Computational Geosciences 2 (1998) 1, 23-36

Transcription

1 A STUDY OF THE MODELLING ERROR IN TWO OPERATOR SPLITTING ALGORITHMS FOR POROUS MEDIA FLOW K. BRUSDAL, H. K. DAHLE, K. HVISTENDAHL KARLSEN, T. MANNSETH Comptational Geosciences 2 (998), Abstract. Operator splitting methods are often sed to solve convection-dision problems of convection dominated natre. However, it is well known that sch methods can prodce signicant (splitting) errors in regions containing self sharpening fronts. To amend this shortcoming, corrected operator splitting methods have been developed. These approaches se the wave strctre from the convection step to identify the splitting error. This error is then compensated for in the dision step. The main prpose of the present work is to illstrate the importance of the correction step in the context of an inverse problem. The inverse problem will consist of estimating the fractional ow fnction in a one-dimensional satration eqation.. Introdction We are here interested in the initial-bondary vale problem associated with nonlinear convection-dision eqations of f() ; () = (x t) 2 (a b) where = (x t) denotes the nknown, f = f() is the x fnction, d = d() is the dision fnction and " > is a scaling parameter. Convection-dision eqations arise in a variety of applications. We here consider two-phase, immiscible and incompressible ow of oil and water in poros media. In this context, is the water satration, f() is the fractional ow fnction and d() is the capillary dision fnction. De to the nonlinear natre of the dierential eqation (), there will in general be no analytical expression for (x t). Hence we mst rely on nmerically techniqes to obtain the soltion. When () is convection dominated, that is, when " is small compared with Key words and phrases. Operator splitting, two-phase ow, satration eqation, modelling error, inverse problem. Brsdal and Mannseth have been spported by the Research Concil of Norwaynder grant 765/42. Karlsen has been spported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).

2 2 BRUSDAL ET AL. other scales in (), conventional methods sally exhibit some combination of diclties, ranging from non-physical oscillations to severe nmerical dision at the trailing end of moving fronts. To overcome sch diclties we make se of operator splitting. In the present context, operator splitting means to split the convection-dision eqation () into ahyperbolic eqation and a parabolic eqation, each of which is solved separately in an alternate fashion. Frthermore, each step of sch an algorithm is flly discretized by accrate and ecient nmerical schemes developed especially for hyperbolic conservation laws and parabolic heat type eqations, respectively. The operator splitting approach has been taken by many athors, [2, 4, 5, 6,7,8, 9,, 2, 5, 6,7] (see the references cited therein for a more complete list of relevant papers). In the case where d(), the physical front widths shold be of order O("). However, it is easy to see that disive splitting errors will lead to front widths of size O( p "t) when the splitting time step, t, is large ( "). This type of splitting error comes from thefactthattheentropy condition forces the hyperbolic solver to throw away certain \information" that controls the strctre of the self-sharpening fronts. It is however possible to compensate for this loss of information, which manifests itself in the form of residal x terms. For example, assme that the soltion of () is a moving front. If the time step is large enogh, the convection step will generate a shock with left and right limits, say, l and r, respectively. We can then identify a residal x term associated with this discontinity f res () = f() ; f c (), where f c () denotes the correct envelope (dictated by the entropy condition) of f() on the interval bonded by l and r. There are several ways to take the residal x term f res () into accont. We can perform a separate correction step after the dision step. Correction is then calclated by solving the \residal" conservation law v t + f res (v) x = over a time interval ( ], where > is some parameter that has to be chosen (see [6] for details). Another approach is to inclde the residal term in the eqation modelling dision, that is, instead of solving the eqation w t ; "(d(w)w x ) x =,we solve w t +(f res (w) ; "d(w)w x ) x =. This eqation contains the necessary information needed to prodce the correct strctre of the front. We will here rely on the latter approach, which we shall refer to as corrected operator splitting (COS). Splitting methods which do not accont for residal x terms will be denoted by OS. The idea of sing a residal x term in the dision step was introdced in [] (and frther developed in [4, 6, 7, 8]) in the context of two-phase ow with an established front as initial data. In this setting the residal term can be determined a priori, which

