc,θ

1) = (nc,ξ

4,nc,θ

4) = n41= (−1,0),

1) = n12= (−sin∆θ

2,nc,θ

3) = n34= (−sin∆θ

2= (ξl+ ξr)sin(θc−∆θ

4= (ξl+ ξr)sin(θc+∆θ

(nc,ξ

2,nc,θ

2) = (nc,ξ

1,nc,θ

2,−cos∆θ

2),

(nc,ξ

3,nc,θ

3) = (nc,ξ

2) = n23= (1,0),

(nc,ξ

4,nc,θ

4) = (nc,ξ

3,nc,θ

2,cos∆θ

2),

rc

1+ rc

2),

rc

3+ rc

2),

rc

2+ rc

3= 2sinθccos∆θ

2ξr,

rc

1+ rc

4= 2sinθccos∆θ

2ξl.(2.29)

16

Page 17

Substituting (2.26-2.29) into (2.22), we get

mcd

dt

sinθccos∆θ

sinθc[1

−1

−sinθccos∆θ

sinθccos∆θ

sinθc(−ξrlrPr+ ξlllPl+ 2AcPm)

−AcPmcosθc+ AcPmcosθc

−sinθccos∆θ

−cos∆θ

1/ρc

uξ

uθ

Ec

c

c

=

2(ξrlrur− ξlllul)

2(ξl+ ξr)lmPmsin∆θ + ξlllPl− ξrlrPr] + AcPmsinθc

2(ξl+ ξr)lmPmsin∆θcosθc+ AcPmcosθc

2(ξrlrurPr− ξlllulPl)

2(ξrlrur− ξlllul)

=

2(ξrlrurPr− ξlllulPl)

2(ξrlrur− ξlllul)

−(ξrlrPr− ξlllPl) + 2AcPm

0

2(ξrlrurPr− ξlllulPl)

= sinθc

cos∆θ

.(2.30)

Since the cell is an equal-sided trapezoid, we have

mc= ρ1Vc= ρ1rcAc= ρ1ξcAcsinθc

(2.31)

where rcand ξcare the values of r and ξ at the cell center respectively.

Thus from (2.30) and (2.31), we have

d

dt

1/ρc

uξ

uθ

Ec

c

c

=

1

ρ1ξcAc

cos∆θ

−(ξrlrPr− ξlllPl) + 2AcPm

0

−cos∆θ

2(ξrlrur− ξlllul)

2(ξrlrurPr− ξlllulPl)

(2.32)

Finally we obtain the modified scheme (2.18) in the following detailed expression

1/ρn+1

uξ,n+1

c

uθ,n+1

c

En+1

c

c

=

1/ρc

uξ

uθ

Ec

c

c

+

∆t

ρ1ξcAc

cos∆θ

−(ξrlrPr− ξlllPl) + 2AcPm

0

−cos∆θ

2(ξrlrur− ξlllul)

2(ξrlrurPr− ξlllulPl)

(2.33)

From the formula (2.33), we can see the cell velocity is radial and the magnitude of all

the conserved variables is independent of the angular position of the cell at the (n + 1)th

time step. The proof of the symmetry preservation property is thus completed.

17

Page 18

3Numerical results in the two-dimensional cylindrical

coordinates

In this section, we perform numerical experiments in two-dimensional cylindrical coordinates.

Purely Lagrangian computation, the ideal gas with γ = 5/3, the initially equal-angle polar

grid and the modified scheme (2.18) are used in the following tests unless otherwise stated.

Reflective boundary conditions are applied to the z and r axes in all the tests. For the

velocity of vertices located at the z coordinate, we obtain it by imposing the boundary

condition of zero normal velocity into the solver (2.9).

3.1Accuracy test

We test the accuracy of the modified scheme (2.18) on a free expansion problem given in [21].

The initial computational domain is [0,1] × [0,π/2] defined in the polar coordinates. The

gas is initially at rest with uniform density ρ = 1 and pressure has the following distribution,

p = 1 − (z2+ r2).

