A multiresolution method for the simulation of sedimentation-consolidation processes Raimund B¨ urger1 and Alice Kozakevicius2 1
2
Departamento de Ingenier´ıa Matem´ atica, Universidad de Concepci´ on, Casilla 160-C, Concepci´ on, Chile. E-Mail:
[email protected] Departamento de Matem´ atica, Universidade Federal de Santa Maria, Faixa de Camobi, km 9, Campus Universit´ ario, Santa Maria, RS, CEP 97105-900, Brazil. E-Mail:
[email protected]
Summary. A multiresolution method for a one-dimensional strongly degenerate parabolic equation modeling sedimentation-consolidation processes is introduced. The method is based on the switch between central interpolation or exact evaluation of the numerical flux combined with a thresholded wavelet transform applied to point values of the solution to control the switch. A numerical example is presented.
1 Introduction The multiresolution method has been devised to reduce the computational cost of high resolution methods for conservation laws, whose solutions are usually smooth on the major part of the computational domain but strongly vary in small regions near discontinuities. The method adaptively concentrates computational effort on the latter regions. It goes back to Harten [8] for conservation laws and was used in [2, 12] for parabolic equations. In this contribution, we construct adaptive multiresolution schemes and present numerical results for the strongly degenerate parabolic equation ut + f (u)x = A(u)xx ,
(x, t) ∈ QT := (0, L) × (0, T ],
(1)
where f, A : R → R are piecewise smooth and Lipschitz continuous, and A(·) is nondecreasing. On intervals [α, β] with A(u) = const. for all u ∈ [α, β], equation (1) degenerates into a conservation law. Equation (1) arises from a sedimentation-consolidation model for suspensions [1]; see [5] for other applications. Since A = 0 on u-intervals of positive length, (1) is called strongly degenerate. Its solutions are in general discontinuous. The multiresolution method reduces the number of exact flux evaluations required by a high resolution scheme. To this purpose, point values or cell averages of the numerical solution are defined on a hierarchical sequence of nested diadic meshes, where the initially given mesh is the finest. The sequence of coefficients for all meshes forms the multiresolution representation of the
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Raimund B¨ urger and Alice Kozakevicius
solution. Since multiresolution coefficients are small on smooth regions, data can be compressed by thresholding, i.e. setting to zero those multiresolution coefficients which are in absolute value smaller than a prescribed tolerance. The multiresolution representation can be used to locate discontinuities of the numerical solution, since multiresolution coefficients measure its local regularity. Harten converted this idea [8] into a sensor to decide at which fine-mesh positions fluxes should be exactly evaluated, or can otherwise be obtained more cheaply by interpolation from coarser scales. Our multiresolution scheme combines the switch between central interpolation and exact evaluation of both convective and diffusive numerical fluxes with a thresholded wavelet (multiresolution) transform applied to solution point values to control the switch. The first alternative is performed on smooth regions, while the second applies near strong variations. Instead of calculating a wavelet transform for cell averages as in [8], we use here the interpolatory framework (point values) to analyze the smoothness of the solution. This slight change improves the efficiency of the algorithm, since the multiresolution representation is cheaply obtained as in [9]. The efficiency of the multiresolution method is measured in terms of the data compression rate and CPU time. Our scheme can be extended to multidimensional problems by a multidimensional wavelet transform [3] or by dimensional splitting (see e.g. [6]). In this work, we consider the zero-flux initial-boundary value problem for a bounded domain Ω := [0, L] with the conditions u(x, 0) = u0 (x),
x ∈ Ω;
f (u) − A(u)x = 0,
x ∈ {0, L}, t ∈ (0, T ]. (2)
