Journal of Computational Physics 231 (2012) 3083–3100

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Journal of Computational Physics

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Modified Eulerian–Lagrangian formulation for hydrodynamic modeling

Shaul Sorek a,b,⇑, Vyacheslav Borisov b

a J. Blaustein Institutes for Desert Research, Zuckerberg Institute for Water Research, Department of Environmental Hydrology & Microbiology,Sde Boker Campus 84990, Israelb Department of Mechanical Engineering, Perlstone Center for Aeronautical Studies, Beer Sheva 84105, Ben-Gurion University of the Negev, Israel

a r t i c l e i n f o

Article history:Received 6 April 2011Received in revised form 6 December 2011Accepted 8 December 2011Available online 20 December 2011

Keywords:AdvectiveDispersive and advective–dispersive PDE’sEulerianEulerian–Lagrangian and modifiedEulerian–Lagrangian formulationNon-divergent PDE formConservative and non-conservativenumerical schemeApparent particle velocityMonotone interpolation for particle trackingConservative and non-conservativenumerical schemesHeat transfer problemDensity driven problem

0021-9991/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.jcp.2011.12.005

⇑ Corresponding author at: J. Blaustein Institutes foMicrobiology, Sde Boker Campus 84990, Israel.

E-mail address: sorek@bgu.ac.il (S. Sorek).

a b s t r a c t

We present the modified Eulerian–Lagrangian (MEL) formulation, based on non-divergentforms of partial differential balance equations, for simulating transport of extensive quan-tities in a porous medium. Hydrodynamic derivatives are written in terms of modifiedvelocities for particles propagating phase and component quantities along their respectivepaths. The particles physically interpreted velocities also address the heterogeneity of thematrix and fluid properties. The MEL formulation is also implemented to parabolic PartialDifferential Equations (PDE’s) as these are shown to be interchangeable with equivalentPDE’s having hyperbolic – parabolic characteristics, without violating the same physicalconcepts. We prove that the MEL schemes provide a convergent and monotone approxima-tion also to PDE’s with discontinuous coefficients. An extension to the Peclet number is pre-sented that also accounts for advective dominant PDE’s with no reference to the fluidvelocity or even when this velocity is not introduced.

In Sorek et al. [27], a mathematical analysis for a linear system of coupled PDE’s and anexample of nonlinear PDE’s, proved that the finite difference MEL, unlike an Eulerianscheme, guaranties the absence of spurious oscillations. Currently, we present notions ofmonotone interpolation associated with the MEL particle tracking procedure and provethe convergence of the MEL schemes to the original balance equation also for discontinuouscoefficients on the basis of difference schemes approximating PDE’s. We provide numericalexamples, also with highly random fields of permeabilities and/or dispersivities, suggestingthat the MEL scheme produces resolutions that are more consistent with the physical phe-nomenon in comparison to the Eulerian and the Eulerian–Lagrangian (EL) schemes.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The numerical solution of the solute transport equation introduces, in general, difficulties mainly because of certain termswhich may conform this equation to become advection dominant. Such cases may be when the advective flux or when thegradient of the diffusion (or dispersion for macroscopic PDE’s) factor, associated with the diffusive flux, become dominantover other factors. The numerical solution of such cases may result in problems of numerical dispersion, spurious oscillationsand over/under shooting errors.

The EL methods have been developed as particle tracking techniques for solving the advection dominated transport prob-lems. All EL schemes are based on the ‘‘advection’’ part solved along characteristic pathlines. An EL least-square collocationmethod was proposed by Bentley and Pinder [6]. The EL collocation method with the use of characteristic method for the

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r Desert Research, Zuckerberg Institute for Water Research, Department of Environmental Hydrology &

3084 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

advective problem is documented by Baptista [3] and by Allen and Khosravani [1]. The performance of the EL methods andsources of errors in such schemes is discussed by Bentley and Pinder [6].

Neuman [14,15], Neuman and Sorek [16], Sorek and Braester [24], and Sorek [21,22,25] also decompose the dependentvariable into advective and residual parts. The resulting advection problem is solved by tracking particles along pathlineswhile the ‘‘dispersive’’ problem is solved on a fixed grid.

None of the above mentioned, incorporate the gradient of the diffusion factor as part of the velocity that drives the par-ticle along its pathline.

Following the formulation of the MEL method (Bear et al. [5]) and its mathematical analysis (Sorek et al. [26]), in Soreket al. [27] the MEL formulation is discussed in details concerning the numerical modeling of coupled PDE’s, its numericalanalysis and an example is provided for the formulation of the set of flow, solute transport and heat transfer equations refer-ring to hydrodynamic processes in porous media. In the case of the MEL formulation for the flow equation, particle’s modifiedvelocity is associated with the gradient of the hydraulic conductivity while, e.g., for the transport equation it accounts for thegradient of the hydrodynamic dispersion.

2. Theory

2.1. The general case

Consider the transport of an extensive quantity, E, of a fluid phase occupying part of the void space of a porous mediumdomain. Let S and e denote the saturation of the fluid phase and specific quantity of E, respectively.

