GENUINELY MULTIDIMENSIONAL NON-DISSIPATIVE FINITE VOLUME SCHEMES FOR TRANSPORT

Transcription

1 GENUINELY MULTIDIMENSIONAL NON-DISSIPATIVE FINITE VOLUME SCHEMES FOR TRANSPORT BRUNO DESPRÉS, FRÉDÉRIC LAGOUTIÈRE Commissariat à l Énergie Atomique, Bruyères-le-Châtel. despres@cmpax.polytechnique.fr despres@cmpax.polytechnique.fr Université Paris-VII and Laboratoire Jacques-Louis Lions, UMR 7598, 75, rue du Chevaleret, 753 Paris lagoutie@math.jussieu.fr Keywords: grids. multidimensional transport, finite volume schemes, anti-dissipative schemes, triangular Abstract We develop a new multidimensional finite volume algorithm for transport equations. This algorithm is both stable and non-dissipative. It is based on a reconstruction of the discrete solution inside each cell, at every time step. The proposed reconstruction, genuinely multidimensional, allows to recover sharp profiles both in the direction of the transport velocity and in the transverse direction. It is an extension of the one-dimensional reconstructions analyzed in Lagoutière 27 and Lagoutière 27. Introduction The present study concerns reconstruction schemes for transport equations. We are especially interested in schemes that are not dissipative, in particular for initial conditions with discontinuities. This leads to consider reconstructions that are not smooth, on the contrary to usual reconstructions. We indeed develop a discontinuous reconstruction scheme which consists in reconstructing the constant-in-cell datum as a discontinuous inside each cell function. Typically, the goal is to develop transport schemes for mass or volume fractions in multi-fluids. These fractions can be discontinuous at interfaces between components or continuous in mixing zones. A former algorithm was already developed in Després and Lagoutière 27. It was based in dimension on the limited downwind scheme equivalent to the Ultra-bee limiter for advection with constant velocity and the multidimensional algorithms where obtained via an alternate direction splitting strategy. Results are satisfying, in particular for inter- This work was partially supported by CEA/DIF Bruyères-le-Châtel faces see results for interface instabilities in dimensions 2 and 3 in the cited reference. The main drawback of this method is precisely the dimensional splitting, which prevents the algorithm form being used on non-cartesian grids. We here present a way to generalize the limited downwind scheme to the case of a general triangular grid. It is based on the geometrical approach followed in Lagoutière 27, Lagoutière 27, that provided a new interpretation of the limited downwind algorithm, in terms of reconstruction scheme. The paper is organized as follows. Sections 2 and 3 present the mathematical and numerical problems and the notations. Then, we recall the reconstruction procedure leading to the limited downwind scheme in dimension section 4. In section 5, the main subject of the paper is addressed: the extension to upper dimensions of the preceding procedure. The presentation deals with dimension 2. Finally, section 6 exhibits numerical results. For the time-being, the most efficient algorithms for pure transport rest upon interface reconstruction: see for example SLIC and Youngs method, Noh and Woodward 976 and Youngs 984, and the method of Mosso Mosso and Cleancy 995 which is a recent promising enhancement. These methods are essentially limited to Cartesian grids. We here try to derive a truly multidimensional reconstruction algorithm.

2 2 Model problem The considered model is the linear transport equation with constant given velocity t ut, x + divaut, x = t ut, x + a x ut, x = for t >, u, = u L R 2, where a is a smooth divergence-free velocity field: divat, x = for every t, x R + R 2. The main issue for the numerical treatment of this PDE problem is the numerical diffusion. This phenomenon, easily understandable in dimension, is much more complex in higher dimension. It shall be decomposed in two different types of diffusion. The diffusion of the first type, which will be called longitudinal diffusion, is the one that occurs in the direction of the velocity. It is the diffusion which is present in one-dimensional classical algorithms. The second type diffusion, transverse diffusion, is typically multidimensional and is due to the fact that the mesh may not be aligned with the velocity. This distinction between to phenomena could appear arbitrary, but is in accordance with the numerical tests. It is illustrated on FIG. 2 and 3 representing numerical solutions obtained with the classical upwind scheme on a square mesh, with, as initial condition, the characteristic function of a square FIG.. The transport velocity is a =, aligned with the mesh for FIG. 2 and a =, diagonal, not aligned with the mesh for FIG. 3 and the boundary condition on [, ] 2 are periodic. The results are displayed for time t = after one period Figure. Initial condition. Figure 2. Upwind scheme. a =,, aligned with the mesh. Time t =. The longitudinal diffusion applies in the direction of transport. There is no transverse diffusion Figure 3. Upwind scheme. a =,, not aligned with the mesh. Time t =. The diffusion which applies orthogonally to the transport direction is brought to light. The difference between longitudinal and transverse diffusions appears clearly in Després and Lagoutière 2 which was a previous attempt to elaborate nondissipative schemes on non-cartesian grids. Based on mono-dimensional techniques, the schemes derived from this work where longitudinally anti-dissipative, but not transversely. The new method here described is based on an interpretation of the anti-dissipative limited downwind scheme which was proposed in Lagoutière 27 previous works are Després and Lagoutière 22 and Després and Lagoutière 27. This paper shows that the limited downwind scheme in dimension can be understood as a reconstruction scheme, decomposed in 3 stages at each time step: a reconstruction stage which, starting from a constant-in-cell datum, constructs a new datum, presenting in each cell one discontinuity that separates two constant values, a transport stage that solves the transport operation with the new datum, a projection stage that computes the mean value of the transported datum in each cell 2

