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COMMUNICATIONS IN COMPUTATIONAL PHYSICS

Vol. 4, No. 5, pp. 1008-1024

Commun. Comput. Phys.

November 2008

AThirdOrderConservativeLagrangianTypeScheme

on Curvilinear Meshes for the Compressible Euler

Equations†

Juan Cheng1,∗and Chi-Wang Shu2

1InstituteofAppliedPhysicsandComputationalMathematics,Beijing100088,China.

2Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

Received 10 February 2008; Accepted (in revised version) 1 March 2008

Available online 8 July 2008

Abstract. Based on the high order essentially non-oscillatory (ENO) Lagrangian type

scheme on quadrilateral meshes presented in our earlier work [3], in this paper we

develop a third order conservative Lagrangian type scheme on curvilinear meshes for

solving the Euler equations of compressible gas dynamics. The main purpose of this

work is to demonstrate our claim in [3] that the accuracy degeneracy phenomenon

observed for the high order Lagrangian type scheme is due to the error from the

quadrilateral mesh with straight-line edges, which restricts the accuracy of the result-

ing scheme to at most second order. The accuracy test given in this paper shows that

the third order Lagrangian type scheme can actually obtain uniformly third order ac-

curacy even on distorted meshes by using curvilinear meshes. Numerical examples

are also presented to verify the performance of the third order scheme on curvilinear

meshes in terms of resolution for discontinuities and non-oscillatory properties.

AMS subject classifications: 65M99

Key words: Lagrangian type scheme, high order accuracy, conservative scheme, curvilinear

mesh, WENO reconstruction, compressible Euler equations.

1 Introduction

In numerical simulations of multidimensional 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

†Dedicated to Professor Xiantu He on the occasion of his 70th birthday.

∗Corresponding author. Email addresses: cheng juan@iapcm.ac.cn (J. Cheng), shu@dam.brown.edu (C.-W.

Shu)

http://www.global-sci.com/1008

c ?2008 Global-Science Press

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J. Cheng and C.-W. Shu / Commun. Comput. Phys., 4 (2008), pp. 1008-10241009

generally, the motion of the grid can also be chosen arbitrarily, this method is called the

Arbitrary Lagrangian-Eulerian method (ALE; see, e.g., [1,7,14]).

Lagrangian methods are widely used in many fields for multi-material flow simula-

tions such as astrophysics and computational fluid dynamics (CFD), due to their dis-

tinguished advantage of being able to capture the material interface sharply. In the

past years, many efforts have been made to develop Lagrangian methods. Some algo-

rithms are developed from the nonconservative form of the Euler equations, see, e.g.,

[10,13,22], while othersstart from the conservative form of the Euler equations, e.g. those

in [1,4,11,12].

Inourpreviouspaper[3], wedevelopedaclass ofLagrangiantypeschemesonquadri-

lateral meshes for solving the Euler equations which are based on the high order essen-

tially non-oscillatory (ENO) reconstruction. Advantages for the schemes in [3] include

their conservativity for the density, momentum and total energy, their essentially non-

oscillatory performance, parameter-free implementation, and their formal high order ac-

curacy both in space and time. However, we also found that our third order Lagrangian

type scheme could only achieve second order accuracy on two dimensional distorted

Lagrangian meshes. We attributed this phenomenon to a fundamental problem in the

formulation of the Lagrangian schemes. Since in a Lagrangian simulation, each cell rep-

resents a material particle, its shape may change with the movement of fluid. Therefore,

a cell with an initially quadrilateral shape may not keep its shape as a quadrilateral at

a later time. It usually becomes a curved quadrilateral. Thus if during our Lagrangian

simulation the mesh is always kept as quadrilateral with straight-line edges, this approx-

imation of the mesh will bring second order error into the scheme. Finally we made a

conclusion that for a Lagrangian type scheme in multi-dimensions, it can be at most sec-

ond order accurate if curved meshes are not used. Meanwhile, we also predicted that our

scheme can be extended to higher than second order accuracy if curvilinear meshes are

used. Inthis paper, weexplorecurvilinear meshestodemonstrateourclaim statedabove.