The analytical solution of the problem is as follows,

R(t) =

√1 + 2t2,

uξ(z,r,t) =

2t

1 + 2t2

1

R3,

1

R5

√z2+ r2,

ρ(z,r,t) =

p(z,r,t) =

?

1 −z2+ r2

R2

?

where R is the radius of the free outer boundary and uξrepresents the value of velocity in

the radial direction.

We perform the test both on an initially equal-angle polar grid and a random polar

grid, see Figure 3.1. For the random polar grid, each internal grid point is obtained by an

independent random perturbation on the angular direction from a equal-angle polar grid

18

Page 19

z

r

0 0.20.40.6 0.81

0

0.2

0.4

0.6

0.8

1

z

r

0 0.20.40.6 0.81

0

0.2

0.4

0.6

0.8

1

Figure 3.1: The initial grid of the free expansion problem with 20×20 cells. Left: equal-angle

polar grid; Right: random polar grid.

which can be expressed as follows

θk,l=

?

K − 1cosθk,l,

rk,l=k − 1

K − 1sinθk,l,

(l − 1)∆θ,

(l − 1)∆θ + c1ck,l∆θ,

zk,l=k − 1

k = 1,K or l = 1,L

else

1 ≤ k ≤ K,1 ≤ l ≤ L

where (zk,l,rk,l) is the z-r coordinate of the grid points with the sequential indices (k,l),k =

1,...,K,l = 1,...,L in the radial and angular directions respectively. K,L represent the

number of grid points in the above mentioned two directions. ∆θ =

π

2(L−1). −0.5 ≤ ck,l≤ 0.5

is the random number, c1is a parameter which is chosen as 0.5 in this test.

Free boundary condition is applied on the outer boundary. Figure 3.2 shows the final

grid. We can clearly observe symmetry in the left figure. The errors of the scheme on these

two kinds of grid at t = 1 are listed in Tables 3.1-3.2 which are measured on the interval

[1

5K,4

5K] × [1,L] to remove the influence from the boundary. From both of the tables, we

can see the expected first order accuracy for all the evolved conserved variables.

3.2Non-oscillatory tests

Example 1 (The Noh problem in a cylindrical coordinate system on the polar grid [19]).

We test the Noh problem which is a well known test problem widely used to validate

Lagrangian scheme in the regime of strong shock waves. In this test case, a cold gas with unit

19

Page 20

z

r

0 0.20.4 0.60.811.21.41.6 1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

z

r

00.20.40.60.811.2 1.41.61.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 3.2: The grid of the free expansion problem with 20 × 20 cells at t = 1. Left:

equal-angle polar grid; Right: random polar grid.

Table 3.1: Errors of the scheme in 2D cylindrical coordinates for the free expansion problem

using K × L initially equal-angle polar grid cells.

K = LNormDensityorderMomentum

20L1

0.97E-2

L∞

0.16E-1

40L1

0.53E-20.88

L∞

0.77E-21.03

80L1

0.28E-20.91

L∞

0.36E-21.07

160L1

0.15E-20.93

L∞

0.19E-20.93

orderEnergy

0.64E-2

0.11E-1

0.34E-2

0.52E-2

0.18E-2

0.28E-2

0.92E-3

0.15E-2

order

0.13E-1

0.19E-1

0.67E-2

0.11E-1

0.35E-2

0.58E-2

0.18E-2

0.30E-2

0.91

0.83

0.95

0.90

0.97

0.94

0.90

1.03

0.94

0.89

0.96

0.93

Table 3.2: Errors of the scheme in 2D cylindrical coordinates for the free expansion problem

using K × L initially random polar grid cells.