The general macroscopic Eulerian balance equation for the phase E quantity, considering a variable saturated (0 < S 6 1)porous medium domain, on the basis of continuum mechanics (Bear and Bachmat [4]), reads

@

@tð/SeÞ ¼ �$ � /S eV � DE

h � $e� �h i

� f Ea!b þ /SqCE; ð1Þ

where / denotes the porosity, V denotes the mass weighted phase velocity vector, DEh denotes the hydrodynamic dispersion

tensor of E. Actually, a fundamental notion of continuum mechanics is that the flux vector JE = eVE be associated with thevelocity vector of E. As this velocity is not a measurable quantity, the flux is modified into JE = eV + e(VE � V) advectiveand dispersive fluxes, respectively. The latter is based on empirical (i.e. constitutive) relations and conveniently is arbitrarilyset to follow a potential flux pattern [i.e. eðVE � VÞ � �DE

h � $e]. Hence, a non-divergent form of the dispersive flux constitu-

tive relation in (1) {i.e. $ � /S DEh � $e

� �h ican be replaced with $ � ð/SDE

hÞ � $eþ /SDEh : ð$$eÞ} is also valid and can not be re-

garded as a physical flaw. In (1) we further have f Ea!b that denotes the transfer of E from the considered a phase to all other

(b) phases across common microscopic interfaces, q denotes the phase mass density, and CE denotes the rate of growth(=source) of E per unit mass of the phase.

The MEL formulation of (1) reads

1/

DV

Dtð/Þ þ 1

SDV

DtðSÞ þ 1

e

DeVDtðeÞ ¼ �$ � V þ DE

h

e: ð$$eÞ � 1

/Sef Ea!b þ

qe

CE; ð2Þ

where DVDt ð::Þ � @

@t ð::Þ þ V � $ð::Þ� �

denotes the hydrodynamic derivative of (..) associated with a particle carrying (..), moving at avelocity V along its characteristic pathline. Since (/Se) expresses the quantity of E per unit volume of porous medium, the lefthand side (l.h.s.) of (2) describes how this quantity varies along the pathline of the traveling particle.

The velocity eV that serves as the characteristic velocity for the propagation of e reads

eV � V � 1/S

$ � ð/SDEhÞ: ð3Þ

This is referred to as a modified velocity accounting for spatial variations in /, S, and DEh.

We note that (3) accounts for the fluid velocity and an apparent velocity term associated with the heterogeneity charac-teristics of the medium. The first is along a path from a higher energy level to a lower one, the latter along a gradient fromhigh hydrodynamic dispersion to a lower one.

Following the Lagrangian concept, e will be carried with a particle along a pathline defined by

DeVDtðfÞ ¼ eV ð4Þ

where f denotes a spatial position vector.We note that omitting the velocity (i.e. V = 0) in (1) and (2) to (4) defines, respectively, the Eulerian and MEL formulations

of a dispersion equation (parabolic PDE).Following (2) we note that the MEL formulation is based on the non-divergent form of (1) the general balance equation.

Hence, PDE’s with advective-dispersive characteristics are interchanged with their corresponding parabolic forms, address-ing the MEL form without violating the same physical concepts.

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3085

Let us demonstrate this (Sorek and Braecster [23]) by considering the Eulerian form of the flow equation of a compressiblefluid flowing through an unsaturated zone,

q/k@U@t¼ $ � ðqK � $UÞ; ð5Þ

in which U denotes Hubbert’s potential, K � kkrql

� �denotes the permeability tensor associated with, k, the intrinsic perme-

ability tensor, kr, the relative permeability, l, the fluid viscosity, and k � q Sðaf þ arÞ � dSdPc

h i� �that denotes a storage coeffi-

cient with af, ar and Pc denoting the fluid and solid matrix compressibilities and the capillary pressure, respectively. The MELform of (5) reads

/k@U@tþ Vp � $U

� �¼ Kr2U; ð6Þ

where, Vp, the apparent velocity of the particle is given by

Vp � �1

q/k$ � ðqKÞ: ð7Þ

Actually, by virtue of (5) and in view of (7) we note that Vp is directly related to, q, Darcy’s specific flux reading

q ¼ /kq2kr

l@U

@ðq2kr=lÞ

� Vp ð8Þ

Following Sorek and Braester [23], Huang et al. [10] rewrite the flow equation using characteristics similar to the advection-dispersion transport equation with an apparent particle velocity. Moreover, as was demonstrated in Sorek et al. [27], any PDEwith dispersive-advective characteristics can be converted into a parabolic equivalent form, without violating its physicalmeaning. The MEL formulation is more general than that of the EL in applying particle tracking schemes also to parabolicPDE’s like (5) that do not directly address the phase velocity. To exemplify this let us, e.g., consider a parabolic PDE thatis associated with a self-adjoint diffusion term in the form

rn @e@t¼ @

@rrnD

@e@r

� �; ð9Þ

in which r denotes the radius distance in (n = 1) polar or (n = 2) spherical coordinates and D denotes a constant diffusion fac-tor. We note that (9) leads to a symmetric system of discrete equations upon implementing the Eulerian numerical method.However, instead of referring to (9) as a diffusion equation with its diffusive flux rnD @e

@r

�, one may physically interpret (9) as

a PDE with advection-diffusion characteristics and D @e@r

�as its apparent diffusive flux (or dispersion when addressing a mac-

roscopic PDE)

@e@t¼ @

@rD@e@r

� �þ Vr

@e@r; ð10Þ

in which

Vr �nDr: ð11Þ

The EL form of (10) and the particle pathline equation read respectively,

DVr

DtðeÞ ¼ @

@rD@e@r

� �;

