Constrained Minimization for Monotonic Reconstruction - CiteSeerX

Constrained Minimization for Monotonic Reconstruction  William J. Rider

Douglas B. Kothe

y

z

casting geometries typically require generally unstructured meshes. We wish to design and use methods that do not rely upon operator splitting-based methodology [8], as complex grid topologies such as triangular/tetrehedral [9], tree-based adaptive [10] and arbitrary connectivity polygonal grids [11] pose signi cant problems for schemes using simple extensions of onedimensional methods. Our starting point are multi-dimensional \k-exact" methods devised by Barth [12, 13, 14, 15, 16]. We specifically embrace Barth's approach for deriving a reconstruction based upon least-squares methodology [17]. Barth applies monotonicity after the minimization process, following principles similar to those set forth by Dukowicz [18, 19] and Zalesak in his multi-dimensional FCT algorithm [20]. The problem, however, is that the de nition and application of monotonicity is not part of minimization process, and therefore remains tied to the one-dimensional process. We show that if monotonicity considerations are recast as constraints in the minimization processes, the resulting reconstruction is truly multi-dimensional, i.e., the dierence between a constrained (monotonic) and unconstrained (nonmonotonic) reconstruction can be interpreted as a geometric \limiter" that is in general a vector. Rather than impose scalar monotonic constraints subsequent to the reconstruction, our monotonicity imposition will assume the form of an inequality constraint and can be interpreted as a vector \correction" to the unconstrained reconstruction. We show that basic onedimensional slope limiter ideas can be recast as constraints, in a multi-dimensional reconstruction. We also discuss a powerful weighted least squares approach that incorporates expected numerical error into the interpolation process. Two-dimensional numerical results are given to substantiate the bene ts of the basic methodology underlying our approach. Another aspect of the many one-dimensional methods is the ability to design the level of numerical dissipation into the methods through the choice of the limiter. This freedom enables the method to possess discrete properties best suited for the physical/mathematical structure of the waves being transported (e.g. using superbee on linearly degenerate characteristics). Below we introduce an approach to applying data dependent weights to a least squares/minimization formalism which recovers much of the functionality of the family of classical TVD limiters. Our method allows a fairly wide degree

Abstract We present a new method for monotonic reconstruction that is based upon application of constrained minimization techniques. Classical \TVD" limiters are naturally incorporated into a multi-dimensional monotonic reconstruction by recasting them as datadependent weights and/or constraints in the minimization process. The method is devoid of any assumptions of grid topology, requiring only local data. We discuss the solution techniques demanded by an overdetermined linear system subject to various monotonicity constraints, and present twodimensional numerical results as evidence for the utility of the methodology.

I. Motivation Over the last twenty years, methods for higher order monotone advection have become accepted and are ubiquitous in the CFD literature [1, 2, 3, 4, 5, 6]. When able to reliably suppress non-physical oscillations, monotone methods have supplanted rst-order upwinding schemes. The design and understanding of these methods for data that varies in one dimension, is well developed [1]. Most often multi-dimensional applications of monotone schemes are derived from an operatorsplit application of the basic one-dimensional method. One motivation for our design of a general multidimensional monotonic reconstruction method is our need to model the high Reynolds number incompressible ows found in the mold- lling stage of gravity-pour casting process [7]. The complex three-dimensional  This work performed under the auspices of the U.S. Department of Energy by Los Alamos National Laboratory under Contract W-7405-ENG-36. This paper is declared work of the U.S. Government and is not subject to copyright protection in the United States. y Applied Theoretical and Computational Physics Division, Hydrodynamic Methods Group, Los Alamos National Laboratory, Mail Stop F663, Los Alamos, New Mexico, 87545, USA, E-mail:[email protected] z Theoretical Division, Fluid Dynamics Group (T{3), Los Alamos National Laboratory, Mail Stop B216, Los Alamos, New Mexico, 87545, USA, E-mail:[email protected]

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Kothe and Rider of exibility in tailoring the dissipation inherent in the the vector b is given by multidimensional interpolation process. 0

II. Methodology

B

b=B B@

II.1 Least Squares Reconstruction

( ?  ) .. . ( ?  ) k

i

n

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1 CC CA ;

Given discrete values of a scalar quantity  at grid points i, a polynomial reconstruction R at an arbitrary the weights are speci ed as 0 1 point (x; y) (for two dimensions) near point i is de ned by BB w. CC .. C ; w=B R (x; y) =  + r?  (x ?
x1 ) + r  (y ? y1 )(1) @ A + r 2  x2 ? x 2 +