K = LNormDensityorderMomentum

20L1

0.94E-2

L∞

0.16E-1

40L1

0.50E-20.91

L∞

0.84E-20.89

80L1

0.27E-20.89

L∞

0.47E-20.86

160L1

0.14E-20.92

L∞

0.25E-20.89

order Energy

0.62E-2

0.11E-1

0.33E-2

0.67E-2

0.17E-2

0.35E-2

0.88E-3

0.19E-2

order

0.12E-1

0.19E-1

0.65E-2

0.12E-1

0.34E-2

0.62E-2

0.17E-2

0.35E-2

0.93

0.75

0.95

0.89

0.97

0.84

0.92

0.69

0.93

0.91

0.95

0.92

20

Page 21

z

r

0 0.10.20.30.4

0

0.1

0.2

0.3

0.4

z

r

00.10.2 0.30.4

0

0.1

0.2

0.3

0.4

Figure 3.3: The results of the Noh problem with 20 × 20 cells at t = 0.6. Left: the original

scheme (2.15); Right: the modified scheme (2.18).

density and zero internal energy is given with an initial inward radial velocity of magnitude

1. The equal-angle polar grid is applied in the1

4-circle computational domain defined in the

polar coordinates by [0,1]×[0,π/2]. The shock is generated in a perfect gas by bringing the

cold gas to rest at the origin. The analytical post shock density is 64 and the shock speed is

1/3. The comparison of the final grid with 20 × 20 cells between the original scheme (2.15)

and the modified scheme (2.18) is given in Figure 3.3 which demonstrates the improvement

of the symmetry property for the modified scheme. Figure 3.4 shows the results of the

modified scheme including the final grid with 200 × 20 cells and density as a function of

radial radius for two different angular zonings (200 × 20, 200 × 40) at t = 0.6. From Figure

3.4, we observe the results are symmetrical and non-oscillatory. The shock location and the

shock magnitude are closer to those of the analytical solution with the grid refinement in

the angular direction, which reflects the convergence trend of the numerical solution toward

the analytical solution.

Example 2 (The spherical Sedov problem in a cylindrical coordinate system on the polar

grid [21]).

We perform our test on the spherical Sedov blast wave problem in a cylindrical coordi-

nate system as an example of a diverging shock wave. The initial computational domain is a

1

4-circle region defined in the polar coordinates by [0,1.125] × [0,π/2]. The initial condition

consists of unit density, zero velocity and zero specific internal energy except in the cells

21

Page 22

z

r

00.20.40.60.81

0

0.2

0.4

0.6

0.8

1

z

r

00.20.4

0

0.2

0.4

radius

density

00.050.10.150.20.250.3

10

20

30

40

50

60

exact

numerical 200*20

numerical 200*40

Figure 3.4: The results of the Noh problem at t = 0.6. Left: initial grid with 200 × 20 cells;

Middle: final grid with 200 × 20 cells; Right: density vs radial radius with 200 × 20 and

200 × 40 cells respectively. Solid line: exact solution; dashed line: computational solution.

z

r

00.20.40.6 0.81

0

0.2

0.4

0.6

0.8

1

z

r

00.20.4 0.60.81

0

0.2

0.4

0.6

0.8

1

Figure 3.5: The results of the Sedov problem with 20×20 cells at t = 1.0. Left: the original

scheme (2.15); Right: the modified scheme (2.18).

connected to the origin where they share a total value of 0.2468. Reflective boundary condi-

tion is applied on the outer boundary. The analytical solution is a shock at radius unity at

time unity with a peak density of 4. Figure 3.5 shows the comparison of the final grid with

20 × 20 cells between the original scheme (2.15) and the modified scheme (2.18). We can

see that the latter has a perfect symmetry property. The final grid, density as a function of

the radial radius and surface of density with 100×30 cells obtained by the modified scheme

are displayed in Figure 3.6. We also observe the expected symmetry in the plot of grid.

The shock position and peak density coincide with those of the analytical solution very well

and there is no spurious oscillation, demonstrating the good performance of the scheme in

symmetry preserving, non-oscillation and accuracy properties.

Example 3 (The one-dimensional spherical Sod Riemann problem).