DVr

DtðrÞ ¼ �Vr : ð12Þ

The parabolic PDE’s of (9) or (10), its advective-diffusive equivalent, will conform to the same MEL form.Furthermore, following Sorek et al. [27] we can generalize and show the possible interchangeability between an advec-

tive-diffusive PDE and its equivalent parabolic form.Consider the advective-diffusive PDE that reads,

@e@tþ V � $e� Dr2e ¼ Q ; ð13Þ

which when multiplying by K(x,y) – 0 can be rewritten to,

@ðKeÞ@tþKw � $e�K$ � ðD$eÞ ¼ Q 0; Q 0 ¼ KQ ; w � V þ $D: ð14Þ

Actually, we wish to conform the advective-diffusive form of (14) to its equivalent PDE characterized by diffusion (i.e. a par-abolic form), namely to obtain

K@e@t� $ � ðKD$eÞ ¼ Q 0: ð15Þ

3086 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

Equating between (14) and (15) is ensured when,

Table 1Choice

Cont

fuk

ðKwþ D$KÞ � $e ¼ 0; ð16Þ

so that K can be evaluated on the basis of (16),

K ¼ K0 exp �Z

XD�1

wdX� �

: ð17Þ

PDE’s, such as (1), (9) or (10), with constant coefficients next to spatial derivatives or without derivatives of these coeffi-cients, are regarded as conservation forms. Such PDE’s are associated with a divergence of the macroscopic advective to-gether with dispersive fluxes (see (1)) commencing from the decomposition of the flux associated with the velocity of theE quantity, as addressed by continuum mechanics. Hence, we expect that the discretized version of such PDE’s will maintainthe conservation (i.e. conservative property) feature.

Samarskii [20] proves that non-conservative difference schemes of balance equations, can be stable but not convergentfor the case of discontinuous coefficients, whereas these non-conservative schemes provide consistent approximation for thecase of continuous coefficients. Yet, stable conservative schemes of advective-dispersive PDE’s expressed in conservationforms such as (1), may exhibit spurious oscillations or numerical dispersions depending on the approximation of their advec-tive flux. To overcome these numerical problems, we can rewrite PDEs like (1) in non-conservative forms and apply the ELscheme (note that the Lagrangian derivative, can be associated with a particle velocity that has not a clear and direct physicalinterpretation) to obtain conservative forms of the discretized PDEs. As was demonstrated in Sorek et al. [27], another pos-sibility for the numerical solution of equations like (1) is to convert their advective-dispersive form into a dispersive conser-vative form.

We note that the MEL approach associates with the Lagrangian derivative an additional apparent velocity commencingfrom the gradient of the diffusion coefficient. In view of Samarskii [20] we prove in the Appendix section that the MELscheme converges to the original balance equation also for discontinuous coefficients.

We refer to (9) or (10) as scalar PDE forms. Sorek et al. [27] prove the superiority of the MEL schemes leading to monotonealgorithms when applied to a set of PDE’s (i.e. vector PDE forms) as well as demonstrate its superiority for a case of nonlinearset of PDE’s. Both the Eulerian numerical scheme of (9) and the MEL numerical scheme of (10), will be compared in theNumerical Examples section.

In view of (2) the Eulerian, EL and MEL forms for the flow (i.e. the hydrodynamic dispersion tensor is exchanged with thepermeability tensor) and the transport equations can be unified to read

1/

DV�

Dtð/Þ þ 1

SDV�

DtðSÞ þ 1

e

DeV�DtðeÞ ¼ �ð1� f � kÞ$ � V þu

DEh

e: ð$$eÞ þ ð1�uÞ

/Se$ � /S DE

h � $e� �h i

� ke

V � $e� 1/Se

f Ea!b þ

qe

CE;

ð18Þ

where

fV� � V� � u/S

$ � ð/SDEhÞ; V� � ð1� f ÞV: ð19Þ

The various formulations that can be obtained are described in Table 1.

2.2. Why MEL?

The phase mass weighted velocity associated with the Eulerian and the EL numerical schemes, do not specifically addressthe heterogeneity of the medium which may take the form of steep gradients of the permeability and/or the dispersion ten-sors. This for the MEL method is specifically accounted for as additional information about the velocity, (3), of the particleand thus affects the solution process. Moreover, in the MEL numerical scheme we account for the influence of, say, perme-ability and/or dispersion gradients, whereas in the case of the Eulerian and the EL schemes such parameters are approxi-mated by the grid nodal points.

The EL formulation is reported to be effective in overcoming numerical dispersion phenomena when simulating problemswhich are dominated by advection. In view of (3), it is argued that the second term on the right hand side (r.h.s.) of (3) canalso be significant in dictating a situation similar to the advection dominant characteristics. Note that this may occur for

of control coefficients.

rol coefficient Eulerian EL MEL

Flow Transport Flow Transport Flow Transport

1 1 / 0 1 00 0 / 0 1 10 1 / 0 0 0

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3087

governing equations with no reference to V, or even when V = 0. Accordingly, as described in Table 1, the EL formulation can-not be implemented to flow problems (i.e. parabolic PDE’s) as in the case of the MEL formulation. We thus suggest that it canbe preferable to simulate advection dominated problems by, (2), the MEL formulation rather then the EL formulation.