; w where the constants x are de ned so that and the solution vector x is given by Z 0 1  (x; y) d =  : r 

BB CC We wish to minimize the weighted L2 error in the rer  B C x=B constructed value of  at points j near i (R ) relative BB r2  CCC : to the discrete values  : @ . A X

? R


.. (2) min w  ? 2 The number of columns (n) in A and the length of vecHere, w are weights for the reconstruction at point j tors w, b, and x depends upon the number of terms that arise from geometric or data considerations. As taken in the reconstruction, and the number of rows example of a geometric weight, points j at greater (m) in A depends upon the number of neighbors j condistances from the reference point i will be weighted sidered for each point i. The system is in general overdeless than those closer (e.g., an inverse-distance weight). termined, i.e., m > n. Data weights can arise from monotonic considerations, Caution must be taken in constructing the local linas shown later. Another way to view (2) is as a weighted ear system of equations. The system, for example, description of the error in the interpolation process. For can become quite ill-conditioned with certain choices suciently smooth data (2) becomes of weights. The construction and solution of this sysX tem of equations requires careful attention to any errors min kw (Truncation Error)k2 that might be introduced in the solution process. For example, a solution via the normal equations with stanwhere for a linear reconstruction in two dimensions, dard methods is usually sucient on regular grids, but  in the more general case this approach can be prone to failure. A QR-factorization can circumvent this probTruncation Error = 12
x2r  lem in cases where the system is ill-conditioned. In this  2 work we use subroutines available with LAPACK [21]. + 2
x
yr  +
y r  In addition, if the minimization problem becomes rankde cient, it can be regularized or solved outright with + H.O.T.: a SVD algorithm. We choose to regularize the solution The expected error is roughly proportional the square with a Tikhonov-type method in which a small parameof the mesh spacing for data that is smooth enough. ter,  ( 1:0  10?6 ), is added to the entry (and retains For the reconstruction (2) can be solved for via a least the entry's sign). squares problem given by Consider the example two-dimensional discrete data

min w (Ax ? b) 2 (3) for  given in Figure 1. We wish to nd a linear reconstruction of this data (i.e., retain only the rst derivawhere the matrix A is given by tive terms in the polynomial expansion), using the least 1 squares system given by (3). Two least squares solutions 0 BB (x1 ?. x1 ) (y1 ?. y1 ) (x2 ?. x2 )
.

CC for the gradient of  are shown in Figure 2, a \centered" . . C ; gradient obtained without weighting (i.e., w = 1) and .. .. .. A=B A \distance squared" gradient obtained with inverse dis@ (x1 ? x1 ) (y1 ? y1 ) (x2 ? x2 )


tance squared weights. These two gradients, which are i

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Kothe and Rider

(-1,1,6)

(0,1,8)

whereas monotonicity constraints follow from consideration of nearby discrete data (not their location) as well as reference cell geometry. First consider a scalar monotonicity constraint, in

(1,1,15)

which a constant  is applied to the interpolant

(-1,0,3)

(0,0,5)



 (x; y) =  +  r  (x ? x1 ) + r  (y ? y1 )

(1,0,13)

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+ r2  x2 ? x2 x

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(0,-1,4)

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;i




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+


;

such that the above reconstruction is monotone with respect to the local data. A typical implementation of this algorithm is as follows:

(x,y, )

Figure 1: The discrete data used to demonstrate fea- Post-Reconstruction Scalar Monotone Limiter tures of the reconstruction procedures discussed in this 1. Find the local minimum and maximum of the depaper. pendent data min and max. 5 2. Find the reconstruction minimum and maximumin max Raw Gradients

, min

and  . Centered Gradient min compute  such that min = min, 3. If min



otherwise  = 1.  max    = min 1; max ?

?

