Abstract. We develop a new cell-centered control volume Lagrangian scheme for solving Euler

Transcription

1 A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry Juan Cheng and Chi-Wang Shu Abstract We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. Based on a local coordinate transform strategy, the scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area weighted schemes that possess the spherical symmetry property, our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density, momentum and total energy. Several two dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry, accuracy and non-oscillatory properties. Keywords: Lagrangian scheme; symmetry preservation; conservative; cell-centered; compressible flow; cylindrical coordinates Institute of Applied Physics and Computational Mathematics, Beijing 88, China. cheng juan@iapcm.ac.cn. Research is supported in part by NSFC grants 934 and Additional support is provided by the National Basic Research Program of China under grant 5CB37. Division of Applied Mathematics, Brown University, Providence, RI 9. shu@dam.brown.edu. Research is supported in part by ARO grant W9NF-8--5 and NSF grant DMS-8986.

2 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 generally, the motion of the grid can also be chosen arbitrarily, resulting in the so-called Arbitrary Lagrangian- Eulerian method (ALE; see, e.g. [3]). The Lagrangian method is widely used in many fields for multi-material flow simulations such as astrophysics and inertial confinement fusion (ICF), due to its distinguished advantage in capturing material interfaces sharply. In the past years, many efforts have been made to develop Lagrangian methods. Some algorithms are 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 / internal energy and velocity are solved on two different control volumes, see, e.g. [,, 4]. The dissipation of kinetic energy into internal energy through shock waves is ensured by an artificial viscosity term [, 5, 7]. Other algorithms start from the cell-centered discretization in which density, momentum and energy are all centered within cells and evolved on the same control volume. Advantages for the cellcentered schemes such as [, 9, 8, 9, 7] include that this kind of methods is easy to remap when necessary and it does not require the addition of an explicit artificial viscosity for shock capturing. Numerical diffusion is added implicitly in the Riemann solution in a very natural way. In Lagrangian simulation, it is a critical issue for a scheme to preserve certain symmetry (for example, cylindrical or spherical) in a coordinate system distinct from that symmetry. The importance of preserving physical symmetries in numerical simulation is well realized. As in the simulation of implosion problem with strong compressions, the preservation of spherical symmetry is significantly important, since the small deviation from spherical symmetry due to numerical errors may be exaggerated by Rayleigh-Taylor or some other instabilities which potentially produces unpredictably large errors. On the other hand, to understand and

3 predict the dynamics of implosion well, it is essential for us to know whether the asymmetric numerical result is due to the discrete error or to the physical intrinsic behavior. Concerning the issue related to spherical symmetry preservation in two dimensional cylindrical coordinates many works have been done mostly in the framework of staggeredgrid hydrodynamics. The most widely used method that maintains spherical symmetry exactly on an equal-angle-zoned grid in cylindrical coordinates is the area-weighted method [6,,, 5, 4, 6]. 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 might violate strict momentum and total energy conservation. This shortcoming of the schemes has been partially corrected in [4] by constructing a compatible area-weighted scheme which can preserve the conservation of total energy. However it seems that their flaw in the loss of momentum conservation has not been solved so far. Differently from the area-weighted scheme, Browne [3] presented a Lagrangian scheme termed integrated total average which is discretized on the true control volume in cylindrical coordinates. This scheme has been proven to be able to preserve the desired spherical symmetry in the two dimensional cylindrical geometry for equal-angle zoning. Unfortunately, this scheme can not keep the momentum and total energy conservation and has the potential trouble in that if the initial mass in neighboring cells has a big difference, large error in acceleration may happen. Margolin and Shashkov use a curvilinear grid to construct symmetry-preserving discretizations for Lagrangian gas dynamics [8]. They use high-order curves to connect the nodes, so that planar, cylindrical and spherical symmetry are exactly preserved even on unequal-angle zoned grids. In [6], Caramana and Whalen study the way to achieve the one-dimensional symmetry, in a twodimensional coordinate system distinct from that symmetry by a simple modification of the gradient operator used to compute the force in a staggered-grid Lagrangian hydrodynamics algorithm. In this paper, we propose a new cell-centered Lagrangian scheme on quadrilateral grids 3

4 for solving compressible Euler equations in cylindrical coordinates. The scheme is based on a genuine volume discretization formulation. It preserves the conservation of mass, momentum and total energy. By using a local coordinate transform strategy, the scheme can preserve one dimensional spherical symmetry in the two dimensional cylindrical geometry for equalangle-zoned initial grids. An outline of the rest of this paper is as follows. In Section, we describe the individual steps of the Lagrangian scheme with the symmetry and conservation properties for Euler equations in cylindrical coordinates. In Section 3, numerical examples are given to demonstrate the performance of the new Lagrangian method. In Section 4 we will give concluding remarks. A cell-centered conservative Lagrangian scheme with the preservation of spherical symmetry in cylindrical coordinates. The 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 ρdv = Ω(t) d dt Ω(t) MdV = pnds Γ(t) d EdV = (.) pu nds dt Ω(t) Γ(t) where ρ is density, p is pressure, M and u are the vectors of momentum and velocity respectively, and E is total energy. n is the unit outward normal to the boundary Γ(t). In this paper, we seek to study the axisymmetric compressible Euler system. Its specific form in the cylindrical coordinates is as follows d dt ρrdrdz = Ω(t) d M dt Ω(t) zrdrdz = pn Γ(t) zrdl d dt Ω(t) M rrdrdz = pn Γ(t) rrdl + pdrdz Ω(t) d Erdrdz = pu dt Ω(t) Γ(t) νrdl (.) 4