We will develop a third order scheme on curved quadrilateral meshes in two space di-

mensions. The reconstruction is based on the high order WENO procedure [8,9] but with

simpler linear weights. The accuracy test and some non-oscillatory tests are presented to

verify our claim. The scheme can also be extended to higher than third order accuracy

if a higher order approximation is used on both the meshes and the discretization of the

governing equations.

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

ual steps of the third order Lagrangian type scheme on curvilinear meshes in two space

dimensions. In Section 3, numerical examples are given to verify the performance of the

new Lagrangian type method. In Section 4 we will give concluding remarks.

2 The third order conservative Lagrangian type scheme on

curvilinear meshes

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1010J. Cheng and C.-W. Shu / Commun. Comput. Phys., 4 (2008), pp. 1008-1024

2.1The 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 the integral form in the Cartesian coordinates as

d

dt

?

Ω(t)UdΩ+

?

Γ(t)FdΓ=0,(2.1)

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 by

U=

ρ

M

E

,

F=

(u−˙ x)·nρ

(u−˙ x)·nM+pn

(u−˙ x)·nE+pu·n

, (2.2)

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 velocity of the control volume boundary Γ(t), n denotes

the unit outward normal to Γ(t). The system (2.1) represents the conservation of mass,

momentum and energy.

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

the following general form

p=p(ρ,e), (2.3)

where e=E/ρ−1

gas, then the equation of state has a simpler form,

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

p=(γ−1)ρe,

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

This paper focuses on solving the governing equations (2.1)-(2.2) in a Lagrangian

framework, in which it is assumed that ˙ x = u, and the vectors U and F then take the

simpler form

2.2The third order conservative Lagrangian type scheme on curvilinear

meshes in two space dimension

U=

ρ

M

E

,

F=

0

pn

pu·n

.(2.4)

The 2D spatial domain Ω is discretized into Nx×Nycomputational cells. Ii+1/2,j+1/2

is a generalized quadrilateral cell, with each edge being a quadratic curve.

curved quadrilateral cell is uniquely identified by its 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)} and the four middle points of its

four quadratic edges {(xi+1/2,j, yi+1/2,j), (xi+1,j+1/2, yi+1,j+1/2), (xi+1/2,j+1,yi+1/2,j+1),

(xi,j+1/2,yi,j+1/2)}. Si+1/2,j+1/2denotes the area of the cell Ii+1/2,j+1/2with i =1,···,Nx,

This

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J. Cheng and C.-W. Shu / Commun. Comput. Phys., 4 (2008), pp. 1008-1024 1011

j = 1,···,Ny. 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

and (ui+1/2,j,vi+1/2,j),(ui,j+1/2,vi,j+1/2) define the fluid velocities at the middle points of

the curvilinear edges of the cell. We consider only the non-staggered mesh, with all the

variables except velocity stored at the cell center of Ii+1/2,j+1/2in the form of cell av-

erages. For example, the values of the cell averages for the cell Ii+1/2,j+1/2denoted by

ρi+1/2,j+1/2, Mx

i+1/2,j+1/2, My

i+1/2,j+1/2and Ei+1/2,j+1/2are defined as follows

ρi+1/2,j+1/2=

1

Si+1/2,j+1/2

??

Ii+1/2,j+1/2

??

??

??

ρdxdy,

Mx

i+1/2,j+1/2=

1

Si+1/2,j+1/2

1

Si+1/2,j+1/2

1

Si+1/2,j+1/2

Ii+1/2,j+1/2

Mxdxdy,

My

i+1/2,j+1/2=

Ii+1/2,j+1/2

Mydxdy,

Ei+1/2,j+1/2=

Ii+1/2,j+1/2

Edxdy,

where ρ, Mx, Myand E are the density, x-momentum, y-momentum and total energy,

respectively.