22

Page 23

z

r

0 0.2 0.4 0.60.81

0

0.2

0.4

0.6

0.8

1

radius

density

00.20.40.60.81

0

1

2

3

4

exact

numerical

z

0

0.5

1

0

0.5

1

density

0

1

2

3

4

Figure 3.6: The result of the Sedov problem with 100 × 30 cells at t = 1.0. Left: final

grid; Middle: density vs radial radius. Solid line: exact solution; dashed line: computational

solution; Right: surface of density.

The modified scheme (2.18) is tested on the Sod Riemann problem in the cylindrical

coordinates.The initial computational domain is a

1

4-circle region defined in the polar

coordinates by [0,20] × [0,π/2]. Its initial condition is as follows

(ρ,uξ,p) = (1.0,0,1.0),0 ≤ ξ ≤ 10

(ρ,uξ,p) = (0.125,0,0.1),10 < ξ ≤ 20.

Reflective boundary condition is applied on the outer boundary. The reference solution

is the converged result obtained by using a one-dimensional second-order Eulerian code in

the spherical coordinate with 10000 grid points. Figure 3.7 shows the numerical results of

the grid and density as a function of the radial radius and the surface of density performed

by the modified scheme with 400 × 10 equal-angle polar cells at t = 1.4. We observe the

good behavior of the scheme in symmetry and the good agreement between the numerical

result and the reference solution.

Example 4 (Kidder’s isentropic compression problem [11, 14]).

This problem is a self-similar isentropic problem which is usually used to validate the

capability of a Lagrangian scheme in simulating a spherical isentropic compression. At the

initial time, the shell with a ring shape constitutes the computational region [ξ1,ξ2]×[0,π/2]

in the polar coordinates, where ξ1= 0.9 is the internal radius and ξ2= 1 is the external

23

Page 24

z

r

05101520

0

5

10

15

20

radius

density

6810121416

0.2

0.4

0.6

0.8

1

reference

numerical

1.00

0.94

0.88

0.82

0.76

0.70

0.64

0.57

0.51

0.45

0.39

0.33

0.27

0.21

0.15

Figure 3.7: The results of the Sod problem at t = 1.4. Left: final grid with 400 × 10 cells;

Middle: density versus radial radius. Solid line: exact solution; dashed line: computational

solution; Right: surface of density in the whole circle region obtained by a mirror image.

radius. The initial density and pressure ρ0,P0are expressed as follows

ρ0(ξ) =

?ξ2

2− ξ2

ξ2

2− ξ2

1

ργ−1

1

+ξ2− ξ2

ξ2

1

2− ξ2

1

ργ−1

2

?

1

γ−1

,P0(ξ) = s(ρ0(ξ))γ

where ρ1= 6.31 × 10−4, ρ2= 10−2, s = 2.15 × 104, γ = 5/3. The pressure P1(t) and P2(t)

are imposed continuously at the internal and external boundary respectively which have the

following representation

P1(t) = P0

1a(t)−

2γ

γ−1,P2(t) = P0

2a(t)−

2γ

γ−1,

where P0

1= 0.1, P0

2= 10 and a(t) =

?

1 − (t

τ)2in which τ = 6.72 × 10−3is the focusing

time of the shell and t ∈ [0,τ) is the evolving time.

Denoting ζ(ξ,t) to be the radius at time t of a point initially located at radius ξ, its

analytical solution is ζ(ξ,t) = a(t)ξ. The analytical solutions of three fundamental variables

for this problem in spherical geometry are as follows

ρ(ζ(ξ,t),t) = ρ0(ξ)ξa(t)−

u(ζ(ξ,t),t) = ξd

2

γ−1,

dta(t),

p(ζ(ξ,t),t) = P0(ξ)ξa(t)−2γ

γ−1.