3 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING 3 means that the convection step is to solve v t + f c (v) x = (where f c is linear in the front region). This fact was exploited in [4, 6, 7, 8, ] in the sense that the modied method of characteristics was sed for the convection step. For general initial data this approach may pt severe restrictions on the time step. However, in [5, 6] itwas observed that by sing front tracking [4] to solve the nonlinear conservation law v t + f(v) x = it is possible to dynamically constrct residal x terms. Conseqently, the COS approach makes sense in general. The calclation of these residal terms is a direct conseqence of the fact that front tracking is based on solving Riemann problems. Instead of front tracking, a second order Godnov method will here be sed in the convection step to illstrate that other techniqes can be tilised in the constrction of residal x terms, see also [5]. In what follows, dierences in the Godnov based OS and COS methods will be illstrated in the context of an inverse problem. The inverse problem will consist of recovering the x fnction sing COS and OS as forward models in the inversion procedre. Note that in general it is not possible to dene the residal x term a priori in the inversion procedre since the representation of the x fnction is changing throghot the procedre. This motivates the se of COS where the residal x terms are constrcted dynamically. The main prpose of the present work is to stdy the inence that the modelling errors in the OS and COS forwards models have on the recovery of the x fnction. The paper is organised as follows. In section 2 we give a description of the forward problem and the operator splitting techniqes sed to solve it. In section 3we discss the setp of the inverse problem. Frthermore, we present and discss nmerical experiments. Finally, in section 4 we make some conclding remarks. 2. The forward problem 2.. Nmerical algorithms. In this section we describe in some detail the operator splitting methods that are sed to solve the forward eqation (). This eqation mst be eqipped with appropriate initial and bondary conditions conditions t= = (x) x=a = a 2 R x=b = b 2 R where we assme that is a piecewise smooth fnction. Let n (x) (x t n ) denote the piecewise linear nmerical soltion of () at time level t n, where =t <t < t N = T is a time discretization of [ T]. Here the fnction n = n (x) is piecewise linear with

4 4 BRUSDAL ET AL. respect to a niform grid with grid cell size x >. We determine n+ from n via the following OS algorithm. Convection: Let v(x t n ), t n = t n+ ; t n >, denote the (entropy weak) soltion of the hyperbolic conservation f(v) v(x ) = n (x): The second order Godnov (slope limiter) method described in [3] is sed to calclate this soltion. The Godnov method ses a niform (local) grid with grid cell size x c and a time step < t c t n chosen according to the CFL condition, i.e., t c is chosen so that max jf j t c x c =. Dision: Let w(x t n )bethe soltion of the parabolic heat type @x = w(x ) = v(x t n ) with the Dirichlet bondary data imposed at x = a b. Nmerically this soltion is fond sing the Galerkin method on a niform grid with grid cell x d x and the sal (piecewise linear) \hat" basis fnctions. The time discretization is done by the backward Eler method with a single time step of length t n. Finally, we set n+ = w(x t n ): In the constant dision case, d(), convergence of the OS algorithm is proved in [7]. Convergence reslts in the case where d() is nonlinear and possibly strongly degenerate are obtained in []. Althogh this algorithm converges as varios discretization parameters tend to zero, it is not diclt to constrct an explicit example (see [6]) which shows that OS is too disive in regions containing self-sharpening fronts, at least when the time step is large ( "). This potential shortcoming has motivated the corrected operator splitting (COS) algorithm, which diers from OS in the way that the dision step is carried ot. Dision revisited (residal x terms): To obtain correct strctre of fronts, a residal x term has to be constrcted. To thisend, we introdce the envelope fnction 8 < the lower convex envelope of f between v l and v r if v l <v r, (4) f c (v v l v r )= : the pper concave envelope of f between v r and v l if v l >v r. Let v = v(x t n ) be the tre soltion of (2) at time t = t n, which is known to be piecewise smooth. Let fy i g denote the discontinity points of v. For a xed i, let ~v i and