2.2.1 Spatial and time discretizations

The conservative semi-discrete scheme for Eqs. (2.1) and (2.4) has the following form on

the 2D non-staggered mesh

d

dt

ρi+1/2,j+1/2Si+1/2,j+1/2

Mx

i+1/2,j+1/2Si+1/2,j+1/2

My

i+1/2,j+1/2Si+1/2,j+1/2

Ei+1/2,j+1/2Si+1/2,j+1/2

=−

?

∂Ii+1/2,j+1/2

ˆFdl=−

?

∂Ii+1/2,j+1/2

ˆfD(U−

ˆfMx(U−

ˆfMy(U−

ˆfE(U−

n,U+

n,U+

n,U+

n,U+

n)

n)

n)

n)

dl. (2.5)

Here U±= (ρ±,M±

total energy at two sides of the boundary. U±

and right component values of the momentum which is normal to the cell boundary, i.e.

M±

y-momentum and total energy across the cell boundary respectively. Here in the La-

x,M±

y,E±) are the values of mass, x-momentum, y-momentum and

n= (ρ±,M±

n,E±), where M±

nare the left

n=(M±

x,M±

y)·n.ˆfD,ˆfMx,ˆfMyandˆfEare the numerical fluxes of mass, x-momentum,

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1012 J. Cheng and C.-W. Shu / Commun. Comput. Phys., 4 (2008), pp. 1008-1024

grangian formulation, we have

ˆfD(Un,Un)=0,

ˆfMx(Un,Un)=pnx,

ˆfMy(Un,Un)=pny,

ˆfE(Un,Un)=pun,

(2.6)

where un=u·n is the normal velocity at the cell boundary.

Similar to [3], we use the high order WENO reconstruction with Roe-type character-

istic decomposition [21] to obtain U±and U±

high order quadrature to construct schemes up to the expected third-order spatial accu-

racy. The Dukowicz flux [5] and the Lax-Friedrichs (L-F) flux are used in this paper for

numerical tests. To save space, in the following we will only list the difference in the

procedure of constructing the scheme on the curved quadrilateral meshes from that on

the straight-line edged quadrilateral meshes. We refer to [3] for the other details.

Since calculating the integral of the numerical flux and determining the reconstruc-

tion polynomials in the scheme (2.5) involve the integration on each curved quadrilat-

eral cell, firstly we should know the accurate description of each curvilinear cell with

quadratically-curved edges by the information of the coordinates of its four vertices

and the four middle points of its four edges. We accomplish this procedure by a map-

ping from a canonical square to a curved quadrilateral cell. For simplicity, here we de-

note the coordinates of the four vertices of the concerned cell as {(xm,ym),m=1,2,3,4},

{(ξm,ηm),m = 1,2,3,4} and the middle points of its four edges as {(x12,y12), (x23,y23),

(x34,y34), (x41,y41)}, {(ξ12,η12), (ξ23,η23), (ξ34,η34), (ξ41,η41)} in the (x,y)-plane and

(ξ,η)-plane respectively (see Fig. 1). Then the shape of this curvilinear cell can be de-

termined by the following interpolation function:

nat the boundary and also use sufficiently

x=

4

∑

m=1

ϕmxm+

∑

n=m+1(mod4)

ϕmnxmn,

y=

4

∑

m=1

ϕmym+

∑

n=m+1(mod4)

ϕmnymn,

(2.7)

where

ϕm=−1

ϕmn=1

4(1+ξmξ)(1+ηmη)(1−ξmξ−ηmη),

2(1+ξmnξ+ηmnη)(1−ξ2

m=1,2,3,4,

mnη2−η2

mnξ2),

n=m+1(mod4), −1≤ξ, η≤1.

For each boundary edge Γmof the 4 curved edges in ∂Ii+1/2,j+1/2, the curvilinear integral