3 see section 4 for precision on the reconstruction stage. The natural extension to the multidimensional transport problem consists in reconstructing the datum in two steps: in a first step, performing a transverse reconstruction, and performing a longitudinal reconstruction in a second step. This is the technique which is proposed in this work. The following describes precisely these two operations. 3 General numerical formalism It is the one of finite volume methods for problem. We consider a mesh of R 2 composed of non-empty open triangles T j j Z such that j Z T j = R 2 and T i T j = for every i, every j i. For each cell T j for j Z, one denotes Kj the set of indices of neighboring cells of T j that is to say cells having a common edge with T j, Kj = { k Z \ {j} s.t. meas T j T k > } where meas denotes the Lebesgue measure in dimension. For j Z and for k Kj T j and T k have an edge in common, one denotes l j,k the length of the common edge, l j,k = meas Tj T k and n j,k the unit normal vector to the common edge outward to T j. We thus have l j,k = l k,j and n j,k = n k,j for every j Z and every k Kj. We then denote by K + j the set of indices of the downwind neighbors of T j and by K j the set of indices of the upwind neighbors of T j : K + j = {k Kj s.t. a,n j,k > }, Let s j denote the surface of cell T j : s j = meas 2 T j where meas 2 denotes the Lebesgue measure in dimension 2. The general form of the here considered schemes is obtained by choosing a time step t > and by integrating the transport equation of on [n t, n + t] T j : u n+ j u n j t + l j,k a n j,k s,n j,ku n j,k + j K j K + j l j,k a n j,k,n j,ku n j,k =. 2 In this formula, the quantities a n j,k are approximate values of the given velocity at, x on the edges and the u n j,k are approximate values of the solution on the edges between times n t and n + t. The upwind scheme is obtained taking u n j,k = un j for k K+ j and u n j,k = un k for k K j. We will propose another definition of these numerical fluxes, intended to provide more precise numerical results. 4 Discontinuous reconstructions in dimension In the following, for every a, b R, a, b denotes the interval [a, b] if a b and the interval [b, a] else, that is to say that we adopt the convention a, b = [mina, b, maxa, b] a K + j = {k Kj s.t. a,n j,k < }. T m T k l j,k n j,k T j Figure 4. Mesh and notations k, l, m Kj k, l K + j m K j T l and one has a, b = b, a. We here recall the principle of discontinuous reconstruction schemes in dimension. Details and proofs lie in Lagoutière 27 and Lagoutière 27. We consider a mesh on R with constant space step x > whose cells are the intervals T j = j /2 x, j + /2 x for j Z. It is shown in Lagoutière 27 that for the transport equation t u + a x u =, the limited downwind scheme out of Després and Lagoutière 22 is equivalent to the 3-stages following scheme. Let u n j be a discrete datum associated j Z with a constant-in-cell function. Reconstruct in each cell the discrete datum in a form with more detail not constant following the algorithm detailed above. 2 Perform the exact transport of this reconstructed datum at velocity a for a time t. 3

