A Semi-Lagrangian Scheme for the Open Table Problem in Granular Matter Theory

Transcription

1 A Semi-Lagrangian Scheme for the Open Table Problem in Granular Matter Theory M. Falcone and S. Finzi Vita Abstract We introduce and analyze a new scheme for the approximation of the two-layer model proposed in [] for growing sandpiles over an open flat table. The method is based on a semi-lagrangian approximation of the nonlinear terms which allows to obtain more accurate results with respect to the standard finite difference approximation. We present several features of the scheme and give some hints on its implementation. Finally we show some tests where it is compared with a previously studied finite difference approach. Introduction Mathematical models for the dynamics of granular materials have recently been studied in several papers (see e.g. []), since a complete and realistic description of many phenomena in this field is far from being achieved. In this paper we restrict ourselves to the rather simple case of growing sandpiles over a flat bounded open table under the action of a given vertical source, neglecting wind effects and avalanches. For this phenomenon two main differential models have been proposed. The first one is the variational model of Prigozhin [], where the surface flow of sand is supposed to exist only at critical slope, that is the maximal admissible slope α for any stationary configuration of sand (here for simplicity it is assumed α = ). For that model there are a sufficiently developed theory and efficient numerical schemes which use duality arguments (see [2]). The second approach has been introduced by Hadeler and Kuttler [] as an extension of the known BCRE model [3] (see also [4]). In their model the pile is obtained summing two distinct Maurizio Falcone Dipartimento di Matematica, SAPIENZA - Università di Roma, falcone@mat.uniroma.it Stefano Finzi Vita Dipartimento di Matematica, SAPIENZA - Università di Roma, finzi@mat.uniroma.it 69

2 7 M. Falcone and S. Finzi Vita layers: the standing layer u and the rolling layer v. The two layers interact during the growth process, the exchange term between u and the v being described by the nonlinear term in the following system of partial differential equations: u t = ( Du )v in Ω (,T) v t = div(vdu) ( Du )v+ f in Ω (,T) () u(,) = v(,) = in Ω, u = on Γ (,T). In () f represents the source, the table is assumed initially empty, and no boundary condition is needed on v (the characteristic lines for the second equation are all directed towards the interior of Ω). Note that this model allows for the movement of rolling grains even at sub-critical slope. It also seems to be better suited for the description of fast processes or small details over the surface (see [2] for a detailed comparison between the two models). However, the mathematical theory for the two-layer model is still incomplete (even a general result of existence for its solutions is lacking). It is interesting to note that the two distinct models have essentially the same set of admissible stationary solutions (u, v) and that a characterization of such equilibria has been given, using the theory of viscosity solutions, in [5]: the maximal equilibrium u is given by the function d(x) measuring at any point x Ω its distance to the boundary, and an integral representation is proved for v along of the transport rays of sand. Anyway, as it was observed in [8], only in particular cases the asymptotic behavior is the same (and then u(x) = d(x)). In general, the effective stationary standing solution u for the model () is not explicitly known. It is also for such a reason that in [8] we proposed a finite difference scheme which preserves the properties of the model at the discrete level, and which is able to give a numerical characterization of stationary solutions for every general source support in a square open table. A disadvantage of this approach is that finite difference schemes on a regular structured grid restrict the numerical discrete flow essentially to the axis directions, breaking the homogeneous character of the real phenomenon (cones of sand are in general approximated by pyramids oriented along the grid axes). This numerical anisotropy is typical of finite difference (FD) schemes and trying to eliminate it is particularly important in view of the extension of schemes to the so-called partially-open table problem, that is when high vertical walls bound portions of the table boundary. In fact, in such a problem an infinite number of transport rays can meet at certains points on the boundary, yielding a discontinuous and even unbounded rolling layer v L (Ω) at the equilibrium, and strong numerical difficulties in its description (see [6]). This has motivated the development of a semi-lagrangian (SL) approach which will be discussed in this paper. By construction, the SL approach mimics the method of characteristics in order to follow in a more accurate way the real direction of sand flow. We introduce the scheme and its properties in Section 2, and in Section 3 we present first numerical tests in the regular case of the open table problem, showing a slight but clear improvement in the accuracy with respect to the standard FD approach. These results let us conjecture that in the partially-open table