Solutions of strongly degenerate parabolic PDEs are in general discontinuous and must be defined as entropy solutions. Recent works on the analysis and numerics of these PDEs include [4, 10, 11], see [5] for further references. 2 The multiresolution scheme Let (G0 , G1 , . . . , GLc ) denote a family of uniform nested grids on I := [a, b], where G0 := (x00 , x01 , . . . , x0N0 ), N0 = 2m , m ∈ N is the finest resolution level, and h0 := (b−a)/N0 . The values of a function u on G0 are the input data. The remaining diadically coarsened grids are obtained as follows: given Gk−1 , we obtain Gk by removing the even-indexed grid points. Therefore Gk−1 \ Gk = k−1 k−1 (x2j−1 )j=1,...,Nk , Gk−1 ∩ Gk = Gk , and xkj = x2j for 0 6 j 6 Nk = 2m−k , k = 1, . . . , Lc . The representation of u on any coarser grid G1 , . . . , GLc can be obtained directly from G0 : ukj = u(xkj ) = u(x02k j ) = u02k j for 0 6 j 6 Nk . To recover the representation of u on Gk−1 from its representation on Gk , we need an interpolation operator I(uk , x) of u on Gk to obtain approximations k−1 for the missing points of Gk−1 . The function value at x2j−1 is obtained from k the (r − 1)-th degree polynomial interpolating (uj−s , . . . , ukj+s−1 ). Therefore s X k−1 k−1 βl ukj+l−1 + ukj−l , = I uk , x2j−1 = u ˜2j−1 l=1
r = 2s,
(3)
Multiresolution method for sedimentation-consolidation processes
3
with β1 = 1/2 for r = 2 and β1 = 9/16, β2 = −1/16 for r = 4. The interpolak−1 k−1 tion errors, known as details or wavelet coefficients, are dkj = u2j−1 − u˜2j−1 for 1 6 j 6 Nk . Thus, from uk := (uk0 , uk1 , . . . , ukNk ) and dk := (dk0 , dk1 , . . . , dkNk ), we can exactly recover the representation of u on Gk−1 . The pair of vectors (uk , dk ) is the multiresolution representation of uk−1 . Applying successively this procedure for 1 6 k 6 Lc , we can recover the values of u on G0 from its values on GLc and the sequence of all details from levels Lc to 1: u0 ↔ (d1 , u1 ) ↔ (d1 , d2 , u2 ) ↔ · · · ↔ (d1 , d2 , . . . , dLc , uLc ) =: uM ,
(4)
where uM is the multiresolution representation of u0 ≡ u. The details dk contain information on the smoothness of u, and will be used to flag the non-smooth parts of the solution in the adaptive numerical method. Standard interpolation results imply that if u at a given point x has p − 1 continuous derivatives and a discontinuity in its p-th derivative, then dkj ∼ (hk )p [u(p) ] for 0 6 p 6 r¯ and dkj ∼ (hk )r¯u(¯r) for p > r¯, for xkj near x, where r¯ := r − 1 is the order of accuracy of the approximation and [·] denotes the k−1 jump. Therefore |d2j | ≈ 2−p¯|dkj |, if the k−th level is fine enough, where p¯ := min{p, r¯}. Thus, away from discontinuities of u, the wavelet coefficients dkj diminish as the levels of resolution become finer. We see that near a discontinuity of the function, the wavelet coefficients remain of the same size for all levels of refinement. Thus, data compression and reduction of computational effort can be attained by discarding wavelet coefficients that are smaller than a prescribed tolerance. This operation is known as thresholding or truncation. To define it, let us denote by trεk the hard thresholding operator with εk as threshold parameter: ( 0 if |dkj | 6 εk , k k ˆ 1 6 j 6 Nk , 1 6 k 6 L c . (5) dj = trεk dj = dkj if |dkj | > εk , Consequently, u ˆM := (dˆ1 , . . . , dˆLc , uLc ) is the thresholded multiresolution representation. Let u˜0 be the data recovered from u ˆM . Harten proved in [8] that ku0 − u ˜ 0 k 6 c1 (ε1 + · · ·+ εLc ) 6 ε, where the constant c1 is independent of Lc . Hence, given a tolerance ε, we can compress data by truncating uM . Clearly, the actual compression rate depends on the chosen strategy (ε1 , . . . , εLc ). In contrast to the evaluation on a sparse point representation [9], we evaluate the differential operator on the uniform fine grid but adapt the manner of flux computation to the significant coefficients of u ˜0 , as is done in [8, 2, 6]. This strategy does not provide memory savings, but we have a better compression rate, and consequently, a smaller number of exact flux evaluations. Finally, the index set of significant coefficients in each time step, Dn , needs to capture the finite speed of propagation of information and the formation of shock waves. For this reason, Harten [8] proposed an algorithm to extend Dn after thresholding, including so-called safety points near positions associated ˜ n . We here utilize with significant details, which yields an extended index set D a version of Harten’s algorithm [8, Alg. (6.1)], see [5, Alg. 2.1] for details.