5 where z and r are the axial and radial directions respectively. M = (M z, M r ), u = (u z, u r ), where M z, u z, M r, u r are the momentum and velocity components in the z and r directions respectively, n = (n z, n r ) is the unit outward normal to the boundary Γ(t) in the z-r coordinates, and u ν = (u z, u r ) n is the normal velocity at Γ(t). The set of equations is completed by the addition of an equation of state (EOS) with the following general form p = p(ρ, e) (.3) where e = E ρ u is the specific internal energy. Especially, if we consider the ideal gas, then the equation of state has a simpler form, p = (γ )ρe where γ is a constant representing the ratio of specific heat capacities of the fluid.. The construction of the scheme Figure. indicates the cylindrical geometry (z, r) we are concerned with. The D spatial domain Ω is discretized into K L computational cells. Figure. shows an equal-anglezoned grid with logical l-lines radially outward and logical k-lines in the angular direction. I k+,l+ is a quadrilateral cell constructed by the four vertices {(z k,l, r k,l ), (z k+,l, r k+,l ), (z k+,l+, r k+,l+ ), (z k,l+, r k,l+ )}. S k+,l+ cell I k+,l+ and V k+,l+ denote the area and volume of the with k =,..., K, l =,..., L respectively, where the volume of the cell refers to that of the circular ring obtained by rotating this cell around the azimuthal z-axis, while for simplicity and without any confusion we omit the factor π which should appear in the integration of the real volume. The fluid velocity ((u z ) k,l, (u r ) k,l ) is defined at the vertex of the grid. In this paper, in order to obtain a scheme with both the spherical symmetry and conservation properties in a D cylindrical geometry, we will discretize the momentum equation in (.) along the local polar and angular directions (, ) rather than in the usual z and r 5

6 r l+ l+ 4 3 l l- k- k k+ k+ Figure.: Equi-angular polar grid for cylindrical geometry. z directions in each cell, where is the radial direction passing through the cell center and the origin. For an equal-angle-zoned grid, the cell shown in Figure. is an equal-sided trapezoid, it has the property that the angles between and the two equal sides of the cell are the same. is the angular direction which is orthogonal to, see Figure.. After completing the momentum discretization, we transform them back to the original cylindrical (z, r) coordinates. This is similar in spirit to the local characteristic decomposition procedure in limiting, where the relevant quantities are first transformed to the local characteristic fields, then the limiting procedure is performed, and afterwards the result is transformed back to the original component space. In the actual implementation, once the local and directions for each cell are determined at the initial time, these directions will not change for later time. To perform this local coordinate transformation, in each cell, the Euler equations in cylindrical coordinates are rewritten in the following form d ρrdrdz = dt I k+,l+ d M dt I k+ rdrdz = pn,l+ I k+ rdl +,l+ I k+,l+ d M dt I k+ rdrdz = pn,l+ I k+ rdl +,l+ I k+,l+ d Erdrdz = pu dt I k+,l+ I k+ ν rdl,l+ p sin ϕdrdz p cos ϕdrdz (.4) where M and M are the component values of momentum in the local and directions respectively, n = (n, n ) is the unit outward normal of the cell boundary I i+,j+, and ϕ is the angle between the radial direction passing through the corresponding point and the axial 6

7 coordinate z. The value of is denoted as the radial radius which represents the distance to the origin. Thus we have z = cos ϕ, r = sin ϕ and = z + r. For the cell-centered scheme, all the variables except velocity are stored at the cell center of I k+,l+ in the form of cell averages. For example, the values of the cell averages for the cell I k+,l+, denoted by ρ k+,l+, M k+,l+, M k+,l+ ρ k+,l+ M k+,l+ M k+,l+ E k+,l+ = = = = V k+,l+ V k+,l+ V k+,l+ V k+,l+ I k+,l+ I k+,l+ I k+,l+ I k+,l+ and E k+,l+, are defined as follows ρrdrdz, M rdrdz, M rdrdz, Erdrdz where V k+,l+ = rdrdz. In our scheme, the cell averages of the above conserved I k+,l+ variables are evolved in time, as a regular finite volume scheme... Spatial discretization We first formulate the semi-discrete finite volume scheme of the governing equations (.4) as d dt = ρ k+,l+ V k+,l+ M k+,l+ V k+,l+ M k+,l+ V k+,l+ Ē k+,l+ I k+,l+ V k+,l+ ˆFdl + (p c ) k+,l+ (p c ) k+,l+ sin(ϕ c ) k+,l+ cos(ϕ c ) k+,l+ s k+,l+ s k+,l+ (.5) where I k+,l+ ˆFdl = I k+,l+ ˆf D (U ν, U + ν ) ˆf M (U ν, U+ ν ) ˆf M (U ν, U + ν ) ˆf E (U ν, U+ ν ) dl (.6) 7