5 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING Figre. The left plot shows an exact soltion (solid line) from the convection step, which contains for discontinities (left and right shock states are shown by circles). The thick solid lines indicate where the corresponding residal x terms are dened when sed in the dision step. The right plots show the residal x terms (solid line) associated with the for discontinities (the pper left plot corresponds to the rst discontinity, the pper right plot to the second and so on). The x fnction is shown as the thick dashed line and the envelope fnction as dotted. ~v i+ denote the left and right limits of v at x = y i, respectively. Choose positions x i and x i+ sch that y i is located somewhere in the interval (x i x i+ ). Then dene the residal x fnction associated with this (ith) discontinity by (see Figre ) 8 < f(v) ; fres(v i f c (v ~v i ~v i+ ) for v 2 [~v i ~v i+ ] t n )= : for v 62 [~v i ~v i+ ]: For convenience, introdce the globally dened residal x term f res (x v t n )=f i res(v t n ), x 2 [x i x i+ ) for some i. Let now w(x t n ) be the weak soltion of the following f res (x w t n ) ; = w(x ) = v(x t

6 6 BRUSDAL ET AL. with the Dirichlet bondary data imposed at x = a b. Nmerically this soltion is fond by a Petrov-Galerkin method with qadratic test fnctions, see e.g. [4, 6, 7,, 5] for details. To nd a residal x term, we need to locate the associated discontinities (shocks) based on the approximate soltion generated by the Godnov method. This is done by introdcing a sitable parameter > and identify a shock whenever the distance between two sbseqent satration vales in the Godnov soltion is greater than, see [5] for details. Remark. The loss of symmetry de to the residal x term does not aect the overrall comptational time for the COS method since the tridiagonal linear systems are solved by direct methods. This claim is conrmed in [5] where the nmerical experiments show that COS' rntime (with one or a few time steps) is less than that of OS (with many time steps). In higher dimensions this loss of symmetry may become an isse since iterative methods have to be sed to solve the reslting linear systems. However, since the eqations are (partially) symmetrized by the se of a Petrov-Galerkin method this isse seems to be of mch less consern than the restriction on the time step imposed for the OS method. Remark. A sitable vale of is problem dependent and the parameter is adjsted by considering the nmber of grid blocks sed in the nmerical comptations. Therefore, changing cold reslt in a less accrate constrction of the residal x fnction. However, since the nmber of grid blocks is the same in each nmerical experiment, the parameter is kept xed Nmerical experiments. For two phase ow in poros media [3] the advective x fnction in () is given by (5) f() = w () w ()+ o () where i = k ri i, is the mobility of phase i, i = w o. Here, k ri, denotes the relative permeability and i the viscosity. The relative permeabilities are represented by third order normalised B-spline expansions [9] in which the coecient vectors are given by ~c w = (: : :52 :) and ~c o = (: :25 :2 :). The knot vector for the B-spline basis is ~y = (: : : :5 : : :). Frthermore, we set d(), = :5 and " =:.

7 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING 7 The satration proles are simlated sing axfnction, f (), which corresponds to the B-spline expansion given above (see Figre 2a)). The initial satration (see Figre 2b)) is 8 >< : ; x : x :3 (x) = :7 ; 7 (x ; :3) :3 x :3 >: : :3 <x:: The Dirichlet bondary data are = at x = and = at x =. In the nmerical experiments the size of the grid blocks is given by x = and x c =. The grids are 2 xed since we here want to stdy the eect of varying the time step. Only one shock frontis present in the soltion, therefore we let the corresponding residal x term be dened on the whole interval [ ]. In the COS simlation we have sed time step to reach T =:2 since with the COS method the correct width of the front is obtained independently of the time step. In the OS simlations the nmber of time steps is and. We refer to these simlations as COS(), OS() and OS(), respectively. In Figre 3 the COS and OS soltions are shown together with the reference soltion, which is generated by a nite dierence scheme with very ne space and time discretization (t = T 48 To measre the relative errors in the (C)OS satration proles we dene (6) r (C)OS = jj~ ref ; ~ (C)OS jj 2 jj~ ref jj 2 : and x = 5 ). We have r COS() = :6, r OS() = :5 and r OS() = :36. Ths the relative errors measred in the L 2 -norm are abot eqally large for the COS() and OS() soltions. From Figre 3 and the relative errors it is seen that COS is the most ecient and accrate algorithm for the problem nder consideration. We refer to [5, 6] for a more detailed comparison of COS and OS for the forward problem. One may ask whether the relative merits of OS and COS will be similar also when the models are applied to solve an associated inverse problem. The importance of neglecting some specic physics may be dierent for the inverse problem than for the forward problem. 3. The inverse problem 3.. Problem formlation. We will attempt to recover the x fnction, f(), from discrete vales of, which are denoted by ~ obs. The forward model is given by eqation () (recall that d() ) with an appropriate soltion algorithm for (x t). The OS()