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For the time discretization of ut = L(u) ≡ −f (u)x + A(u)xx we use an explicit 3-step third-order Runge-Kutta TVD (RKTVD) scheme [7]. A general nRK -step explicit RKTVD scheme has the form (0)
uj
= unj ,
(i)
uj =
i−1 X (k) αik uj + ∆tβik Lj u(k) ,
(nRK )
un+1 = uj j
,
(6)
k=0
i = 1, . . . , nRK , where Lj (u) contains the flux and diffusion terms. We distinguish between the interior operators L1 , . . . , LN0 −1 and the boundary operators L0 and LN0 , which include the boundary conditions. Point values of the initial solution of (1) are given on a uniform fine grid ˜ n is considered already built. Then a G0 , ui = u(xi ), and the index set D conservative semi-discrete scheme is given by ¯ 1/2 /∆x, u˙ 0 (t) = L¯0 u(t) := − F¯1/2 − D ¯ j+1/2 − D ¯ j−1/2 ) /∆x, (7) u˙ j (t) = L¯j u(t) := − F¯j+1/2 − F¯j−1/2 − (D ¯ N −1/2 /∆x, u˙ N0 (t) = L¯N0 u(t) := F¯N0 −1/2 − D 0
where k = 1, . . . , N0 , u(t) := (u0 (t), . . . , uN0 (t)), and the numerical fluxes ¯ i+1/2 contain the advective and diffusive terms, respectively. F¯i+1/2 and D n ˜ If i ∈ D , then we use a Lax-Friedrichs splitting [13] with a third-order ENO interpolation for F¯i+1/2 and add a fine-grid finite difference of the dif˜ n , the numerical flux is approximated by interpolation fusion term. If i ∈ / D of fluxes previously evaluated on a coarser level. On our finite domain, the k−1 L /16, where ) = I˜k,j interpolator (3) is replaced by I L (F¯ k , x2j+3/2 k ¯ ¯k ¯k ¯k j = 0, 5F1/2 + 15F3/2 − 5F5/2 + F7/2 , L k k k k ˜ ¯ ¯ ¯ ¯ Ik,j := −Fj−1/2 + 9Fj+1/2 + 9Fj+3/2 − Fj+5/2 , j = 1, . . . , Nk − 2, ¯k FNk −7/2 − 5F¯Nk k −5/2 + 15F¯Nk k −3/2 + 5F¯Nk k −1/2 , j = Nk − 1.
˜ n . ThereBy construction, all positions from the coarsest level Lc are in D fore all fluxes on level Lc are always exactly evaluated. The u-values required for the flux computation are taken from the finest level, k = 0. k = (fˆ+ )ki+1/2 + (fˆ− )ki+1/2 , The convective fluxes in (7) are given by F¯i+1/2 + i = 0, . . . , N0 − 1, where f (ui ) = (f (ui ) + αui )/2 and f − (ui ) = (f (ui ) − αui )/2 for 0 ≤ i ≤ N0 , where α := maxu |f ′ (u)| [13] and (fˆ+ )ki+1/2 , (fˆ− )ki+1/2 are approximations obtained by the ENO interpolator of each splitting component f + and f − , evaluated at cell boundaries. The diffusive fluxes at level k k are calculated by Di+1/2 := (A(u02k i+1 ) − A(u02k i ))/∆x. The stability condition is the same as that of the difference finite scheme on the finest grid. i.e. CFL := maxu |f ′ (u)|(∆t/∆x)+2 maxu |a(u)|(∆t/∆x2 ) ≤ 1. Since the ENO interpolator needs six points to search the least oscillatory four-point stencil for the flux calculation, we extrapolate the solution across the boundaries of I. We summarize the flux computation procedure as follows.
Multiresolution method for sedimentation-consolidation processes
5
Algorithm 1 Lc Lc , as stated above, for i = 0, . . . , NLc − 1. and Di+1/2 1. Compute F¯i+1/2 2. do k = Lc , Lc − 1, . . . , 1 (compute fluxes for all other levels) do j = 0, . . . , Nk − 1 ¯ k−1 ← D ¯k F¯ k−1 ← F¯ k , D 2j+1/2
j+1/2
2j+1/2
j+1/2
˜ n then (flux/diffusion terms are computed explicitly) if (j, k) ∈ D k−1 k−1 k−1 F¯2j+3/2 ← (fˆ+ )2j+3/2 + (fˆ− )2j+3/2 0 ¯ k−1 ← A u0k−1 D 2 (2j+1)+1 − A u2k−1 (2j+1) /∆x 2j+3/2 else (flux/diffusion terms are computed by interpolation) k−1 k−1 ¯ k , xk−1 ¯ k−1 ← I L D F¯2j+3/2 ← I L F¯ k , x2j+3/2 , D 2j+3/2 2j+3/2 endif enddo enddo The final multiresolution scheme for calculating the approximate solutions u1,0 , . . . , uN ,0 , where N is the number of time steps, is the following algorithm. Algorithm 2 ˜ 0 [5, Algorithm 2.1] 1. Create the intial set of significant positions D 2. do n = 1, . . . , N (0) j = 0, . . . , N0 uj ← un,0 j , do i = 1, . . . , nRK do k = 0, . . . , i − 1 if βik 6= 0 then (k) (k) using u0 , . . . , uN0 as input data for Algorithm 1, calculate 0 ¯0 −D L¯0 u(k) ← − F¯1/2 j+1/2 /∆x, 0 0 ¯0 ¯0 − D − F¯j−1/2 L¯j u(k) ← − F¯j+1/2 j+1/2 − Dj−1/2 /∆x, j = 1, . .. , N0 − 1, ¯0 L¯N0 u(k) ← F¯N0 0 −1/2 − D N0 −1/2 /∆x endif enddo (i) (i) calculate u0 , . . . , uN0 by (6), with Lj replaced by L¯j enddo (n ) un+1,0 ← uj RK , j = 0, . . . , N0 , compute un+1 j M , ˜ n+1 using [5, Alg. 2.1]; apply data compression to un+1 determine D M enddo 3 Sedimentation-consolidation processes We limit ourselves here to batch settling of a suspension of initial concentration u0 = u0 (x) in a column of height L, where u0 (x) ∈ [0, umax ] and umax is a maximum solids volume fraction. The relevant initial and boundary conditions are (2). The unknown is the solids concentration u as a function of