8 and ˆf D (U ν, U ν ) =, ˆf M (U ν, U ν ) = pn r ˆf M (U ν, U ν ) = pn r ˆf E (U ν, U ν ) = pu ν r (.7) ˆf D, ˆf M, ˆf M and ˆf E are the numerical fluxes for mass, -momentum, -momentum and total energy across the cell boundary respectively. U ± ν = (ρ±, M ± ν, E± ) are the values of mass, normal component of momentum and total energy at both sides of the cell boundary, where M ± ν = (M ±, M ± ) n. (p c) k+,l+ is the pressure at the cell center. (ϕ c ) k+,l+ the angle between the local direction and the z coordinate. is set to be The first step for establishing the scheme is to determine the line integral term on the right side of Eq. (.5). Suppose the cell boundary I k+,l+ consists of M edges. The line integral concerned with the flux in Eq. (.5) is discretized by the following formula, I k+,l+ ˆFdl where l m is the length of the cell edge m. M m= ˆF(U m+ ν, U m ν ) l m (.8) m+ ˆF(Uν, U m ν ) is a numerical flux at the edge m. To determine the fluxes ˆF = ( ˆf D, ˆf M, ˆf M, ˆf E ) at the edges of the quadrilateral shaped cell, we first need to identify the values of the primitive variables at each side of the edge, that is U m± ν, m =, 4. Here we use the information of the left and right cell average values of the conserved variables. We define the direction of each edge of the cell to be the increasing direction of the index k or l, for example the direction of the edge with two endpoints (k, l) and (k +, l) is from (k, l) to (k +, l). Thus, for example, for Edge in the cell I k+,l+ in Figure., we obtain U ν = (ρ k+,l+, (M ν ) k+,l+, E k+,l+ ) and U + ν = (ρ k+,l, (M ν ) k+,l, E k+,l ) at the two sides of Edge. Next, we will compute the fluxes given the primitive states {U m± ν, m =, 4} at each side of the cell boundary. Here as an example, we use the Dukowicz numerical flux [] to obtain the fluxes at the cell boundary by the following procedure. Most of other commonly used numerical fluxes are also suitable for our scheme which can be proven to have the symmetry preservation property. 8

9 The velocity u m ν at Edge m is obtained by solving the following semi-quadratic equation, ρ m+ A m+ u m ν u m min (u m ν u m min) + ρ m A m u m ν u m max (u m ν u m max) + p m+, p m, = (.9) where u m min = u m+ ν c m+ /A m+, u m max = u m ν + c m /A m ; p m+, = p m+ 4 ρm+ (c m+ ) /A m+, p m, = p m 4 ρm (c m ) /A m. (.) Here ρ m±, u m± ν, p m± are the left and right values of density, velocity and pressure at Edge m respectively, c ± are the left and right values of the sound speed at Edge m and A ± are parameters directly related to the shock density ratio in the limit of strong shocks. We refer to [] for the details of the definition of A ±. In particular, for an ideal gas, A ± = (γ + )/. After we have calculated the velocity u m ν, the pressure p m at Edge m is easily obtained by the following equation, p m = (pm, +p m+, )+ ρm+ A m+ u m ν u m min (u m ν u m min) ρm A m u m ν u m max (u m ν u m max) If p m is found to be negative which predicts cavitation, then we set p m =. (.) Thus we can get the fluxes ˆf D, ˆfM, ˆf M and ˆf E at Edge m as follows, where (n m, nm ) and rm e ˆf D (U m ν, U m+ ν ) =, ˆf M (U m ν, U m+ ν ) = p m n m rm e ˆf M (U m ν, U m+ (.) ν ) = p m n m rm e ˆf E (U m ν, U m+ ν ) = p m u m ν rm e are the unit outward normal direction and the r coordinate of the middle point of Edge m respectively. We have now finished the determination of the flux term in Eq. (.5). As to the source term in Eq. (.5), (p c ) k+,l+ (p c ) k+,l+ is denoted as = (p + p 3 ) (.3) where p and p 3 are the values of pressure at Edges and 3 of Cell I k+,l+ (see Figure.) obtained in the above flux determination procedure. Based on the above manipulation, the 9

10 scheme can preserve the conservation for all the conserved variables, since it is discretized on the true volume and the numerical flux across each cell boundary is single-valued for the update of its two neighboring cells. Meanwhile, we can also prove that the scheme can keep the spherical symmetry property if equi-angular polar initial grids are used. The proof will be given in the last subsection of this section... The determination of the vertex velocity The vertex velocity of the mesh is determined in the same way as in [8]. We first obtain the tangential and normal velocities along each edge, specifically, the tangential velocity of the vertex (or edge center) along the edge is defined as a simple average of that in both sides. The normal velocity is obtained by the value of velocity through the Dukowicz approximate Riemann solver along the normal direction of the edge. Then the velocity of the vertex is set to be the arithmetic average of the velocities along the edges which share this vertex...3 Time discretization The time marching for the semi-discrete scheme (.5) can be accomplished by the Euler forward method. Thus the fully discretized scheme can be written as follows ρ n+ V n+ ρ n V n k+,l+ k+,l+ k+,l+ k+,l+ M,n+ k+,l+ V n+ M,n k+,l+ k+,l+ V n k+,l+ M,n+ k+,l+ V n+ M,n k+,l+ k+,l+ V n k+,l+ E n+ k+,l+ V n+ M = t n m= E n k+,l+ k+,l+ ˆF(U m,n ν ) l m,n + V n k+,l+ (p c ) n k+,l+ (p c ) n k+,l+ sin(ϕ c ) n s n k+,l+ k+,l+ cos(ϕ c ) n s n k+,l+ k+,l+. (.4) Here the variables with the superscripts n and n + represent the values of the corresponding variables at the n-th and (n + )-th time steps respectively. The scheme (.4) is consistent with the Euler equations (.4) and has first order accuracy in space and time.

11 The time step t n is chosen as follows where l n k+,l+ t n = λ min k=,...,m,l=,...,n ( ln /c n ) (.5) k+,l+ k+,l+ is the shortest edge length of the cell I k+,l+, and c n k+,l+ is the sound speed within this cell. The Courant number λ in the following tests is set to be.5 unless otherwise stated. In our numerical tests, the scheme in Eq. (.4) can work well in most cases. However in some of the more demanding tests, the scheme is observed to show some instability. It seems that a good way to enhance stability is to use the second order TVD Runge-Kutta method [4] for the time discretization. Then the scheme shows very satisfactory stability in all our numerical tests. We refer the reader to [8] for the implementation details on high order Runge-Kutta methods in a Lagrangian scheme. The usage of higher order TVD Runge-Kutta time discretization simply to enhance stability, not to increase the order of accuracy, can also be found in [7]...4 Proof of the spherical symmetry preservation property In this section, we will prove our scheme (.4) can keep the spherical symmetry property computed on an equal-angle-zoned initial grid. Once this conclusion is proven to be correct, it can be easily extended to the scheme with the same spatial discretization and the high order Runge-Kutta time discretization. Theorem: The scheme (.4) can keep the one dimensional spherical symmetry property computed on an equal-angle zoned initial grid. That is, if the solution has one 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 scheme (.4) can keep the spherical symmetry at (n + )th-step, if the solution is known to be of spherical symmetry at nth-step. Notice that, for the Lagrangian solution, symmetry preserving refers to the evolution of both the conserved variables and the grids.