8 8 BRUSDAL ET AL..9 a).9 b) f().5 (x,) x Figre 2. The x fnction, f (), in plot a) and the initial satration, (x), in plot b) a) Reference (x) COS() OS() b) Reference (x) OS() x x Figre 3. The reference, COS() and OS() soltions of the forward problem in plot a). The reference soltion and OS() soltion in plot b). and COS() soltion algorithms are selected since these gave similar accracy when applied to the forward problem. The discrete observed vales will be sampled from the ne-grid soltion of the forward problem at time T = :2. The x fnction sed when calclating the ne-grid soltion is f () (seesection 2.2). Objective fnctions, which measre the least sqares distance between observed and corresponding calclated vales of ~, are formed J (C)OS (f) =k~ obs ;~ (C)OS (f)k 2 2. Recovery

9 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING 9 of f () is carried ot throgh minimisation of the objective fnctions. Representing the x fnction by a B-spline expansion leads to a nite dimensional parameter estimation problem for the coecients in the expansion. This is solved by a Levenberg-Marqardt algorithm [8] Soltion criteria. Dierent parametrisations of the x fnction lead to dierent parameter estimation problems with soltions corresponding to dierent fnctional spaces for the estimated x fnction. Hence, having solved the parameter estimation problem does not necessarily mean that one has solved the nderlying inverse problem. Frthermore, inverse problems are known to be inherently ill-posed. Soltion non-existence, soltion non-niqeness and instability of the soltion with respect to data may occr. This shows the need for soltion criteria to decide whether a soltion to the parameter estimation problem is also a soltion to the original inverse problem. Often, inverse problems where the modelling errors are negligible compared to the error in the observed data are considered. When the noise in the observed data is a reslt of additive independent random errors (e.g., reslting from independent measrements), it is possible to apply statistically based soltion criteria [, 2] to test if the soltion to a parameter estimation problem can be accepted as a soltion to the nderlying inverse problem. These criteria concern the vale of the objective fnction and the randomness of the elements in the residal vector at soltion. Frthermore, if a soltion to this kind of inverse problem has been fond, it is possible to give approximate error bonds for the soltion. The inverse problem stdied here is qite dierent. We consider a problem where the modelling error dominates the error in the observed data. In fact, the whole prpose is to stdy the eect of two dierent modelling errors on the recovery of the x fnction. Contrary to measrement errors, modelling errors can not be expected to be additive independent random errors. Hence, neither the above soltion criteria nor the error bonds can be applied. Presently, we do not know enogh abot the natre of the modelling error to pt forth any alternative soltion criteria covering this case. Therefore, the estimated x fnctions presented later will be soltions to specic parameter estimation problems, while we can not claim that any inverse problem has actally been solved. However, it is still of interest to stdy the inence of the modelling errors in OS and COS when attempting to recover the x fnction.