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time t and depth x. The suspension is characterized by the hindered settling function f (u) and the integrated diffusion coefficient A(u), which models the sediment compressibility. The function f (u) is assumed to satisfy f (u) > 0 for u ∈ (0, umax ) and f (u) = 0 for u ≤ 0 and u ≥ umax . A typical example is f (u) = v∞ u(1 − u)C for u ∈ (0, umax ), C > 0, f (u) = 0 otherwise,
(8)
where v∞ > 0 is the settling R uvelocity of a single particle in unbounded fluid. Moreover, we have A(u) = 0 a(s)ds, where a(u) := f (u)σe′ (u)/(∆̺ gu). Here, ∆̺ > 0 is the solid-fluid density difference, g is the acceleration of gravity, and σe′ (u) is the derivative of the effective solid stress function σe (u). We assume that the solid particles touch each other at a critical concentration 0 ≤ uc ≤ umax , and that σe (u), σe′ (u) = 0 for u ≤ uc and σe (u), σe′ (u) > 0 for u > uc . This implies that a(u) = 0 for u ≤ uc , such that for this application, (1) is indeed strongly degenerate parabolic. A typical function is σe (u) = 0 for u ≤ uc , σe (u) = σ0 [(u/uc)β − 1] for u > uc ,
(9)
with σ0 > 0 and β > 1. In our numerical example, the suspension is characterized by the parameters v∞ = 2.7 × 10−4 m/s, C = 21.5, umax = 0.5, σ0 = 1.2 Pa, uc = 0.07, β = 5, ∆̺ = 1660 kg/m3 and g = 9.81 m/s2 . 4 Numerical results We consider a suspension of concentration u0 ≡ 0.06 in a column of depth L = 0.16 m. Figure 1 shows the numerical solution. The finest and coarsest levels are N0 := 211 and NLc := 23 , respectively. The thresholding strategy is ε1 = 1.9 × 10−7 and εk = 2.99εk−1 for k ≥ 2. We used the parameters CFL = 0.075, ∆t = 0.0491898 h, ∆x = L/N0 and a final time t = 4000 s. The CPU time for this simulation was 503 min (user time) against 1852 min when all fluxes are calculated on G0 without multiresolution. The example involves the formation of a stationary type-change interface (the sediment level). Figure 1 also displays the grid positions of the significant wavelet coefficients of ˜ n , at which the solution. The marked positions are the current elements of D fluxes are evaluated explicitly. At unmarked positions, these terms have been obtained by a simple cubic interpolation. Figure 1 illustrates how the scheme concentrates significant multiresolution coefficients near the downwards propagating shock (Figures 1 (a) and (b)), near the parabolic-hyperbolic type change interface, and near x = 0 and x = L. Figure 2 shows the number of significant wavelet coefficients of the solution in each time step of the simula˜ n . This simulation starts tion and the corresponding compression rate N0 /#D from a very high compression rate, since the initial solution is constant all over the domain, having a discontinuity near the boundary. As time evolves, the solution varies rapidly, and through the multiresolution analysis, this causes a variation of the density of significant positions.
Multiresolution method for sedimentation-consolidation processes (a) t = 160 s
(b) t = 1920 s
8 7 6 5 4 3 2 1
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MRS solution uc
0.14
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0 significant positions 0
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u
(d) t = 2880 s
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(c) t = 2720 s
0.06
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MRS solution uc
0 significant positions 0
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significant positions 0
(e) t = 3200 s
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 u
(f) t = 3680 s
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Fig. 1. Numerical solution to the batch settling problem, including significant positions of the wavelet coefficients of the solution per transformation level.
Acknowledgements RB acknowledges support by Fondecyt project 1050728 and Fondap in Applied Mathematics. AK has been supported by FAPERGS, Brazil, by the ARD project 0306981, and Fondap in Applied Mathematics.
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(b)
512
32 amount of significant positions 256
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compression rate no compression
384
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90000 iteration
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60000
90000
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180000
iteration
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