12 For the convenience of notation, we adopt the convention that variables without the superscript n + are those at the n-th time step. Assume that at the nth-step the grid is a polar grid with equal angles (see Figure.) and the cell averages of the conserved variables including density, momentum and total energy are symmetrical on this grid, namely these variables in the cells with the same k indices are identical. For simplicity, in the following we will omit the subscript l + for the variables which are independent of the l index, for example, ρ n k+,l+ M,n k+,l+ M,n k+,l+ E n k+,l+ s n k+,l+ = ρ k+, = M k+, = M k+ =, = E k+, = s k+, for k =,...K, l =,..., L. (.6) Next we denote some variables concerning the grid geometry. For simplicity, in the following text, we omit the subscript (k +, l + ) for the quantities corresponding to the cell edge if it will not bring any confusion. ϕ is denoted as the angle between any two neighboring l lines which is a constant for the equi-angular polar grid under consideration. The value of at the middle point of each edge is defined as m e, m =, 4, the length of each cell edge is denoted as { l m, m =, 4}. Since the grid is symmetrical, m e and l m, for m =, 4, are independent of the l index, thus we denote them as e = 3 e = k+, e = k+, e 4 = k, l = l 3 = l k+, l = l k+, l 4 = l k. (.7) Considering an arbitrary cell I k+,l+, for the convenience of proof, in the following, we will project all the variables relative to the determination of the cell s flux and velocity to its local - coordinates. For example, the outward normal direction n = (n, n ) of the cell s four edges in the local - coordinates are as follows, (n, n ) = ( sin( ϕ), cos( ϕ)), (n, n ) = (, ), (n 3, n3 ) = ( sin( ϕ), cos( ϕ)), (n 4, n4 ) = (, ).

13 In the following, we will first prove symmetry of the grid at the (n + )-th step. We will look at the velocity on each cell edge first. For this purpose we need to determine U m± ν = (ρ m±, Mν m±, E m± ) and Mt m±, for m =, 4, where Mt m± is the tangential momentum at the two sides of Edge m. (ρ m, M m ν, E m ) denote density, normal momentum and total energy at the edge which are obtained from U m± ν through the approximate Riemann solver. The pressure, normal velocity and tangential velocity at Edge m are defined as p m, u m ν, um t respectively, where u m t Using (.6), we have = m (M t /ρ m + Mt m+ /ρ m+ ), u m ν = M ν m/ρm. ρ ± = ρ k+, E ± = E k+ ρ = ρ k+ E = E k+, ρ 3± = ρ k+ E 3± = E k+ ρ 4 = ρ k E 4 = E k,, ρ + = ρ k+ 3,, E + = E k+ 3,,, ρ 4+ = ρ k+,, E 4+ = E k+. As to the momentum in Eq.(.4), since it is discretized in the local, directions, we should first project the momentum of the neighboring cells in their local, directions to those of Cell I k+,l+. By simple manipulation we have, M = M k+, M =, M + = M k+ cos( ϕ), M + = M k+ sin( ϕ), M = M k+, M =, M + = M k+ 3 M 3 = M k+, M + =, cos( ϕ), M 3+ = M k+, M 3+ =, =, M 4 = M k, M 4 M 4+ = M k+, M 4+ =. M 3 = M k+ sin( ϕ), 3

14 Thus, and M ν = (M, M ) (n, n ) = sin( ϕ)m k+,, Mν + = (M +, M + ) (n, n ) = sin( ϕ)m k+ Mν = (M, M ) (n, n ) = M k+, Mν + = (M +, M + ) (n, n ) = M k+ 3, M 3 ν = (M 3, M 3 ) (n 3, n3 ) = sin( ϕ)m k+, M 3+ ν = (M 3+, M 3+ ) (n 3, n3 ) = sin( ϕ)m k+ M 4 ν = (M 4, M 4 ) (n 4, n4 ) = M k, Mν 4+ = (M 4+, M 4+ ) (n 4, n4 ) = M k+,, Mt = (M, M ) (cos( ϕ)), sin( ϕ))) = cos( ϕ)m k+, Mt + = (M +, M + ) (cos( ϕ)), sin( ϕ))) = cos( ϕ)m k+, Mt = Mt + =, M 3 t = (M 3, M 3 ) (cos( ϕ)), sin( ϕ))) = cos( ϕ)m k+ Mt 3+ = (M 3+, M 3+ ) (cos( ϕ)), sin( ϕ))) = cos( ϕ)m k+, Mt 4 = Mt 4+ =. Then we have, U ν = (ρ k+, sin( ϕ)m k+, E k+ ), U + ν = (ρ k+ U 3 ν = (ρ k+, sin( ϕ)m k+, E k+ ), U 3+ ν = (ρ k+ U ν = (ρ k+, M k+, E k+ ), U + ν = (ρ k+ 3 ν = (ρ k, M k, E k ), U 4+ ν = (ρ k+ U 4, M k+ 3, M k+, sin( ϕ)m k+, sin( ϕ)m k+, E k+ 3 ),, E k+ )., E k+ ),, E k+ ), By the Dukowicz approximate Riemann solver, we obtain u ν = u3 ν =, p = p 3 and u ν, u 4 ν, p and p 4 are independent of the l index since U ± ν and U 4± ν are independent of the l index. The tangential velocities at each edge are as follows, u t = (Mt /ρ + Mt + /ρ + ) = cos( ϕ)m k+ /ρ, k+ u 3 t = 3 (Mt /ρ 3 + Mt 3+ /ρ 3+ ) = cos( ϕ)m k+ /ρ, k+ u t = (Mt /ρ + Mt + /ρ + ) =, u 4 t = 4 (Mt /ρ 4 + Mt 4+ /ρ 4+ ) =. 4 4