10 BRUSDAL ET AL. We will se the same parametrisations of the x fnction when rnning the OS() model as when rnning the COS() model, thereby ensring that corresponding OS and COS estimates belong to the same fnctional space Nmerical experiments. The observed data are generated at T = :2 by a nite dierence scheme sing the x fnction, f (), and the ne grid given in section 2.2. From the reslting satration prole eqally spaced data are sampled. This constittes the observation data sed in the parameter estimation. The size of the grid cells sed in OS() and COS() is xed x =. For a given order, m, of the B-spline basis fnctions and a given nmber of internal knots, k, in the knot vector, the nmber of coecients in each relative permeability crve is m + k. The nmber of elements in the knot vector is 2m + k. Here the order, m, is xed m = 3. The rst and last coecients in the B-spline expansions of each relative permeability crve are also xed, c w = c m+k o =: and c m+k w = c o =:, so that the total nmber of parameters to be estimated is 2(m + k) ; 4=2(k +). In the nmerical experiments, 2, 3 and 4, the exibility of the estimated fractional ow fnction is systematically increased by adding knots in the knot vector (increasing k). The nmber of knots is k =,k =2,k = 3 and k = 7, respectively. Increased exibility in the x fnction allows the x fnction to compensate for modelling errors in the OS and COS models. Hence, we expect the minimm vale of the objective fnction, J (C)OS (f), to decrease with increased exibility in the x fnction while the relative errors in the estimated x fnctions may increase. The COS and OS estimated x fnctions and the corresponding minimm vales of J (C)OS (f) willbe compared at each level of exibility. The knot vector, initial gess on the nknown parameters, and the estimated parameters from the experiments can be fond in Table, 2, 3 and 4, respectively. The minimm vales of the objective fnction, J (C)OS (f), and the relative errors in the estimated x fnctions are given in Table 5. In Figre 4b) the observed satrations and (C)OS simlated satrations corresponding to the minimm vale of J (C)OS are given for experiment. We omit plotting the satration proles for experiment 2, 3 and 4, since they are visally identical to the plot in Figre 4b). In Figre 5 and 6 the tre x fnction, f (), and the estimated x fnctions, f (C)OS (), are plotted.

11 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING.9 a).9 b).8.7 Flx fnction, f * () Initial f().8.7 Observed (x) COS() OS().6.6 f() x Figre 4. The tre x fnction, f (), and the initial gess on the x fnction in plot a). In plot b) the observed satration, obs, and simlated satrations, (C)OS,inexperiment..9 a).9 b) f().5 f() Tre flx fnction, f * () COS() OS().3.2 Tre flx fnction, f * () COS() OS() Figre 5. The tre x fnction, f (), and estimated x fnctions, f (C)OS (), in experiment (plot a)) and experiment 2 (plot b)). The minimm vales of the objective fnction, J (C)OS (f), are decreasing with increased exibility of the fractional ow fnction, whereas in most cases the relative error of the x fnction is increasing. We also have J COS() < J OS() and that the relative error in the x fnction is greater with OS() than COS() at each level of exibility. We interpret these reslts as follows: In the convection step the speed of a point ^ on a satration crve is given by f (^). The OS() soltion contains nmerical dision

12 2 BRUSDAL ET AL..9 a).9 b) f().5 f() Tre flx fnction, f * () COS() OS().2 Tre flx fnction, f * () COS() OS() Figre 6. The tre x fnction, f (), and estimated x fnctions, f (C)OS (), in experiment 3 (plot a)) and experiment 4 (plot b)). Table Experiment no. Knot vector ~y =(: : : :5 : : :) Initial parameters ~c w =(: :3 :82 :) ~c o =(: :45 :2 :) Estimated parameters ~c w =(: :98 :399 :) COS() ~c o =(: :66 :27 :) Estimated parameters ~c w =(: : :725 :) OS() ~c o =(: :326 :9 :) sch that low-satration points on the front travel too fast. To compensate for this the inversion algorithm seeks to decrease the speed of these points by deforming the estimated x fnction in the lower satration region, cf. Figre 5 and 6. The ability to do so, withot aecting the higher satration regions too mch, increases with increased exibility in the fnctional representation of the x fnction. Observe that not even the deformed x fnction in Figre 6b) did bring the minimm vale of J OS() below that of J COS() in experiment. Ths, the modelling errors in OS() are clearly more serios than those of COS(), as far as the crrent inverse problem is concerned. In an attempt to remove nmerical dision in the OS simlations, the nmber of time steps is increased to 2. With 2 time steps the width of the front in the OS soltion is of