We suggest that in view of (1) for a transport equation (similar arguments hold upon referring to the flow equation),when considering the dispersive flux /SðDE

h � $eÞ for a heterogeneous medium (with steep gradients of /SDEh), it may be that

$ � ð/SDEhÞ

h i� $e

��� ���� /SDEh : $$e

��� ���. This will conform (1) to become advection dominant. The extent to which a PDE withadvection – dispersion characteristics is considered to be advection dominant, is commonly related to, Pe, the Peclet number.In view of (3), we suggest to expand the definition of the Peclet number to read (e.g. in a one dimensional, 1-D, cartesiancoordinate system)

Pe �Lc

DEh

V � 1/S

@

@xð/SDE

hÞ���� ����; ð20Þ

in which Lc denotes a characteristic length. By virtue of (20) we note that a high Peclet number (Pe� 1), which is associatedwith an advective dominant equation, can be resulting from steep gradients of DE

h also when V = 0. The latter can be referredto the case of the flow equation for which DE

h is exchanged with the permeability coefficient. Hence, (20) can be used for both,a flow and/or a transport equation.

2.3. Interpolation

The numerical scheme for the EL and MEL formulation is based on the decomposition of the differential operator and thedependent variables, applying the combined numerical approximation at a fixed frame of reference together with particletracking procedure addressing the Lagrangian derivative. The spatial distribution of the load carried by the particles is di-rectly related to particles velocity along their pathlines. Accurate evaluation of this velocity, especially when accountingfor discontinuities in its ingredients, is essential for obtaining reliable numerical predictions. In view of (3) discontinuitiescan be associated with the phase velocity concerning the EL formulation (i.e. discontinuities in quantities with C0 continuity)and also gradients of phase properties concerning the MEL formulation (i.e. discontinuities in quantities with C0 and C1 con-tinuity). Discussion on the suitability of the MEL scheme in addressing the divergence of discontinuous potential flux patternappears in Bear et al. [5]. In Sorek et al. [27] we had demonstrated that the implicit MEL numerical scheme is monotone for alinear coupled system of PDE’s. Its particle tracking procedure should also be based on a monotone interpolation procedureto overcome possible spurious oscillations. Such a 1-D monotone C1 piecewise cubic interpolation, that specifically addressesdiscontinuities for general tabulated prescribed data, is presented next.

Consider the problem of constructing a 1-D monotone interpolation for tabulated values, a data set given by X ¼ xi; sigNi¼1.

This in return means to construct a curve s(x) 2 C1(X) which is monotone increasing (decreasing) at any interval (xi,x-i+1)"i < N so that s(xi) = si. It can be done by the algorithm developed by Fritsch and Carlson [8] for the monotone piecewisecubic interpolation. In this algorithm the derivatives of the interpolant must be initialized. The initialization can be per-formed by, e.g., the classic cubic spline interpolation (e.g. Press et al. [18]) as it belongs to the C2 continuity (i.e.sðxÞ 2 C2ðXÞ) on the basis of which we thus expect s = s(x) to be monotone C1 piecewise cubic interpolation. Moreover, itis desirable to modify the derivatives s0i in such a way that s0i would be the solution to the following mathematical program-ming problem:

X

i

s0i � s0i �2 ! min

s0i

; ð21Þ

in which s0i and s0i denote the first derivatives (at the i node) with respect to x of the piecewise cubic and the prescribed cubicspline functions, respectively. By virtue of (21) we estimate s0iði ¼ 1;NÞ and with the known si the monotone C1 piecewisecubic is established. The presentation of the algorithm that approximates appropriately (21), is beyond our scope. The re-sults, however, of implementing a C1 piecewise cubic interpolation when compared with the classic cubic spline interpola-tion, are depicted in Fig. 1. We note (Fig. 1) that the constructed function produces monotone interpolation while yielding C2

approximations at some sections. However, as our interpolation also yields a C1 approximation we will refer to it, in whatfollows, as being in general the second order approximation.

Calculation of nodal first derivatives and construction of a monotone interpolant for 2-D or 3-D spaces can be based onthe procedure outlined from (21).

3. Numerical Examples

3.1. Comparisons with analytical solutions

3.1.1. Scalar PDE’sConsider the case of heating a sphere for which (9) represents the non-dimensional heat equation with T denoting tem-

perature (e � T;n = 2). Let D = 1, the solution domain X = {0 < r < 1; t > 0}, the initial and boundary conditions be, respectively,given by,

Fig. 1. Interpolation of a 1-D tabulated function.

3088 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

Tðr;0Þ ¼ 0; Tð1; tÞ ¼ 1;@

@rTð0; tÞ ¼ 0 ð22Þ

For the discretized form of the spherical coordinates let us consider the grids Xt = {tk,k = 0,1, . . .}; Xr = {ri, i = 0,1, . . . ,N};X = Xs �Xr. Let h (=const.) denote the spatial intervals and s (=const.) the time increments, and let ri � ih; riþ0:5 �ðiþ 0:5Þh; tk � ks; ui � Tðri; tkþ1Þ; �ui � Tðri; tkÞ; ur � ur;i � ðuiþ1 � uiÞ=h; u�r � u�r;i � ðui � ui�1Þ=h; u�t � u�t;i � ðui � �uiÞ=s.