We test the modified scheme (2.18) on the problem with 80 × 40, 160 × 80, 320 × 160

cells respectively. The final time is set to be t = 0.99τ. Figure 3.8 shows the initial and final

24

Page 25

z

r

00.2 0.40.6 0.81

0

0.2

0.4

0.6

0.8

1

z

r

00.020.040.060.080.1 0.120.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time

radius

0.0020.0040.006

0.2

0.4

0.6

0.8

1

exact

numerical

Figure 3.8: The results of the Kidder problem with 40×20 grids. Left: initial grid; Middle:

final grid at t = 0.99τ; Right: trajectory of external boundary compared with the exact

solution. Solid line: exact solution; dashed line: computational solution.

radius

density

0.130.135 0.14

0

0.5

1

1.5

2

2.5

3

3.5

exact

numerical 80*40

numerical 160*80

numerical 320*160

radius

velocity

0.130.1350.14

-1040

-1020

-1000

-980

-960

-940

-920

exact

numerical 80*40

numerical 160*80

numerical 320*160

radius

pressure

0.130.135 0.14

0

50000

100000

150000

exact

numerical 80*40

numerical 160*80

numerical 320*160

Figure 3.9: The results of the Kidder problem at t = 0.99τ with three different zonings.

Left: density vs radial radius; Middle: velocity vs radial radius; Right: pressure vs radial

radius. Solid line: exact solution; dashed line: computational solution.

grids and the time evolution of the position of the external boundary with 40 × 20 grids.

Figure 3.9 shows the results of density, velocity and pressure at the final time. From these

figures, we can see the perfect symmetry in the grid. The trajectory of the external boundary

coincides with the analytical solution quite well. The numerical solutions of density, velocity

and pressure converge to the analytical solutions asymptotically.

Example 5 (Implosion problem of Lazarus [12]).

The implosion problem of Lazarus is a self-similar problem. At the initial time, a sphere

of unit radius with unit density and zero specific internal energy is driven by the following

inward radial velocity

uξ(t) =

−αf

(1 − ft)1−α

(3.1)

25

Page 26

z

r

00.1 0.20.30.4

0

0.1

0.2

0.3

0.4

radius

density

00.10.20.30.4

0

5

10

15

20

25

30

reference

reference

30 angular cells t=0.74

30 angular cells t=0.8

60 angular cells t=0.74

60 angular cells t=0.8

t=0.74

t=0.8

z

0

0.1

0.2

0.3

0.4

r

0

0.1

0.2

0.3

0.4

density

5

10

15

20

25

Figure 3.10: The results of the Lazarus problem. Left: final grid with 200×30 cells at t = 0.8;

Middle: density vs radial radius at t = 0.74, 0.8 with 200×30 and 200×60 cells respectively.

Solid line: reference solution; dashed line: computational solution; Right: surface of density

at t = 0.8 with 200 × 30 cells.

where α = 0.6883545, f = 1 − εt − δt3, ε = 0.185, δ = 0.28.

We test the problem on a grid of 200 × 30 cells in the initial computational domain

[0,1]×[0,π/2] defined in the polar coordinates. The numerically converged result computed

using a one-dimensional second-order Lagrangian code in the spherical coordinate with 10000

cells is used as a reference solution. We display the results of the modified scheme (2.18)

using 200 × 30 and 200 ×60 cells in Figure 3.10. In the plot of grid, we notice the expected

symmetry. In the plot of density, we observe the non-oscillatory and accurate numerical

solution and the convergence tendency of the numerical results toward the reference solution.

Example 6 (Coggeshall expansion problem [9]).

This is a two-dimensional adiabatic compression problem proposed by Coggeshall. We

attempt to apply it to test the performance of the modified scheme (2.18) on a truly two-

dimensional problem. The computational domain consists of a quarter of a sphere of unit

radius zoned with 100 × 10 cells. The initial density is unity and the initial velocity at the

grid vertices is given as (uz,ur) = (−z/4,−r). The specific internal energy of a cell is given

as e = (3zc/8)2, where zcis the z coordinate of the cell center. Figure 3.11 shows the results

of the grid and density plotted as a function of the radial radius along each radial line at

the time of 0.8 when the analytical density is expected to be flat with a value of 37.4. From

the figures, we can observe the numerical result agrees with the analytical solution except

26

Page 27

z

r

0 0.20.4 0.60.81

0

0.2

0.4

0.6

0.8

1

z

r

0 0.20.40.60.811.2 1.4

0

0.2

0.4

0.6

0.8

1

radius

density

00.1 0.2

0

20

40

60

80

100

Figure 3.11: The results of the Coggeshall problem at t = 0.8. Left: initial grid; Middle:

final grid; Right: density versus the radial radius.

for the small region near the origin.