2

1

5. Choose  to be the smallest of the available choices from steps 3 and 4.. 0 0 2 4 6 8 10 By relying upon the methodology for solving least y squares problems [17], we can extend the utility of this approach. The formulation of a monotonicity satisfyFigure 2: Gradients computed from various pairs of the ing interpolation with an inequality constraint is quite raw discrete data and the regular and distance squared similar. The least squares problem is modi ed by inweighted least squares solutions. The grey shaded re- equalities that preserve the monotonicity of the recongion indicates that the gradients are monotone. struction. The minimization is recast as



min w (Ax ? b) 2 subject to C x = d:

(4) similar, lie inside of a gradient space formed by points resulting from \raw" ( nite dierence) gradients com- If none of the inequalities are violated then C is null. At most the rank of C is equal to x. For example, if puted from a set of eight nearest-neighbor triangles. there are two unknowns then up to two constraints can be active and their solution will determine the system. II.2 Monotonic Constrained Minimization Were one constraint active then a minimization would When designing monotonic reconstruction methods, the take place. monotonicity constraint should be considered a discrete We nd the active constraint in a process similar to data concept, in contrast to a high-order reconstruction scalar algorithm. The dierence is that more than one method, which is accuracy-driven. Therefor the prin- constraint can be used to determine the overall limiter. ciple to follow is reconstruction follows from considera- The process proceeds as follows: T

Constrained Minimization Limiter

tion of nearby discrete data and their physical location,

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Kothe and Rider can take the view that the data is being ltered to accentuate certain properties in the data. The practical eect will be to either remove or add dissipation from the solution. In this way much of the exibility inherent in one-dimensional high-resolution methods can be applied to multidimensional methods on arbitrary grids.

1. Find the local minimum and maximum of the dependent data min and max. 2. Compute the smallest positive dierence
min = ?
min max min  ?  ;  ?  . 3. Find the reconstruction minimumand maximumin max

, min

and  . 4. Check to see if the reconstruction maxima and minima violate the monotonicity constraint. If min this is an active constraint, and  ? min

>
 min max if  ?  >
 this is an active constraint. 5. If any of the constraints are active, solve the linear system (4). For simple rectangular mesh cells several simpli cations can be made. The constraints that can be active are h  r  + h  r   2
min; h  r  ? h  r   2
min; and ?h  r  + h  r   2
min; where  = sign (r ) and  = sign (r ). For our example data (Figure 1), min = 1 and max = 15 hence
min = 4 and the constraint line is given by r   r  = 8. The equation can be applied to the minimization problem in the following way: suppose we are computing the following interpolant, x

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II.3 Extending Classical TVD Limiters

As a starting point, we should note that we have already de ned multidimensional extensions of a monotone Fromm's (or centered) method1. Two other important limiters are the minmod2 and superbee3 limiter. By making the weights data dependent, these limiters can easily be implemented. Each of these multidimensional extensions reduces to the corresponding onedimensional limiter if the ow becomes one-dimensional (mesh aligned on an orthogonal grid). The minmod limiter in one dimension chooses the minimum of the available slopes, and the superbee chooses the maximumwith the condition that the choice be monotonicity preserving. One way to generalize this is to make the choice based on the average slope. Others [22, 23] have done the same, but with the interpretation of superbee as the largest of the available slopes. These choices are implemented by setting the weights to zero if the data does not meet the selection criteria. Thus the minmod limiter generalizes by choosing the data that is smaller than (or equal to) the average and the superbee by choosing data that is larger (or equal to). For a two-dimensional linear reconstruction algorithm would be organized as follows:

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Multidimensional Minmod or Superbee Limiters

 + r  (x ? x1 ) + r  (y ? y1 ) ; i

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;i

1. Solve Equation (3) with (geometrically weighted) centered data. 2. For each cell in the stencil compute   = (x1 ? x1 ) r  + (y1 ? y1 ) r . 3. For the minmod limiter if j
j > j j, then w = 0, otherwise it is unchanged. 4. For the superbee limiter if j
j < j j, then w = 0, otherwise it is unchanged. 5. Solve the reweighted least square problem. 6. Impose the monotonicity constraints on the resulting solution. For the example problem, we show the solution for the minmod and superbee limiters applied in multidimensional least squares fashion. In this case the

and a constraint h r  + h r  =
lim: x

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We have two options in applying this constraint to the interpolation problem: the method of weighting and algebraic elimination. The method of weighting takes some constant  and add the constraint equation to the matrix A as a new row and an entry in b as

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 h r  + h r  ?
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For algebraic elimination one solves for one of the unknowns in terms of the other(s) and then a reduced system is minimized. In this example we could take lim r  =
 ?h h r  : 1 In the standard description of TVD limiter, () =   The result of this process is a vector scaling of the in- minmod 12 (1 + ) 2 2 , where minmod returns the minimum terpolant such that the constraint is satis ed and the of its arguments provided they are all the same sign, otherwise error is minimized. zero is returned. 2 ( ) = minmod(1 ). Within the constrained least squared minimization 3 ( ) = minmod[max(1 ) 2 2 ] framework several variations can be constructed. One y