3 A Semi-Lagrangian Scheme for the Open Table Problem 7 problem the SL approach could be of interest, despite of its time-consuming feature. Experiments in this direction are actually under consideration. A detailed presentation of SL methods for evolutive nonlinear partial differential equations of the first order can be found in [7]; see also [9] for a convergence result of a first order SL scheme to the viscosity solution. 2 The semi-lagrangian scheme and its properties Let us start from the first equation in () and observe that it can be equivalently written in one dimension as ( ) u t = v( u x ) = v + min au x, (2) a B(,) where B(,) denotes the closed unit ball of IR, that is the interval [,]. In this way the nonlinear term u x is written as the minimum over the directional derivatives of u and is actually computed comparing the values for a =, a = and a =. Note that in two dimensions we can use exactly the same formula to compute u. At the discrete level, if we introduce in Ω = (α,β) a uniform grid of points G = {x i : x i = α + i x,i =,..,N}, (2) can be written as u n+ i = u n i + tv n i ( u n (x i + a t) u n i + min a {,,} t ), (3) where, as usual, u n i denotes the approximate value for u(x i,n t). If t x and a, then the point z i (a) x i + a t is not a node of the grid G and the value of u n in it has to be computed, for example, by linear interpolation. For the equation in v, we have also second order terms. In fact, v t = v x u x + vu xx v( u x )+ f ; (4) let us use the standard second order central difference D 2 u n i to approximate u xx (x i ) whereas u x at any node is replaced by the maximal (in absolute value) finite difference in the left and right directions, Du n i. A simple trick is to replace the nonlinear term at the node x i by the previously computed difference (u n+ i u n i ). Finally, for v x we use the discrete directional derivative in the direction of u x. After a simplification, we finally obtain the following discretization of (4) v n+ i = v n (x i + tdu n i ) (un+ i u n i )+ t(vn i D2 u n i + f i). (5) Of course, also the term v n (x i + tdu n i ) requires a local reconstruction by interpolation. In order to complete the scheme we need to add initial and boundary conditions u i = v i = i ; u n = un N = n. (6)

4 72 M. Falcone and S. Finzi Vita The extension of this approach to IR 2 is straightforward.we assume for simplicity that Ω = (,) (,), that is the unit square table, and we introduce a uniform grid of points G = {x i, j = (i x, j x),i, j =,..,N}, where x = /N is again the space discretization step. Then the analogue of (3)-(5) and (6) is ( u n+ i, j = u n i, j + tv n i, j + ) min t a B(,) (un (x i, j + a t) u n i, j), v n+ i, j = v n (x i, j + tdu n i, j ) (un+ i, j u n i, j )+ t(vn i, j D2 u n i, j + f (7) i, j), u i, j = v i, j = i, j ; un i, j = n, if x i, j Ω, where u n i, j denotes the approximate value for u(x i, j,n t). In order to compute the value of u n (x i, j + a t) we use now a bilinear interpolation with respect to the four vertices of the cell containing x i, j + a t. The same local reconstruction is required for the term v n (x i, j + tdu n i, j ). The minimum term in (7) is approximated by comparing the values of u n on a finite set of directions (for example the eight directions θ k = kπ/4, k =,..,8, plus the origin, that is the node under consideration). D 2 u n i, j denotes the standard five-points second order difference which replaces u(x i, j ). 2. Properties of the SL scheme Here we present some features of the above scheme, showing that it preserves the physical properties of the continuous model. At the moment we are able to give a complete proof of that only in the one-dimensional case, but these properties are confirmed by all the 2D experiments. Further details and complete proofs will be presented in a forthcoming paper. Theorem. Let f in Ω and ( ) t x min 2,, (8) C f where C is a positive constant. Then, the sequences u n and v n defined in (3)-(5)-(6) satisfy the following properties for any n:. positivity and monotonicity in u: u n u n+ ; 2. positivity in v: v n ; 3. sub-critical slope in u: Du n. Proof. The structure of the proof is the same of that of Theorem 3. of [8]. The above properties are proved by induction. Here we only discuss some details which are characteristic of the semi-lagrangian scheme. In particular, for the nonlinear term in (3) we remark that min a {,,} u n (x i + a t) u n i t, n,i ; (9)

