Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes
ABSTRACT In , 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  to the first order control volume scheme of Maire in  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.
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ABSTRACT: In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in Dumbser, Enaux, and Toro (2008). In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.07/2012;
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ABSTRACT: Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their dis-tinguished advantage in capturing material interfaces automatically. In some of these ap-plications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, signif-icant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretiza-tions of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equa-tions and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrody-namic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.
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ABSTRACT: In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a nonlinear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.04/2013;
Improvement on spherical symmetry in two-dimensional
cylindrical coordinates for a class of control volume Lagrangian
Juan Cheng1and Chi-Wang Shu2
In , 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  to the first order control volume
scheme of Maire in  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 email@example.com. 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.
firstname.lastname@example.org.Research is supported in part by ARO grant W911NF-08-1-0520 and NSF
University, Providence,RI02912. E-mail:
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
and Shashkov used a curvilinear grid to construct symmetry-preserving discretizations for
Lagrangian gas dynamics . In our recent work , 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  does not satisfy the geometric conservation
In , 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 . 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  which verify the robustness of the schemes.
In this paper, we attempt to apply the strategy proposed in  on Maire’s first order con-
trol volume Lagrangian scheme  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
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.
2 The improvement on the cell-centered control vol-
ume Lagrangian scheme of Maire in cylindrical co-
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
Γ(t)Pu · nds
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
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
Ω(t)ρuzrdrdz = −?
Γ(t)u · nrdl
Ω(t)ρurrdrdz = −?
Γ(t)Pu · nrdl
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.
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
2.2The control volume scheme of Maire in cylindrical coordinates
2.2.1Notations and assumptions
We will mostly use the notations in . 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