15 We can now get the conclusion that the pressure, normal velocity and tangential velocity at the cell edges are symmetrical in the angular direction. For simplicity, we denote the pressure, normal velocity and tangential velocity at the edge connecting the points (k, l) and (k, l + ) to be p k, (u ν ) k and (u t ) k for any k [, K], l [, L ], and denote those at the edge connecting the points (k, l) and (k +, l) to be p k+, (u ν ) k+ k [, K ], l [, L] respectively, namely, for Cell I k+,l+, p = p 3 = p k+, p = p k+, p 4 = p k, u ν = (u ν) k+ =, u ν = (u ν) k+, u 3 ν = (u ν) k+ u t = u3 t = (u t) k+, u t =, u4 t and (u t ) k+ =, u 4 ν = (u ν) k, for any =. (.8) By now, we can conclude that the magnitude of the velocity at the cell edges in the radial direction with the same k index is identical and its direction is the edge s tangential direction, while the magnitude of the velocity at the cell edges in the angular direction with the same k index is identical and its direction is the edge s normal direction, see Figure.. All the values of the velocity are independent of the l index. Denote the value of the composite velocity at the edge connecting the points (k, l) and (k, l + ) to be u k for all k [, K], l [, L ], and the value of the composite velocity at the edge connecting the points (k, l) and (k +, l) to be u k+ for k [, K ], l [, L]. Since the velocity at the vertex is obtained by the arithmetic average of the four velocities along its four connecting edges (see Subsection..), we have, at the vertex (k, l) in Figure., u k,l = u k A + u kb + uk+ C + u kd (.9) = u k A + u k+ C + u k ( B + D) (.) where A, C are the unit tangential vector and B, D are the unit normal vector of the relative edges. Since the grid is equi-angular, we have A = C, B + D = αc, where α = cos ϕ is a 5

16 r l+ l+ A D B C l l- k- k k+ z Figure.: the sketch of velocities at the edges connected to the vertex (k, l). constant for the given equi-angular polar grids. Thus we get u k,l = (u k + u k+ + αu k ) C. (.) Eq.(.) indicates that the velocities at the vertices with the same k index are identical and their directions are along their radial directions. Thus we have finished the proof of the symmetry of the grid at the (n + )-th step. Next we will prove that the symmetry can also be maintained for the evolved conserved variables at the (n + )-th step. Using the symmetry property of grid at (n + )-th step, the area and volume of Cell I k+,l+ at the (n + )-th step can be denoted simply as follows, s n+ k+,l+ = s n+, V n+ k+ k+,l+ = (r c ) n+ s n+ k+,l+ k+ = ( c ) n+ k+ sin(ϕ c ) l+ s n+ k+ where r c and c are the values of r and at the cell center respectively. We have omitted the subscript l + of those variables which are independent of the l index, and we have omitted the subscript k + and superscript n + of ϕ c since it is independent of the k and n indices. 6

17 Thus, the term at the left hand of Eq.(.4) can be rewritten as follows, ρ n+ V n+ ρ n V n k+,l+ k+,l+ k+,l+ k+,l+ M,n+ k+,l+ V n+ M,n k+,l+ k+,l+ V n k+,l+ M,n+ k+,l+ V n+ M,n k+,l+ k+,l+ V n k+,l+ E n+ k+,l+ = sin(ϕ c ) l+ V n+ E n k+,l+ k+,l+ ρ n+ k+,l+ M,n+ k+,l+ M,n+ k+,l+ E n+ k+,l+ V n k+,l+ ( c ) n+ s n+ k+ k+ ρ k+ ( c ) n+ s n+ M k+ k+ k+ ( c ) n+ s n+ M k+ k+ k+ ( c ) n+ k+ s n+ k+ E k+ ( c ) k+ s k+ ( c ) k+ s k+ ( c ) k+ s k+ ( c ) k+ s k+ (.) The flux term in Eq.(.4) can be expressed as follows M ˆF(U m ν ) l m m= = = p sin( ϕ)r e l + p re l p 3 sin( ϕ)r3 e l3 p 4 re 4 l4 p cos( ϕ)r e l + p 3 cos( ϕ)r3 e l3 p u νre l + p u νre l + p 3 u 3 νre l p 4 u 4 νre l 4 4 p sin( ϕ) l (r e + r 3 e) + p r e l p 4 r 4 e l 4 p l (r 3 e r e ) cos( ϕ) p u ν r e l + p 4 u 4 ν r4 e l4 where U m,n ν, l m,n are simplified as U m ν, lm. Using (.7), (.8) and the following equalities. (.3) r e = k+ sin((ϕ c ) l+ ϕ), re = k+ sin(ϕ c ) l+, re 3 = k+ sin((ϕ c ) l+ + ϕ), re 4 = k sin(ϕ c ) l+, 7