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Kothe and Rider 5

5 Minmod

Scalar Monotonicity

Superbee

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Figure 3: Computed multidimensional minmod and su- Figure 4: The standard scalar and constrained limiter applied to the non-monotone superbee gradients. Also, perbee least squares gradients for the example data. shown are the two constraints that are relevant to the reconstruction. Only r  + r  = 8 is active. minmod limiter results in a gradient that is smaller in magnitude than any of the raw gradients. The 5 minmod gradient uses the following four data points: Raw Gradients (?1; ?1) ; (?1; 0) ; (?1; 1) ; (0; ?1). It should be noted Superbee that the raw gradients shown do not span all the pos4 Gradient Weighted sible gradients arising from reconstructing this discrete data. The superbee limiter provides a gradient that is near the size of the largest raw gradients. The su3 perbee gradient uses the following four data points: (0; 1); (1; ?1); (1; 0); (1; 1). Both of these gradients are x shown along with the centered reconstruction in Fig2 ure 3. In Figure 4 we show the result of applying the monotonicity constraints to the superbee reconstruction. One 1 inequality constraint is active: r +r  = 8, although a second constraint is shown, the superbee gradient does not violate it. As can be observed the scalar limiter simply uniformly scales the gradient onto the constraint line 0 and diers from the scalar monotonicity solution. 0 2 4 6 8 10 y Two other important limiters are the harmonic mean limiter [24]4 and van Albada's limiter [25]5 . By choosing the weights to be inversely proportional to the absolute Figure 5: Computed gradient weighted and superbee value of size of the data, the harmonic mean limiter is least squares gradients for the given data. generalized. This simply requires the weight vector be scaled by w := w = j
j. If this choice is the inverse of the size of the data squared, the van Albada limiter acteristics of the interpolant to a given application or is generated. Again, this requires the weight vector be situation. For our example data, the gradient weighted scaled, in this case by w := w = (
)2. These limiters and superbee gradients are shown in Figure 5. This may be useful in ne-tuning the resolution and char- shows that the gradient weighting is nearly as steep as superbee. For the inverse gradient weighting the proper 4 ( ) = ( + ) (1 + ). comparison is with the minmod limiter. This is shown ? + 2
?1 + 2
. 5 ( )= in Figure 6. x

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Kothe and Rider on their reliability. The above generalization of limiters can be viewed in this context. In [30] the least squares formalism is discussed with relation to the expected error. There a linear interpolant is determined for some data using the model,

5 Raw Gradients Minmod Example Data Inverse Gradient Weighted

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y = a + bx + 
y;

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where the last term, 
y, is the expected error. Since we are essentially approximating a Taylor series expansion, the general form of the expected error is known (where the function is smooth). As noted earlier, for a linear expansion the error terms scale with the square of the distance, thus the geometric portion of the weight should be inversely proportional to the distance squared. This relation will change as the order of the interpolant increases. As a result, the norm of the residual is a reasonable estimate of the truncation error of the reconstruction.

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III. Results

Figure 6: Computed inverse gradient weighted and minmod least squares gradients for the given data.

We will present the results of the above described methods on two test problems: a smooth double sine wave, and a circular region both on a periodic grid. The sine wave test will show the order of accuracy of the method and the circular region will show the behavior of the method at discontinuities and the distortion of a smooth body. Table 1 shows the error for the sine wave test on a series of grids 16  16 through 64  64. The ne grid error and the order of convergence are shown. We can draw several conclusions from these results: the multidimensional reconstructions are superior and constraint based limiters improve the solution although the improvement with the new limiters is not profound. We show the methods described above in Figures 7 through 9. The solutions are computed with a monotone unsplit dierencing method similar to the one in [31]. Each is computed at a CFL number of one-half on a 50  50 grid. We believe the close-ups of the details of the solution provide critical details to judge the delity of the calculations In general the purely multidimensional methods provide a signi cantly less grid-dependent solution. The general features of the one-dimensional limiters have been replicated in a consistent two-dimensional manner. The superbee limiter shows small 3-4 cell transitions and the Huynh compressive limiter is even sharper. While the constraint-based limiting is somewhat superior to scalar limiting, its superbee implementation shows some evidence of increased grid dependence. This may be due to a decrease in dissipation, and the interface thickness while consistent in thickness in both methods is sharper with the constraint-based limiting.