Following the numerical scheme for (1) represented in Morton and Mayers [13] and accounting for (22), we obtain for (9)a difference scheme with a second order accuracy in the form

r2i�0:5ui�1 � bi þ r2

i�0:5 þ r2iþ0:5

�ui þ r2

iþ0:5uiþ1 ¼ �bi�ui; i ¼ 1; . . . ;N � 1; ð23Þ

� 1þ h2

6s

!u0 þ u1 ¼ �

h2

6s�u0; uN ¼ 1; ð24Þ

where

bi �r3

iþ0:5 � r3i�0:5

3sh; i ¼ 1; . . . ;N � 1 ð25Þ

We note that (23) has a conservative property as the scheme matrix is symmetric. Using the balance scheme, represented inSamarskii [20], we can obtain a non-conservative scheme approximating (10) that reads

cr2

i�0:5

r2i

ui�1 � 1þ cr2

i�0:5 þ r2iþ0:5

r2i

� �ui þ c

r2iþ0:5

r2i

uiþ1 ¼ ��ui; ð26Þ

where c = s/h2.A second order approximation for the boundary condition at r = 0 can be estimated using the last term of (10) together

with (11),

2r@T@r

���� ¼ limd!0

2d

@T@r

����r¼d

� @T@r

����r¼0

� �¼ 2

@2T@r2 ð27Þ

In view of (10) and (27), the heat equation at r = 0 can be written in the form

@T@t¼ 3

@2T@r2 ð28Þ

Introducing a fictive grid node at r�1 = �h⁄, h⁄ > 0 and approximating (28) at the stencil surrounding the i = 0 node (Ganzhaand Vorozhtsov [9]), we obtain

u�t ¼3�h0

u1 � u0

h� u0 � u�1

h�

� �; �h0 ¼ 0:5ðhþ h�Þ ð29Þ

When h⁄? 0 we obtain from (29)

u0 � �u0

s¼ 6

hu1 � u0

h� @T@r

����r¼0

� �ð30Þ

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3089

In view of the boundary condition at r = 0 in (22) we obtain from (30) the approximation of the boundary condition at r = 0that coincides with (24).

In discretizing the MEL form of the heat Eq. (9), we apply a backward particle tracking method for the non-dimensionalLagrangian derivative equation of the pathline equation in (12), accounting for (11). Assuming that at the time level tk + s aparticle is situated at the grid node i (i.e., r = ri), we find (i.e. for n = 2, D = 1) that at the time level tk the particle was at thenon-dimensional backward spatial position bri given by

bri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

i þ 4sq

ð31Þ

To estimate the backward value of bui � T(bri, tk) we will apply: (i) a, C1, piecewise cubic interpolation which, in general, willgenerate a second order of accuracy and (ii) a linear interpolation Strang and Fix [29] which, in general, will generate a firstorder of accuracy. Approximating the MEL heat Eq. (12) at the X grid, we obtain

ui � bui

s¼ ur � u�r

h; i ¼ 1; . . . ;N � 1 ð32Þ

Approximating a second accuracy order for the boundary condition at r = 0, we obtain a conservative scheme that reads

cui�1 � ð1þ 2cÞui þ cuiþ1 ¼ �bui; i ¼ 1; . . . ;N � 1; ð33Þ� ð1þ 2cÞu0 þ cu1 ¼ �bu0; uN ¼ 1; ð34Þ

where c � s/h2.We had implemented Thomas algorithm Anderson et al. [2] for the Eulerian scheme No. 1 based on (23) and (24), Eulerian

scheme No. 2 using (26) with (24), and for the MEL scheme, (33) and (34). The results of these simulations with h = 0.04 ands = 0.01 are depicted in Fig. 2. Estimate of the value bui � T(bri, tk) for the MEL scheme, was by C1 piecewise cubicinterpolation.

The results of simulations with h = 0.04 and s = 0.001 are depicted in Fig. 3. Estimate for the value bui � T(bri, tk) for theMEL scheme, was by C1 piecewise cubic interpolation as well as by linear interpolation. We note from Figs. 2 and 3 thatall schemes converge. The Eulerian schemes provide practically the same results, the MEL scheme with C1 piecewise cubicinterpolation provides a similar accuracy as the Eulerian schemes, while the MEL scheme with linear interpolation is some-what less accurate. Moreover, in view of Figs. 2 and 3, we conclude that while, as expected, Eulerian schemes are known tobe robust for discretization of the divergence over a potential flux pattern of parabolic PDE’s (e.g. the heat Eq. (9)), the ELformulations habitually are not implemented to such PDE’s even if these can conform to their equivalent (10) advective-dif-fusion forms, and the MEL scheme for these PDE’s is at least comparable in its resolutions to that of the Eulerian ones. More-over, in the case of relatively high value of time, the MEL scheme (with C1 piecewise cubic interpolation) is more accuratethan that of the Eulerian scheme.

3.1.2. PDE’s with discontinuous coefficientsConsider the following diffusive PDE

@

@xDðxÞ @U

@x

� �¼ 0; 0 < x < 1; Uð0Þ ¼ 1; Uð1Þ ¼ 0; ð35Þ

for which its D(x) coefficient is discontinuous in the form of a step function given by

DðxÞ ¼DL; 0 6 x < n

DR; n < x 6 1

�; n ¼

ffiffiffi2p

3ð36Þ

The analytical solution, U, to (35) and (36), subject to the continuity conditions, reads (e.g. Samarskii [20]),

UðxÞ ¼1� a0x; 0 6 x 6 n; a0 � 1= jþ ð1þ jÞn½ �ja0ð1� xÞ; n 6 x 6 1; j � DL=DR

�ð37Þ

The numerical solution, u, to (35) and (36) will compare the performance of the Eulerian and MEL schemes. Following nota-tions similar to those that appear in example subsection 3.1.1, we write the first Eulerian scheme (Samarskii [20]) which isconservative, convergent and produces a first order accuracy,