4 3 Project the computed solution to obtain a new constant-in-cell datum and to be able to iterate these 3 operations. It now remains to describe the reconstruction operation, the two others being trivial. Let u n j be given. We define the discrete function ux = j Z un j χ [j /2 x,j+/2 xx. The al- j Z gorithm is defined by if u n j / u n j, un j+, that is to say if un j is a local extremum, we do not reconstruct the datum in cell T j, if u n j un j, un j+, we define u n j,l = u n j, u n j,r = un j+, d n j = un j+ un j u n j+, un j and the reconstructed datum is defined inside T j by ûx = u n j,l if x [j /2 x, j /2 + d n j x, ûx = u n j,r if x [j /2 + d n j x, j + /2 x. Notice that when u n j un j, un j+, formula 3 is welldefined. In particular, the denominator in the definition of d n j is not. More precisely, one always has dn j,, so that j /2 + d n j x T j. Furthermore, another essential property of this reconstruction is that it is conservative: d n j un j,l + dn j un j,r = un j. Of course, other values for u n j,l and un j,r are admissible, but this choice leads to the most anti-dissipative scheme under constraints of L stability and decreasing of the total variation, cf. Lagoutière 27. At last, notice that not to reconstruct the datum is equivalent to take u n j,l = un j,r = un j and gives the Godunov upwind scheme. The discontinuous reconstruction is illustrated on FIG Discontinuous reconstructions in dimension 2 The two-dimensional extension is not obvious. The genuine two-dimensional nature of the problem, due to the choice of a non-cartesian grid, leads to the fact that the finite volume method 2 cannot be seen as an exact Godunov one which would be composed of an exact transport step preceded or not by a reconstruction followed by a projection on the mesh since only edge fluxes are taken into account, with the exception of node fluxes. In order to separate the problems of longitudinal and transverse diffusions see the introduction of this paper, we propose to perform the reconstruction in 2 steps, each one being a one-dimensional reconstruction. In order to simplify the presentation, let us assume that a is a time and space constant. Let u n j be a discrete datum at time j Z step n. The first reconstruction, transverse, boils down to cut some cells in two in the direction of the velocity and to modify the value of the datum in each of these two subcells. The cell T j has at least one downwind neighbor and at most two thanks to the divergence-free hypothesis. If it has only one downwind neighbor, we do not perform the transverse reconstruction we do not cut the cell. Let us now assume that T j has two downwind neighbors, T k and T l. It has then one upwind neighbor, T m. We consider the intersection point of the two edges relative to the downwind neighbors and cut T j along the line passing on this intersection point and parallel to a. The two induced subcells are denoted T j,k and T j,l : T j,k has T k as downwind neighbor, and T j,l has T l as downwind neighbor. This cutting is illustrated on FIG. 6. a T k T j,k T j T j,l T l u n j+ u n j u n j u n r j u n l j j j j + dn j x d n j x Figure 5. Discontinuous reconstruction. T m Figure 6. Transverse reconstruction. We denote s j,k and s j,l the surfaces of cells T j,k and T j,l. One of course has s j,k + s j,l = s j and s j,k > and s j,l >. 4

5 We now have to assign to each of the sub-cells a value of the reconstructed solution. The adopted principle is quite similar to the one in dimension, except that the locus of the discontinuity is determined by the geometrical aspect and not the local values of the datum: it is the line parallel to a defined just above. The aim is to define a value u n j,k in the cell T j,k and a value u n j,l in the cell T j,l by maximizing u n j,l un j,k to guarantee the antidissipativity and making certain that s j,k u n j,k +s j,lu n j,l = s j u n j to guarantee the conservativity. On the other side, following the same rules as in dimension, we impose that the triplet {u n l, un j,l, un j,k } has the same monotony as the pair {u n l, un j } and that the triplet {un j,l, un j,k, un k } has the same monotony as the pair {u n j, un k }. These constraints imply in particular that the datum in T j would not be reconstructed if u n j was a local extremum in the transverse direction. The algorithm is if u n j / u n k, un l if un j is a local extremum in the transverse direction, we do not reconstruct u in the cell T j that is to say, u n j,l = un j,k = un j, if u n j un k, un l, if s j u n j s j,lu n l /s j,k [u n j, un k ], we define { u n j,l = un l, u n j,k = s ju n j s j,lu n l /s j,k ; if s j u n j s j,ku n k /s j,l [u n j, un j ], we define { u n j,l = s j u n j s j,k u n k/s j,l, u n j,k = un k. Lemma. The Courant-Friedrichs-Lewy CFL condition on the time step is not degraded by the transverse reconstruction. Proof. The standard condition for the upwind scheme for the cell T j is k K t + j l j,ka, n j,k, s j i.e., in the particular studied case, t l j,ka, n j,k + l j,l a, n j,l s j. 4 One has s j,k = l j l j,k a, n j,k 2 and and s j,l = l j l j,l a, n j,l, 2 s j = s j,k + s j,l = l j l j,k a, n j,k + l j,l a, n j,l 2 The two inequalities of 5 and inequality 4 thus rewrite they are equivalent. 2 t l j, Remark. Once the solution is reconstructed, the transport of it is related to one-dimensional transport since T j,l and T j,k have only one upwind and one downwind neighbor: respectively, T m, T l and T m, T k. When T j has only one downwind neighbor, we do not perform the transverse reconstruction and can consider the problem as monodimensional, performing virtually the cutting but assigning the value u n j to each sub-cell. After this transverse reconstruction, the algorithm is more classical, lying on the fact that the transport is mono-dimensional, as noticed in remark. We thus can use the algorithm of our own choice. In the following, we pitch on the limited downwind one. This can also be understood as a second reconstruction, longitudinal. 6 Numerical results We present a few results obtained with the algorithm described herein. Different velocity fields are used: translation and rotation fields. For all the test-cases, the spatial domain is [, ] 2. The triangular mesh has been generated by the software FreeFem++: see FIG. 7. The CFL condition for sub-cells T j,k and T j,l are t l j,ka, n j,k s j,k and t l j,la, n j,l s j,l. 5 Let us denote l j the length of the segment separating T j,k and T j,l : l j = mes Tj,k T j,l. Figure 7. Example of Freefem++ mesh, used for the numerical tests here, 3766 triangles. 5