Yet another small modi cation of the weighting used to extend the harmonic mean and van Albada limiters can be used to implement a L1 minimization rather than the L2 minimization used for the least squares formalism. This is accomplished through using an iteratively reweighted least squares calculation where the weights are inversely proportional to the residual [26].. Thus, the van Leer limited scheme can be used to compute the rst guess, then the inverse of the residual can be used to weight the data and the solution is found again. Another important concept is slope steepening (related strongly to arti cial compression [27, 28]). Onedimensional piecewise linear schemes have been expressed in a complete manner by Huynh [29] who introduced an interesting slope steepener. We rst discuss this in a simpli ed setting from that given by Huynh, but then move the concept to a purely multidimensional implementation. Huynh de nes the scheme using a constant  to determine the compression. The left and right slopes are dierenced and multiplied by  then made monotone. In the standard form     Q (r) = minmod max 21 (1 + r) ;  j1 ? rj ; 2; 2r : In? multidimensions
we replace the term,  j1 ? rj, by  max ? 2 + min . The other terms are also substituted as the implementation of the van Leer scheme in multiple dimensions suggests. We now discuss weighting in a broader sense. The use of least squares methods in computing the functional dependence of data is well known. It seems sensible to apply more or less weight to data points depending 6

Kothe and Rider

Table 1: Order of accuracy for various reconstruction methods for a double sine wave advection. Method Fine Grid Error L1 Fine Grid Error L1 Order L1 Order L1 1-D Fromm 1:03  10?3 1:19  10?2 1.97 1.42 2-D Fromm 4:28  10?4 1:13  10?2 2.68 1.53 ? 4 ? 2 2-D Constrained Fromm 4:02  10 1:13  10 2.72 1.51 2-D Minmod 6:30  10?3 3:81  10?2 1.68 1.14 ? 3 ? 2 2-D Harmonic 7:91  10 3:64  10 1.60 1.25 2-D L1 7:24  10?4 1:72  10?2 2.36 1.38

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Figure 7: Advection of a circle by a unsplit advection scheme using various Fromm-van Leer methods.

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Figure 8: Advection of a circle by a unsplit advection scheme using various Superbee methods.

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Figure 9: Advection of a circle by a unsplit advection scheme using various multidimensional slope reconstruction methods. 8

Kothe and Rider

IV. Conclusions

[12] T. J. Barth and D. C. Jesperson. The design and application of upwind schemes on unstructured meshes. In 27th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 1989. AIAA{89{0366. [13] T. J. Barth and P. O. Frederickson. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. In 28th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 1990. AIAA{90{0013. [14] T. J. Barth. Aspects of unstructured grids and nite-volume solvers for euler and navier-stokes equations, 1995. VKI/NASA/AGARD Special Course on Unstructured Grid Methods for Advection Dominated Flows AGARD Publication R-787. [15] T. J. Barth. Recent developments in high order K-exact reconstruction on unstructured meshes. In 31st Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 1993. AIAA{93{0668. [16] T. J. Barth. Parallel CFD algorithms on unstructured meshes, 1995. VKI/NASA/AGARD Lecture Series on Parallel Computing. AGARD Publication R-807, available via T. J. Barth's WWW Home Page http://oldwww.nas.nasa.gov/ ~barth/home.html1. [17]  A. Bjork. Numerical Methods for Least Squares Problems. SIAM, 1996. [18] J. K. Dukowicz. New mathods for conservative rezoning (remapping) for general quadrilateral meshes in rezoning workshop 1983. Technical Report LA{10112{C, Los Alamos National Laboratory, 1984. [19] J. K. Dukowicz and J. W. Kodis. Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations. SIAM Journal on Scienti c and Statistical Computing, 8:305{321, 1987. [20] S. T. Zalesak. Fully multidimensional uxcorrected transport algorithms for uids. Journal of Computational Physics, 31:335{362, 1979. [21] E. Andserson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK Users' Guide. SIAM, 1992. [22] L. J. Durlofsky, B. Engquist, and S. Osher. Triangle based adaptive stencils for the solution of hyperbolic conservation laws. Journal of Computational Physics, 98:64{73, 1992. [23] P. Batten, C. Lambert, and D. M. Causon. Positively conservative high-resolution convection

We have presented several extensions of existing methods for reconstructing functions for the purpose of constructing a Godunov algorithm. We have demonstrated that these methods are genuinely multidimensional and naturally extend to arbitrary grids. Furthermore, the procedures improve the accuracy and quality of solutions. These methods are also more exible than existing multidimensional methods. This is fertile ground for further investigation.

References

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