Example 7 (Spherical Sedov problem on the Cartesian grid)

We test the spherical Sedov blast wave in a cylindrical coordinate system on the initially

rectangular grid. The initial computational domain is a 1.125 × 1.125 square consisting of

30×30 uniform cells. The initial density is unity and the initial velocity is zero. The specific

internal energy is zero except in the cell connected to the origin where it has a value of

0.2468. Figure 3.12 shows the results of the original scheme (2.15) and the modified scheme

(2.18). From the figures, we can observe the results from the modified scheme are more

satisfactory and more symmetrical even on the non-polar grid.

Example 8 (Spherical Noh problem on the Cartesian grid [19]).

At last, we test the spherical Noh problem on a Cartesian grid to verify the robustness

of the scheme. This problem is a very severe test for a Lagrangian scheme computing on a

Cartesian grid, since in this case the grid near the axes is easy to be distorted which has been

addressed in [5]. The initial domain is [0,1] × [0,1]. The initial state of the fluid is uniform

with (ρ,uξ,uθ,e) = (1,−1,0,10−5), where uξ,uθare the radial and angular velocities at the

cell center. Reflective boundary conditions are applied on the left and lower boundaries.

Free boundary condition is used on the right and upper boundary. The analytical solution

is the same as that in Example 1. Figure 3.13 shows the results of the original scheme (2.15)

and the modified scheme (2.18) with 50 × 50 initially uniform rectangular cells at t = 0.6.

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z

r

0 0.20.4 0.60.81

0

0.2

0.4

0.6

0.8

1

z

r

00.20.40.60.81

0

0.2

0.4

0.6

0.8

1

radius

density

0.511.5

0

0.5

1

1.5

2

2.5

3

3.5

exact

numerical

radius

density

0.511.5

0

0.5

1

1.5

2

2.5

3

3.5

exact

numerical

Figure 3.12: The results of the Sedov problem with 30 × 30 grids at t = 1.0. Left: the

original scheme (2.15); Right: the modified scheme (2.18). Top: final grid. Bottom: density

vs radial radius.Solid line: exact solution; symbols: computational solution.

From these figures, we can see that there is no grid distortion along the axes, the spherical

symmetry is preserved better and the shock position is correct for the modified scheme,

which demonstrate the robustness of the modified scheme in this problem on the Cartesian

grid.

4 Concluding remarks

In this paper we apply the methodology proposed in our previous work [8] on Maire’s first

order control volume Lagrangian scheme [14]. The purpose of this work is to improve the

scheme’s property in symmetry preservation while maintaining its original good properties

including geometric conservation law (GCL) and conservation of mass, momentum and total

energy.The modified scheme is proven to have one-dimensional spherical symmetry in

the two-dimensional cylindrical geometry for equal-angle-zoned initial grids. Several two-

dimensional examples in the cylindrical coordinates have been presented which demonstrate

28

Page 29

z

r

00.20.4

0

0.2

0.4

d

55

50

45

40

35

30

25

20

15

10

5

z

r

00.2

0

0.2

z

r

00.2 0.4

0

0.2

0.4

d

55

50

45

40

35

30

25

20

15

10

5

z

r

0 0.2

0

0.2

Figure 3.13: Grid and density contour for the Noh problem with 50 × 50 Cartesian cells at

t = 0.6. Top: the original scheme (2.15); Bottom: the modified scheme (2.18). Left: whole

grid; Right: zoom on the region with shock.

the good performance of the modified scheme in symmetry, non-oscillation and robustness.

The improvement of the modified scheme in accuracy constitutes our future work.

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