18 we have M ˆF(U m ν ) l m m= = = p k+ sin(ϕ c ) l+ k+ l k+ sin( ϕ) sin(ϕ c ) l+ p k+ l k+ + p k+ k+ l k+ sin(ϕ c ) l+ k+ sin( ϕ) cos(ϕ c ) l+ p k k l k sin(ϕ c ) l+ p k+ (u ν ) k+ k+ l k+ sin(ϕ c ) l+ p k (u ν ) k k l k sin(ϕ c ) l+ k+ l k+ sin( ϕ) + p k+ k+ l k+ p k k l k ) p k+ s k+ cos(ϕ c ) l+. (.4) (p k+ (u ν ) k+ k+ l k+ p k (u ν ) k k l k ) ( p k+ sin(ϕ c ) l+ Using the equality (.3), the source term in Eq.(.4) takes the following form (p c ) n sin(ϕ k+,l+ c ) n s n k+,l+ k+,l+ (p c ) n cos(ϕ k+,l+ c ) n s n k+,l+ k+,l+ = p k+ s k+ sin(ϕ c ) l+ p k+ s k+ cos(ϕ c ) l+ Thus the scheme (.4) can be written as sin(ϕ c ) l+ = t + p k+ p k+ s k+ s k+ ρ n+ k+,l+ M,n+ k+,l+ M,n+ k+,l+ E n+ k+,l+ sin(ϕ c ) l+ sin(ϕ c ) l+ cos(ϕ c ) l+ = t sin(ϕ c ) l+ ( c ) n+ s n+ k+ k+ ρ k+ ( c ) n+ s n+ M k+ k+ k+ ( c ) n+ s n+ M k+ k+ k+ ( c ) n+ k+ ( p k+ sin(ϕ c ) l+ p k+ s n+ k+ k+ E k+ ( c ) k+ s k+ ( c ) k+ s k+ ( c ) k+ s k+ ( c ) k+ s k+ l k+ sin( ϕ) + p k+ k+ l k+ p k k l k ) p k+ s k+ cos(ϕ c ) l+ (p k+ (u ν ) k+ k+ l k+ p k (u ν ) k k l k ) (.5) k+ l k+ sin( ϕ) p k+ k+ l k+ + p k k l k + p k+ p k+ (u ν ) k+ k+ l k+ + p k (u ν ) k k l k s k+, 8

19 ρ n+ k+,l+ M,n+ k+,l+ M,n+ k+,l+ E n+ k+,l+ + t p k+ = ( c ) n+ k+ l k+ k+ s n+ k+ ρ k+ ( c ) k+ s k+ M k+ ( c ) k+ E k+ ( c ) k+ s k+ s k+ sin( ϕ) p k+ k+ l k+ + p k k l k + p k+ p k+ (u ν ) k+ k+ l k+ + p k (u ν ) k k l k s k+ (.6). By now, we can make the conclusion that (ρ n+, M,n+ k+,l+ k+,l+, M,n+ k+,l+, E n+ k+,l+ ) only depend on the k index and are independent of the l index. Thus we have proved the symmetry preservation property of the scheme. 3 Numerical results in the two dimensional cylindrical coordinates In this section, we perform numerical experiments in two dimensional cylindrical coordinates. Purely Lagrangian computation and the ideal gas with γ = 5/3 are used in the following tests unless otherwise stated. Reflective boundary conditions are applied to the z and r axes in all the tests. 3. Accuracy test We test the accuracy of our scheme on a free expansion problem given in [3]. The initial computational domain is [, ] [, π/] defined in the polar coordinates. The gas is initially at rest with uniform density ρ = and pressure has the following distribution, p = (z + r ). 9

20 Table 3.: Errors of the scheme in D cylindrical coordinates for the free expansion problem using K L initially equi-angular polar grid cells. K = L Norm Density order Momentum order Energy order L.E-.4E-.E- L.5E-.7E-.45E- 4 L.E-.86.E-.95.6E-3.5 L.8E-.9.39E-.86.4E L.57E-3.94.E-.96.3E-3.95 L.5E-.9.E-.87.3E L.3E E E-3.88 L.75E-3.98.E E-3.98 The analytical solution of the problem is as follows, R(t) = + t, u (z, r, t) = t + t z + r, ρ(z, r, t) = R, 3 p(z, r, t) = ( ) z + r R 5 where R is the radius of the free outer boundary and u represents the value of velocity in the radial direction. Free boundary condition is applied on the outer boundary. The errors of the scheme at t = are listed in Table 3. which are measured on the interval [ K, 4 K] [, L] to remove 5 5 the influence from the boundary. In Table 3., we can see the expected first order accuracy for all the evolved conserved variables. 3. Non-oscillatory tests Example (The Noh problem in a cylindrical coordinate system []). 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, the initial state of the fluid is uniform with (ρ, e) = (, 5 ) and an inward radial velocity of magnitude. The equi-angular polar grid is applied in the -circle computational domain defined in the polar 4 R

21 r r exact angular cells 4 angular cells.6.4. density z..4 z...3 radius Figure 3.: The result of the Noh problem at t =.6. Left: initial grid with cells; Middle: final grid with cells; Right: density vs radial radius with and 4 cells respectively. Solid line: exact solution; dashed line: computational solution. coordinates by [, ] [, π/]. 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 /3. Figure 3. shows the initial grid and the final grid at t =.6 with cells and density as a function of radial radius at t =.6 for two different angular zonings (, 4). In the plot of grid, we observe the symmetry is perfectly preserved. In the plot of density, we observe no spurious oscillation near the discontinuity region. The shock location and the shock magnitude are closer to the analytical solution with the refinement of the grid in the angular direction which reflects the convergence trend of the numerical solution toward the analytical solution. Example (The one dimensional spherical Sod problem). The Riemann problem proposed by Sod is tested in the cylindrical coordinates. The initial computational domain is a -circle region defined in the polar coordinates by [, ] [, π/]. 4 Its initial condition is as follows (ρ, u, p) = (.,,.), (ρ, u, p) = (.5,,.), <. 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