Di�0:5u�x;i �

x;i ¼ 0; u0 ¼ 1; uN ¼ 0; Di�0:5 ¼ 0:5ðDi þ Di�1Þ ð38Þ

where

Di ¼DL; i 6 n

DR i P nþ 1

�; xn < n ¼

ffiffiffi2p

3< xnþ1: ð39Þ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0T

t=0.01

t=0.05

t=0.1

t=0.15

t=0.2

Analytical solution

MEL, C' piecewise cubics

MEL, linear interpolation

Eulerian scheme-1

Eulerian scheme-2

Fig. 3. Spherical heat conduction. Eulerian and MEL schemes (same schemes as in Fig. 2) compared to the analytic solution. Space increment h = 0.04, timeincrement s = 0.001.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

T

t=0.01

t=0.05

t=0.1

t=0.15

t=0.2

Analytical solution MEL

Eulerian scheme-1 Eulerian scheme-2

Fig. 2. Spherical heat conduction. Eulerian (scheme-1, conservative form; scheme-2, non-conservative form) and MEL (conservative scheme and backwardvalue by C1 piecewise cubic interpolation) schemes compared to the analytic solution. Space increment h = 0.04, time increment s = 0.01.

3090 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3091

Rewriting (35) in the following non-conservative form

@D@x

@U@xþ DðxÞ @

2U@x2 ¼ 0; ð40Þ

we write the second Eulerian scheme

Diuiþ1 � 2ui þ ui�1

h2 þ Diþ1 � Di�1

2huiþ1 � ui�1

2h¼ 0; ð41Þ

which is divergent, Samarskii [20], in the case that D(x) is a discontinuous coefficient.The limiting analytical solution to (41), i.e. the limiting function UðxÞ as h ? 0, can be written in the form (Samarskii [20])

UðxÞ ¼ limh!0

uðx;hÞ ¼1� a0x; 0 6 x 6 n

la0ð1� xÞ; n 6 x 6 1

�ð42Þ

where

a0 �1

lþ ð1þ lÞn ; l � 3þ j5� j

5j� 13jþ 1

ð43Þ

The MEL scheme approximating (40) can be written in the form

ui � bui

s¼ Di

uiþ1 � 2ui þ ui�1

h2 ; bui � �uðbxiÞ; u0 ¼ 1; uN ¼ 0; ð44Þ

where

bxi ¼xi; i < n; i P nþ 1

xn þ as; i ¼ n; as 6 h

xnþ1; i ¼ n; as P h

8><>: ; a ¼ DR � DL

hð45Þ

The prove that the MEL scheme (44) is convergent with a first order accuracy is given in the Appendix A section.Comparison of the numerical results for the two Eulerian and one MEL schemes with a step function diffusion coefficient

DL = 1, DR = 4, and a spatial increment of h = 0.1 is depicted in Fig. 4. The MEL scheme was implemented using a time intervalof s = s0 � h2/(DR � DL) (i.e. based on approximation for the average apparent velocity jrDj ’ (DR � DL)/h) and the estimationof bui was by linear interpolation. The case of h = 0.02 is depicted in Fig. 5, in which for the MEL scheme we applied s = s0, s0/2, 2s0.

We note that the MEL scheme converges for s 6 s0, the second Eulerian scheme is not converging while the first Eulerianscheme and the MEL one are consistently of the same order of accuracy.

3.2. Density driven flow regime

Consider the vertical averaged 2-D areal model for density driven flow regime (Sorek et al. [28]) for which we will com-pare numerical resolutions obtained by an upwind Eulerian (e.g. Morton [12]) scheme with first order approximation for theadvective terms, a Monotone Eulerian (e.g. Samarskii [20]) scheme with second order approximation of the advective terms,the EL and the MEL (see formulation in Bear et al. [5], Sorek et al. [26] and Sorek et al. [27]) schemes. Particle tracking pro-cedure for the El and the MEL schemes is based on (21) (considering the 1-D C1 piecewise cubic interpolation function be-tween the i node and its adjacent i + 1 node) to obtain the nodal values of the first derivatives and a first order bicubicinterpolant (Press et al. [18], 106–108) which was found to be free of spurious oscillations. The following four examples con-sider the extent of temporal and spatial solute distribution. The first three cases are variations on the same theme consid-ering density dependent fluid and accounting for different pumping operations and field characteristics. The fourth case(addressing the same heterogeneities as the third case) for a constant density fluid, aims at comparing among the solutionsobtained by different numerical schemes deviating from those obtained for a variable density flow.

3.2.1. First caseConsider a phreatic aquifer represented by a horizontal area spanning between 0 6 x 6 1000 m and 0 6 y 6 1000 m and

discretized by constant spatial grid increments of Dx = Dy = 50 m. Elevation of the aquifer bottom is at b = �80 m from sealevel and the pumping (QP) operation period be 0 6 t 6 3650 days.

Additional parameters, and their specific meaning (Sorek et al. [28]), associated with the 2-D areal (vertically averaged)model are:Fluxes of qh = qb = 0 m/day (assumed essentially vertical) at the upper (h(x,y, t)) and lower (b(x,y)) aquifer elevations, respec-tively, and concentrations Ch = Cb = 0 kg/kg at these elevations.Addressing a reference medium, we prescribe its fresh water density to be q0 = 1000 kg/ m3 and its porosity as /0 = 0.4.