5 A Semi-Lagrangian Scheme for the Open Table Problem 73 moreover, since the intermediate values are computed in this case by linear interpolation and t x, we have for example that u n (x i + t) u n i t = un i+ un i x At the same time, if for example Du n i >, we can rewrite. () v n (x i + tdu n i ) = vn i + t x (vn i+ vn i )Dun i, () and the proof of positivity for v n follows the same arguments used for the FD scheme in [8], yielding the bound 2 t x. The stronger stability bound (8) has to be assumed in order to achieve the gradient constraint in every interval [x i,x i+ ]. An accurate discussion on admissible configurations allows to rewrite the different quotient in (9) always in terms of the slope of u n inside the interval under consideration. Then, considering the difference (u n i+ un i ) as defined by (3), one gets sub-criticality if ( t vn i ) x, which holds for the uniform boundedness of v n in terms of the source f. 3 Numerical results In this section we present some numerical tests in two dimensions to compare the SL method with the old FD method. For computation we used MATLAB 7. on a Processor Intel Pentium M74,.73GHz with 8Gb RAM. Test : constant source on all of Ω We assume f.5 in all the domain Ω = (,) (,). In this case the ridge set S (the set where d is singular) is given by the two diagonals of the square and, following [5], the stationary solutions are given by u(x) = d(x) and τ(x) v(x) = on S, v(x) = f(x+ tdu(x)) dt in Ω \ S, (2) where τ(x) denotes the distance of x from the set S along the transport ray through x. In Fig. we show the computed solutions u and v for x =.2, t =. (the stopping criterion for the equilibrium was: u n+ u n 6 ). We show in Table the results of the two schemes at the same final time when t/ x=.4, comparing the L and L norms of the error for u. It can be seen that the two methods are both approximately of first order. Nevertheless, the SL scheme is slightly more accurate in the L (+2%) and the L (+3%) norms respectively. Naturally, it is also much more expensive due to the minimum term computation at any iteration.

6 74 M. Falcone and S. Finzi Vita strato stazionario strato rotolante Fig. Test : Numerical results for u and v Table Errors for Test at CFL=.4 method x L error L error FD SL FD SL FD SL Test 2: constant source on a connected subdomain of Ω Assume now that the constant source is concentrated in a small square inside Ω (see Fig. 2), and it is zero elsewhere. An explicit formula for u is not known in this case (with some efforts v could be still computed by formula (2)), and Fig. 2 shows the stationary solution u as computed by the SL algorithm. In Fig. 3 the level lines found for u are compared with those found by the FD scheme u.6.8 Fig. 2 Test 2: The source support and the computed standing layer u

7 A Semi-Lagrangian Scheme for the Open Table Problem 75 curve di livello schema SL curve di livello schema DF Fig. 3 Test 2: Level curves for u with the SL and the FD schemes Test 3: constant source on a disconnected subdomain of Ω Assume now that the constant source is concentrated in two distinct small squares inside Ω, and it is zero elsewhere. Fig. 4 shows this source support together with the stationary solution u as computed by the SL algorithm. In Fig. 5 the level lines found for u by the two algorithms are compared u.6.8 Fig. 4 Test 3: The source support and the computed standing layer u Conclusions The above results seem to show the SL scheme, although enough expensive, can be more accurate than the corresponding FD scheme in the description of singularity regions for the solution. This can be seen by comparing the higher level sets of the two numerical solutions, those corresponding to the crests of the piles. This feature

8 76 M. Falcone and S. Finzi Vita curve di livello schema SL curve di livello schema DF Fig. 5 Test 3: Level curves for u with the SL and the FD schemes can be very important for more complicated problems and geometries, like for the wall and the silos problems, where hardest singularities can arise (see [6]). Acknowledgements We thank Alessia Pacella for the numerical experiments. References. Aranson, I.S., Tsimring, L.S.: Patterns and collective behavior in granular media: theoretical concepts. arxiv:cond-mat/5749 (25) 2. Barrett, J.W., Prigozhin, L.: Dual formulations in critical state problems. Interfaces and Free Boundaries 8, (26) 3. Bouchaud, J.-P., Cates, M.E., Ravi Prakash, J., Edwards, S.F.: A model for the dynamics of sandpile surfaces. J. Phys. I France 4, (994) 4. Boutreux, T., de Gennes, P.-G.: Surface flows of granular mixture, I. General principles and minimal model. J. Phys. I France 6, (996) 5. Cannarsa, P., Cardaliaguet, P.: Representation of equilibrium solutions to the table problem for growing sandpiles. J. Eur. Math. Soc. (JEMS) 6, (24) 6. Crasta, G., Finzi VIta, S.: An existence result for the sandpile problem on flat tables with walls. To appear; preprint available at (28) 7. Falcone, M., Ferretti, R.: Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comp. Physics 75, (22) 8. Falcone, M., Finzi Vita, S.: A finite difference approximation of a two-layer system for growing sandpiles. SIAM J. Sci. Comput. 28, 2 32 (26) 9. Falcone M., Giorgi T.: An approximation scheme for evolutive Hamilton-Jacobi equations. In: W.M. McEneaney, G. Yin, Q. Zhang (eds.), Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, Birkhäuser, (999). Hadeler, K.P., Kuttler, C.: Dynamical models for granular matter. Granular Matter 2, 9 8 (999). Prigozhin, L.: Variational model of sandpile growth. Euro. J. Appl. Math. 7, (996) 2. Prigozhin, L., Zaltzman, B.: Two continuous models for the dynamics of sandpiles surface. Physical Review E 63, 455 (2)