6 For all P the test-cases, the Courant number k K max j Z t + j l j,ka,n j,k s j takes the value.. 6. Translation field Here we consider the veloc- translation of a square ity at, x, y = This first test is the translation of the characteristic function of a square, u x, y = χ [.3,.7] 2x, y. The boundary conditions are periodic in x and y. Initial condition and results at time t = after one revolution are reported in FIG. 8 and FIG. 9 reports the result by the upwind scheme Rotation field Rotation of a square at, x, y = The velocity field is here 2πy 2πx and the initial condition is the same as in the preceding test. The final time is t =. Figure presents the result Figure. Rotation of a square with 5874 cells. Zalesak s test-case the preceding test, at, x, y = The velocity field is the same as in 2πy 2πx and the initial condition is taken from the original paper Zalesak 979: u x, y = χ Z x, y Figure 8. Translation of a square with 5874 cells Figure 9. Translation of a square with 5874 cells, upwind scheme. where with Z = C \ R C = { x, y; x y }, D = {x, y; x.5 /4 and y 7/2}. Initial condition and result computed with 5874 cells at time t = lie in FIG.. Then, FIG. 2 represents the same for a mesh made of 2368 cells, and FIG. 3 for a mesh composed of cells. 6

7 which means that at time t = the field is reversed. The initial condition is the characteristic function of a rectangle: u x, y = χ[.2,.7] [.45,.55]x, y The exact solution at time 2 coincides with the initial condition. We observe the initial condition FIG. 4 and the results at time t = FIG. 5 with 5874 cells and finally the result at time t = 2 with 5874 cells FIG. 6, 2368 cells FIG. 7 and cells FIG. 8. The result given by the upwind scheme with cells is reported on FIG Figure. Zalesak s test-case with 5874 cells Figure 4. Initial condition with 5874 cells Figure 2. Zalesak s test-case with 2368 cells Figure 5. Numerical solution with 5874 cells at time t = Figure 3. Zalesak s test-case with cells Deformation of a rectangle The velocity field is now not constant in time:! 3πy.52 a = sgn t, 3πx Figure 6. Numerical solution with 5874 cells at time t = 2. 7

8 References Després B. and Lagoutière F. 2: Generalized Harten formalism and longitudinal variation diminishing schemes for linear advection on arbitrary grids. M2AN Math. Model. Numer. Anal. 35, no. 6, pp Després B. and Lagoutière F. 22: Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. J. Sci. Comput,. 6 2, no. 4: pp Figure 7. Numerical solution with 2368 cells at time t = 2. Després B., Lagoutière F. 27: Numerical resolution of a two-component compressible fluid model with interfaces, to appear in Progress in Computational Fluid Dynamics Lagoutière F. 27: Stability of reconstruction schemes for scalar hyperbolic conservation laws, in revision Lagoutière F. 27: Non-dissipative reconstruction schemes satisfying entropy inequalities, in revision. Mosso S. and Cleancy S. 995: A geometrical derived priority system for Young s interface reconstruction. LA-CP-95-8, LANL report. Figure 8. Numerical solution with cells at time t = 2. Noh W. F. and Woodward P. R. 976: SLIC Simple Line Interface Calculation. Springer Lecture Notes in Physics, 25: pp D. L. Youngs, An interface tracking method for a 3D eulerian hydrodynamics code, Technical Report 44/92/35, AWRE Zalesak S. T. 979: Fully multidimensional fluxcorrected transport algorithms for fluids, Journal of Computational Physics, 3, pp Figure 9. Numerical solution with cells at time t = 2 with the upwind scheme. 7 Final comments We developed a new method for the numerical transport in dimension 2. The approach is truly multidimensional in the sense that is does not resume to a mono-dimensional reconstruction of interfaces. The numerical results show the anti-dissipative behavior of the algorithm. The results are nevertheless not perfect. Indeed, the discontinuity lines may be degraded in long time see FIG., 2, 3. Next study will concern the extension on general meshes non-triangular and in dimension 3, and the application to gas dynamics equations. 8