13 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING 3 Table 2 Experiment no. 2 Knot vector ~y =(: : : :3 :5 : : :) Initial parameters ~c w =(: :8 :456 :82 :) ~c o =(: :67 :32 :2 :) Estimated parameters ~c w =(: : :35 :79 :) COS() ~c o =(: :562 :258 :92 :) Estimated parameters ~c w =(: : :29 :67 :) OS() ~c o =(: : :22 :2 :) Table 3 Experiment no. 3 Knot vector ~y =(: : : :3 :5 :7 : : :) Initial parameters ~c w =(: :8 :456 :664 :892 :) ~c o =(: :67 :32 :49 :2 :) Estimated parameters ~c w =(: : :25 :35 :52 :) COS() ~c o =(: :338 :52 :63 :4 :) Estimated parameters ~c w =(: : :2 :255 :587 :) OS() ~c o =(: : :66 :64 :4 :) Table 4 Experiment no. 4 Knot vector ~y =(: : : :5 : :2 :3 :5 :6 :7 : : :) Initial parameters ~c w =(: :3 :89 :77 :29 :456 :62 :7 :892 :) ~c o =(: :945 :842 :697 :53 :32 :92 :22 :2 :) Estimated parameters ~c w =(: : : :34 :89 :89 :269 :39 :477 :) COS() ~c o =(: : : :7 :4 :28 :8 :47 :39 :) Estimated parameters ~c w =(: : : : : : :296 :352 :587 :) OS() ~c o =(: : : : :4 :4 :94 :58 :34 :) the order of the physical front, ". We repeat the nmerical experiments, 2, 3 and 4 sing OS(2). The minimm vales of the objective fnction, J OS (f), and the relative errors of

14 4 BRUSDAL ET AL. Table 5 Experimental reslts Objective fnctions, J (C)OS (f) Experiment no COS() :96 :34 :27.23 OS() :25 :245 : OS(2) :2 :9 :82.8 Relative error in f (C)OS () Experiment no COS() :889 :252 : OS() :449 :728 : OS(2).365 :3 : x 3.9 a) b).7 2 f() Objective fnction.5 COS() OS() OS(2).2 Tre flx fnction, f * () OS() OS(2) Relative error Figre 7. The tre x fnction, f (), and OS() and OS(2) estimated x fnctions in plot a). The minimm vales of the objective fnction, J (C)OS (f), verss the relative errors in the estimated x fnction, f (C)OS (), in plot b). the estimated x fnctions have decreased compared with the OS() reslts (see Table 5 and Figre 7b)). Figre 7a) shows the tre x fnction, f (), and the OS() and OS(2) estimated x fnctions in experiment 3. The OS(2) estimated x fnction is considerably better than the one achieved with OS(). This spports or view that the deformation in the OS() estimated x fnctions is de to nmerical dision.