Compressibility of water with regard to changes in concentration is vqc ¼ 0:025 and matrix compressibility (i.e. change of

porosity due to change in pressure) is v/P ¼ 10�7.

0.0 0.2 0.4 0.6 0.8 1.0

Space coordinate (x)

0.0

0.2

0.4

0.6

0.8

1.0

Dep

ende

ntva

riab

le(u

)

Analytic solutionMEL schemeEulerian scheme-1

Limit solution forEulerian scheme-2

Eulerian scheme-2

Fig. 4. Steady state diffusion equation with a diffusion discontinuous coefficient. Eulerian and MEL schemes, space increment h = 0.1, compared to theanalytic solutions.

0.0 0.2 0.4 0.6 0.8 1.0

Space coordinate (x)

0.0

0.2

0.4

0.6

0.8

1.0

Dep

ende

ntva

riab

le(u

)

Analytic solution

Eulerian scheme-1

Limit solution forEulerian scheme-2

Eulerian scheme-2

MEL scheme, 0τ τ=MEL scheme,

0 / 2τ τ=MEL scheme,

02τ τ=

Fig. 5. Steady state diffusion equation with a discontinuous coefficient. Eulerian and MEL schemes, space increment h = 0.02, compared to the analyticsolutions.

3092 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

The specific yield is Sy = 0.3, the relation between aT and aL the transversal and longitudinal dispersivities, respectively, isgiven by aT = 0.1aL and the retardation factor is Rd = 1.Dimensionless fitting factors associated with the vertical averaged model are chosen to be m = 3.8; X0 = 1(0.5 6X0 6 1) andthe additional apparent dispersivity is aA = 3 m.

The area is subdivided into two distinct zones.

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

X (sea border) m

0

100

200

300

400

500

600

700

800

900

1000

Ym

MEL, bicubic interpolation,

first order approximation EL, first order

approximation

E, first order approximation

E, second orderapproximation, refined grid

Fig. 6. First case – variable density flow regime. Simulation results of groundwater levels after 10 yrs of pumping.

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3093

The first {0 6 x 6 1000 m;300 6 y 6 1000 m}, is characterized by aL = 1.0 m and hydraulic conductivity (with reference toq0) of K = 0.1 m/day and one pumping well Q1(x1 = 0,y1 = 600) = 250 m3/day. The second {0 6 x 6 1000 m;0 6 y 6 300 m}, ischaracterized by aL = 0.1 m and hydraulic conductivity of K = 1.0 m/day and a second pumping well Q2(x2 =500,y2 = 200) = 750 m3/day.

Initial Conditions: h(x,y,0) = 0 m; C(x,y,0) = 0 kg/kgBoundary Conditions:

@h@x¼ 0;

@C@x¼ 0 at x ¼ 0 and x ¼ 1000; 0 6 y 6 1000

h ¼ 0; C ¼ 1 at 0 6 x 6 1000; y ¼ 0h ¼ 1; C ¼ 0 at 0 6 x 6 1000; y ¼ 1000

Groundwater levels obtained by the four different schemes are depicted in Fig. 6. As expected, a sharper decrease in ground-water level is noticed near the pumping well Q1(0,600) in comparison to the one Q2(500,200). We note that away from thewells (dispersive dominant regions) the second order Eulerian scheme, with the refined grid, produces better resolution thenthe first order one, and converges to the MEL solution. However, the second order Eulerian scheme results with numericaldispersion, noticed by the disharmony of the contours shape, in zones close to the pumping wells where advection (i.e.hyperbolic feature of the PDE) is dominant. Both first order approximation schemes, the EL and the upwind Eulerian producesimilar numerical resolutions. We thus conclude that the MEL produced the most reliable results.

3.2.2. Second caseIn this example we demonstrate the potential of the MEL to reliably simulate different regional pumping scenarios. We

chose a longer pumping time period, 0 6 t 6 7300 days, during which two pumping scenarios were implemented. During thefirst time period, 0 6 t 6 3650 days , the wells were shutoff (Q1 = Q2 = 0) after which during the period 3650 6 t 6 7300 days,they were operated at an intensity of Q1 = 250 m3/day and Q2 = 750 m3/day. Other parameters, initial and boundary condi-tions are as mentioned in Section 3.2.1, excluding the boundary condition

0 100 200 300 400 500 600 700 800 900 1000

X (sea border) m

0

100

200

300

400

500

600

700

800

900

1000

Ym

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

MEL, bicubic interpolation, first order approximationE, first order approximation

Fig. 7. Second case – variable density flow regime. Simulation results of groundwater levels after 20 yrs with 10 yrs of pumping.

Fig. 8. Third case – variable density, heterogeneous domain, flow regime. Surface of the longitudinal dispersivity 0.01 6 aL 6 0.2 m, characteristic also tothe hydraulic conductivity (0.1 6 K 6 2.0 m/day) surface.

3094 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

h ¼ 1; C ¼ 1 at 0 6 x 6 1000; y ¼ 1000

In this example we implemented a longer evolution period, subject to different pumping operations. At first, hydraulic con-ditions drive the salt into the domain. The pumping is then turned on when groundwater levels had already reached somespatial distribution. Hence, although employing the same pumping period (10 yrs) as mentioned in Section 3.2.1, the overallgroundwater level distribution depicted in Fig. 7 displays relatively moderate gradients in comparison to those obtained bythe conditions of the example discussed in Section 3.2.1. Note that hereinafter we compare between two schemes of firstorder approximation to demonstrate the resolution differences between the Eulerian and the MEL schemes.