22 r 5 density reference numerical z radius Figure 3.: The result of the Sod problem at t =.4. Left: final grid with 4 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. the spherical coordinate with grid points. We display in Figure 3. the numerical results of the grid and density as a function of the radial radius and the surface of density performed by our scheme with 4 equi-angular polar cells at t =.4. We observe the symmetrical behavior of our scheme and good agreement between our numerical result and the reference solution. Example 3 (The Sedov problem in a cylindrical coordinate system [3]). We present the result of the Sedov blast wave in a cylindrical coordinate system as an example of a diverging shock wave. The initial computational domain is a -circle region 4 defined in the polar coordinates by [,.5] [, π/] consisting of 3 equi-angular polar grids. The initial density is unity and the initial velocity is zero. The internal energy is zero except in the cells connecting the origin where they share a total value of.468. Reflective boundary condition is applied on the outer boundary. The analytical solution is a shock at radius unity at time unity with a peak density of 4. The final grid, density as a function of the radial radius and surface of density are displayed in Figure 3.3. We observe the expected symmetry in the plot of grid. The position of shock and peak density are in quite satisfactory agreement with the analytical solution without any spurious oscillations which demonstrates the good performance of the scheme in symmetrical, non-oscillatory and accuracy properties.

23 r 4 exact numerical density density z radius.5 z.5 Figure 3.3: The result of the Sedov problem with 3 cells at t =.. Left: final grid; Middle: density vs radial radius. Solid line: exact solution; dashed line: computational solution; Right: surface of density. Example 4 (Implosion problem of Lazarus[5]). The implosion problem of Lazarus is a problem with self-similar solution. A sphere of unit initial radius with zero specific internal energy and unit density is driven by an inward radial velocity given by u (t) = where α = , f = εt δt 3, ε =.85, δ =.8. αf ( ft) α (3.) We use a grid of 3 equal-angle polar cells in the initial computational domain [, ] [, π/] defined in the polar coordinates. The numerically converged result computed using a one-dimensional second-order Lagrangian code in the spherical coordinate with cells is used as a reference solution. Here the Courant number initially is taken as a smaller value at., and after t =. it returns to the normal value.5. Figure 3.4 shows the results of the grid and surface of density using 3 grids at t =.8 and density as a radial radius function at the typical time t =.74,.8 with two different angular zoned grids ( 3, 6) respectively. In the plot of grid, we notice that symmetry is preserved quite well as expected for the scheme. In the plot of density, the non-oscillatory and accurate numerical solution and the convergence tendency of the numerical results toward the reference solution are observed. Example 5 (Kidder s isentropic compression problem [4, 6]). 3

24 r reference t=.74 reference t=.8 3 angular cells t= angular cells t=.8 6 angular cells t=.74 6 angular cells t=.8 5. density 5 density z radius Figure 3.4: The result of the Lazarus problem. Left: final grid with 3 cells at t =.8; Middle: density versus radial radius at t =.74,.8 with 3 and 6 cells respectively. Solid line: reference solution; dashed line: computational solution; Right: surface of density at t =.8 with 3 cells. 5.. z r.3.4 This problem involves a self-similar isentropic compression of a shell filled with prefect gas which is usually used to validate the ability of a Lagrangian scheme in computing a spherical isentropic compression. The shell initially is a ring shaped region with the internal radius and the external radius. The initial density and pressure ρ, p are defined as follows, ρ () = ( ρ γ + ρ γ ) γ, p () = s(ρ ()) γ where =.9, =., ρ = 6.3 4, ρ =, s =.5 4, γ = 5/3. The pressure p (t) and p (t) are imposed continuously at the internal and external boundary of the shell respectively which have the following representation, p (t) = p γ a(t) γ, p (t) = p γ a(t) γ, where p =., p = and a(t) = ( t τ ) in which τ = is the focusing time of the shell and t [, τ) is the evolving time. Denote ζ(, t) to be the radius at the time t of a point initially located at radius, its analytical solution is ζ(, t) = a(t). The analytical solutions of three fundamental variables 4

25 r r exact numerical 4* radius z z time Figure 3.5: The result of the Kidder problem with 4 grids. Left: initial grid; Middle: final grid at t =.99τ; Right: trajectory of external boundary compared with the exact solution. Solid line: exact solution; dashed line: computational solution. for this problem in spherical geometry are as follows, ρ(ζ(, t), t) = ρ ()a(t) γ, u(ζ(, t), t) = d dt a(t), p(ζ(, t), t) = p ()a(t) γ γ. We test the problem initially on the computational region [.9, ] [, π/] in the polar coordinates with 4, 8 4, 6 8 grids respectively. The final time is set to be t =.99τ. Figure 3.5 shows the initial and final grids and the time evolution of the position of the external boundary with 4 grids. Figure 3.6 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 6 (Coggeshall expansion problem []). This is a two-dimensional adiabatic compression problem proposed by Coggeshall. The computational domain consists of a quarter of a sphere of unit radius zoned with grids. The initial density is unity and the initial velocity at the grid vertices is given as (u z, u r ) = ( z/4, r). The specific internal energy of a cell is given as e = (3z c /8), where z c is the z coordinate of the cell center. Figure 3.7 shows the results of the grid and density 5