15 OPERATOR SPLITTING AND CORRECTED OPERATOR SPLITTING 5 4. Conclsion Wehave compared COS and OS models with respect to accracy in the context of solving the one-dimensional satration eqation of poros media ow. In line with previos work, the nmerical experiments indicate that the COS method is mch more accrate than the OS method. For the problem considered here the OS approach needed time steps to achieve the same accracy in the L 2 -norm as COS did with timestep. We have also stdied the COS and OS models in the context of an inverse problem where we recover the x fnction in the one-dimensional satration eqation from discrete satration data. The nmerical experiments indicate that even thogh the COS() and OS() models solve the forward problem with similar accracy, and ths in that sense contain modelling errors of eqal size, the modelling error in OS() is the most serios one for this inverse problem. References [] Y. Bard. Nonlinear Parameter Estimation. John Wiley and Sons, New York City, 98. [2] J. T. Beale and A. Majda. Rates of convergence for viscos splitting of the Navier-Stokes eqations. Math. Comp., 37(56):243{259, 98. [3] G. Chavent and J. Jare. Mathematical models and nite elements for reservoir simlation, volme 7 of Stdies in mathematics and it's applications. North Holland, Amsterdam, 986. [4] H. K. Dahle. Adaptive characteristic operator splitting techniqes for convection-dominated dision problems in one and two space dimensions. PhD thesis, Department of Mathematics, University of Bergen, 988. [5] H. K. Dahle. ELLAM-based operator splitting for nonlinear advection dision eqations. Technical Report 98, Department of Mathematics, University of Bergen, 995. [6] H. K. Dahle, M. S. Espedal, R. E. Ewing, and O. Svareid. Characteristic adaptive sbdomain methods for reservoir ow problems. Nmerical Methods for PDEs, 6:279{39, 99. [7] H. K. Dahle, M. S. Espedal, and O. Svareid. Characteristic, local grid renement techniqes for reservoir ow problems. International J. for Nmerical Methods in Engineering, 34:5{69, 992. [8] H. K. Dahle, R. E. Ewing, and T. F. Rssell. Elerian-Lagrangian localized adjoint methods for a nonlinear advection-dision eqation. Compt. Methods. Appl. Mech. Engrg., 22:223{25, 995. [9] C. N. Dawson. Godnov-mixed methods for advective ow problems in one space dimension. SIAM J. Nm. Anal., 28(5):282{39, Oct. 99. [] M. S. Espedal and R. E. Ewing. Characteristic Petrov-Galerkin sbdomain methods for two-phase immiscible ow. Compt. Methods. Appl. Mech. Engrg., 64:3{35, 987.

16 6 BRUSDAL ET AL. [] S. Evje and K. H. Karlsen. Viscos splitting approximation of mixed hyperbolic-parabolic convection-dision eqations. Report No. 2, 997/98. Institt Mittag-Leer. Available at the URL [2] R. E. Ewing. Operator splitting and Elerian-Lagrangian localized adjoint methods for mltiphase ow. In Whiteman, editor, The Mathematics of Finite Elements and Applications VII MAFELAP, pages 25{232. Academic press, San Diego, CA, 99. [3] J. B. Goodman and R. J. LeVeqe. A geometric approach to high resoltion tvd schemes. SIAM J. Nm. Anal., 25:268{284, 988. [4] H. Holden, L. Holden, and R. Hegh-Krohn. A nmerical method for rst order nonlinear scalar conservation laws in one-dimension. Compt. Math. Applic., 5(6{8):595{62, 988. [5] K. H. Karlsen, K. Brsdal, H. K. Dahle, S. Evje, and K.-A. Lie. The corrected operator splitting approach applied to a nonlinear advection-dision problem. To appear in Compt. Methods. Appl. Mech. Engrg. [6] K. H. Karlsen and N. H. Risebro. Corrected operator splitting for nonlinear parabolic eqations. Preprint, University of Bergen, Norway, 997. Available at the URL [7] K. H. Karlsen and N. H. Risebro. An operator splitting method for nonlinear convection-dision eqations. Nmer. Math., 77(3):365{382, 997. [8] D.. W. Marqardt. An algorithm for least sqares estimation of nonlinear parameters. SIAM J. Appl. Math., :43{44, 963. [9] L. L. Schmaker. Spline Fnctions: Basic Theory. John Wiley and Sons, New York, 98. [2] S. Weisberg. Applied Linear Regression. John Wiley and Sons, New York, 985. (Kari Brsdal) Department of Mathematics, University of Bergen, Johs. Brnsgt. 2, N-58 Bergen, Norway address: kari.brsdal@mi.ib.no (Helge K. Dahle) Department of Mathematics, University of Bergen, Johs. Brnsgt. 2, N-58 Bergen, Norway address: helge.dale@mi.ib.no (Kenneth Hvistendahl Karlsen) Department of Mathematics, University of Bergen, Johs. Brnsgt. 2, N-58 Bergen, Norway address: kenneth.karlsen@mi.ib.no (Trond Mannseth) RF Rogaland Research,, Thormhlensgt. 55, N-58 Bergen, Norway address: trond.mannseth@rf.no