26 r r 8 6 exact numerical 4* numerical 8*4 numerical 6* exact numerical 4* numerical 8*4 numerical 6*8 5 exact numerical 4* numerical 8*4 numerical 6*8 density 4 velocity pressure radius radius radius Figure 3.6: The result of the Kidder problem at t =.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 density z z radius Figure 3.7: The result of the Coggeshall problem at t =.8. Left: initial grid; Middle: final grid; Right: density versus the radial radius. plotted as a function of the radial radius along each l line at the time of.8 when the analytical density is expected to be flat with a value of From the figures, we can observe the numerical result is in agreement with the analytical solution except for the small region near the origin. This example serves to test the performance of the scheme on a problem with an initial equal-angle-zoned grid evolving into a non-symmetric grid. Example 7 (Cylindrical Noh problem on the Cartesian grid []). We test the Noh problem with the Cartesian grid in a cylindrical coordinate system to verify the performance of our scheme on non-polar grid. The problem domain is [, ] [, ]. The initial state of the fluid is uniform with (ρ, u z, u r, e) = (,,, 5 ). Reflective boundary conditions are applied on the left, right and lower boundaries. Inflow boundary condition is used on the upper boundary. The shock is generated in a perfect gas by bringing 6

27 r r exact numerical.6.4. density z z...3 r Figure 3.8: The result of the Noh problem at t =.6. Left: initial grid; Middle: final grid; Right: density as a function of r. Solid line: exact solution; dashed line: computational solution. the cold gas to rest at a rigid wall (r = ). The analytical post shock density is 6 and the shock speed is /3. Figure 3.8 shows the Lagrangian simulation results of our schemes at t =.6 with grids. We observe good performance of our scheme in this problem on the Cartesian quadrilateral grid. 4 Concluding remarks In this paper we have described a new cell-centered Lagrangian schemes for solving Euler equations in cylindrical coordinates. The scheme can preserve the symmetry property for one dimensional spherical problem computed on an equal-angle-zoned initial grid. Compared with many current Lagrangian type schemes, our scheme has the distinguished feature in being able to guarantee both symmetry and conservation properties for all conserved variables including mass, momentum and total energy. Several two dimensional examples in the cylindrical coordinates have been presented which demonstrate the good performance of the scheme. The strategy proposed in this paper is hopefully applicable to certain other existing schemes to obtain the symmetry property without loss of their original good properties. This generalization, as well as the improvement of the scheme in accuracy and robustness, constitute our future work. 7

28 References [] D.J. Benson, Momentum advection on a staggered mesh, Journal of Computational Physics,, 99, [] D.J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Computer Methods in Applied Mechanics and Engineering, 99, 99, [3] P.L. Browne, Integrated gradients: a derivation of some difference forms for the equation of motion for compressible flow in two-dimensional Lagrangian hydrodynamics, using integration of pressures over surfaces, Los Alamos National Laboratory Report LA- 587-MS, 986. [4] E.J. Caramana, D.E. Burton, M.J. Shashkov and P.P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, Journal of Computational Physics, 46, 998, 7-6. [5] E.J. Caramana, M.J. Shashkov and P.P. Whalen, Formulations of Artificial Viscosity for Multidimensional Shock Wave computations, Journal of Computational Physics, 44, 998, [6] E.J. Caramana and P.P. Whalen, Numerical Preservation of Symmetry Properties of Continuum Problems, Journal of Computational Physics, 4, 998, [7] J.C. Campbell and M.J. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, Journal of Computational Physics, 7,, [8] J. Cheng and C.-W. Shu, A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, Journal of Computational Physics, 7, 7, [9] J. Cheng and C.-W. Shu, A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equation, Communications in Computational Physics, 4, 8,

29 [] S. V. Coggeshall and J. Meyer-ter-Vehn, Group invariant solutions and optimal systems for multidimensional hydrodynamics, Journal of Mathematical Physics, 33, 99, [] B. Després and C. Mazeran, Lagrangian gas dynamics in two dimensions and lagrangian systems, Archive for Rational Mechanics and Analysis, 78, 5, [] J.K. Dukowicz, A general, non-iterative Riemann solver for Godunov method, Journal of Computational Physics, 6, 985, [3] C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, 4, 974, [4] R.E. Kidder, Laser-driven compression of hollow shells: power requirements and stability limitations, Nuclear Fusion,, 976, 3-4. [5] R. Lazarus, Self-similar solutions for converging shocks and collapsing cavities, SIAM Journal on Numerical Analysis, 8, 98, [6] P.-H. Maire, A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, Journal of Computational Physics, 8, 9, [7] P.-H. Maire, R. Abgrall, J. Breil and J. Ovadia, A cell-centered Lagrangian scheme for compressible flow problems, SIAM Journal of Scientific Computing, 9, 7, [8] L.G. Margolin and M.J. Shashkov, Using a curvilinear grid to construct symmetrypreserving discretizations for Lagrangian gas dynamics, Journal of Computational Physics, 49, 999, [9] C.D. Munz, On Godunov-type schemes for Lagrangian gas dynamics, SIAM Journal on Numerical Analysis, 3, 994,

30 [] J. von Neumann and R.D. Richtmyer, A method for the calculation of hydrodynamics shocks, Journal of Applied Physics,, 95, [] W.F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, Journal of Computational Physics, 7, 987, 78-. [] W.D. Schulz, Two-dimensional Lagrangian hydrodynamic difference equations, Methods of Computational Physics, 3, 964, -45. [3] L.I. Sedov, Similarity and Dimensional Methods in Mechanics, Academic Press, New York, 959. [4] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes, Journal of Computational Physics, 77, 988, [5] A. Solov ev and M. Shashkov, Difference scheme for the Dirichlet particle method in cylindrical method in cylindrical coordinates, conserving symmetry of gas-dynamical flow, Differential Equations, 4, 988, [6] M.L. Wilkins, Calculation of elastic plastic flow, Methods of Computational Physics, 3, 964, -63. [7] Z. Xu and C.-W. Shu, Anti-diffusive flux corrections for high order finite difference WENO schemes, Journal of Computational Physics, 5, 5,