0 100 200 300 400 500 600 700 800 900 1000

X (sea border)m

0

100

200

300

400

500

600

700

800

900

1000

Ym

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

MEL, bicubic interpolation, first order approximation

E, first order approximation

Fig. 9. Third case – variable density, heterogeneous domain, flow regime. Simulation results of groundwater levels after 20 yrs with 10 yrs of pumping.

0 100 200 300 400 500 600 700 800 900 1000X (sea border)m

0

100

200

300

Ym

MEL, bicubic interpolation, first order approximation

0 100 200 300 400 500 600 700 800 900 1000

X (sea border)m

0

100

200

300

Ym

E, first order approximation

Fig. 10. Third case – Variable density, heterogeneous domain, flow regime. Simulation results of relative concentration profiles after 20 yrs with 10 yrs ofpumping.

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3095

0 100 200 300 400 500 600 700 800 900 1000X (sea border)m

0

100

200

300

Ym

MEL, bicubic interpolation, first order approximation

0 100 200 300 400 500 600 700 800 900 1000X (sea border)m

0

100

200

300

Ym

E, first order approximation

Fig. 11. Fourth case – Constant density, heterogeneous domain, flow regime. Simulation results of relative concentration profiles after 20 yrs with 10 yrs ofpumping.

3096 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

3.2.3. Third caseAs before, two pumping scenarios are deployed, however during the second period the wells are operated with the same

pumping flux intensity, Q1 = Q2 = 500 m3/day. The heterogeneous domain is characterized by the realization (see Fig. 8) of aregular random (subroutine ‘‘random’’ of the IMSL library with a modified version of the random number generator by Parkand Miller [17] ; Pseudorandom numbers with uniform density over a prescribed interval are obtained by scaling the outputfrom ‘‘random’’) distribution of the dispersivities (0.01 6 aL 6 0.2 m) and of the hydraulic conductivity (0.1 6 K 6 2.0 m/day).Other parameters, initial and boundary conditions are as mentioned in Section 3.2.2. The seemingly numerical difficultyemerging from the heterogeneity characteristics in evaluating the apparent velocity terms of (3) associated with particletracking for the flow and transport problems, can be appreciated in view of, say, Fig. 8. We note the deviation betweenthe results obtained by the MEL and the Eulerian schemes (Figs. 9 and 10). The MEL scheme (as expected) demonstrates con-centration contours focusing towards the pumping well (Fig. 10) while the extent of the salt intrusion is significantly lessthen that predicted by the Eulerian scheme. Based on the conclusions derived from the previous examples we view theMEL scheme as a more reliable solution.

3.2.4. Fourth caseThe previous examples of density driven flow follow the finding of the mathematical analysis for a linear system of cou-

pled PDE’s (Sorek et al. [27]) concerning the superiority of MEL schemes. Let us therefor examine the difference between thenumerical resolution of the Eulerian and MEL schemes for the case that the flow PDE is decoupled from the transport PDE.We thus employ the same conditions as in Section 3.2.3 yet consider the flow of fresh water (i.e. constant liquid density) anda rigid matrix. We note (Fig. 11) that the numerical predictions of the Eulerian and MEL schemes are of significant less devi-ation when compared to those obtained for the coupled set of PDE’s. Yet, migration of solute when considering the Eulerianscheme (Fig. 11) is to a somewhat greater distance and slightly more dispersed in comparison to the resolution of the MELscheme.

4. Conclusions

A general approach is presented using the MEL formulation to rewrite the governing coupled PDE’s of hydrological phe-nomena. Such models can describe the simultaneous transport of any number of extensive quantities in a porous mediumdomain that contains multiple multi-component phases and a deformable solid matrix, under non-isothermal conditions.

In the MEL formulation, each balance equation has a non-divergent form such that its l.h.s. represents temporal changes(expressed as a linear combination of material derivatives of all intensive variables) and the r.h.s. takes the form of a linear

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3097

combination of Laplacians of the same state variables. The original Eulerian PDE’s that are conformed into MEL PDE’s areinterchangeable and represent the same physical outcome.

Each material derivative involves a velocity of propagation of the particle of the corresponding extensive quantity. Thesevelocities are derived as a combination of the phase velocity, and an additional apparent velocity (with possible physicalinterpretation) accounting for the heterogeneity in matrix properties (such as permeability and porosity), fluid and matrixcompressibilities, etc.

The MEL mathematical model can also be implemented to flow problems for which the fluid velocity does not explicitlyappear in the governing parabolic PDE.

We prove that MEL schemes provide a convergent and monotone approximation also to PDE’s with discontinuouscoefficients.

We provide examples concerning scalar PDE’s as well as PDE’s with discontinuous coefficients, for which the performanceof the Eulerian and MEL schemes are compared against analytical solutions. Both schemes are consistently similar in theirnumerical resolutions.

The prediction to a density driven flow problem was investigated by four different schemes: (i) Upwind Eulerian with firstorder approximation for the advective terms; (ii) Monotone Eulerian with second order approximation of the advectiveterms and a refined grid; (iii) EL with first order approximation of the advective terms and (iv) MEL with first order approx-imation of the advective terms and, unlike the EL scheme, particle tracking for both the flow and transport equations. Wenote that away from the pumping wells (dispersive dominant regions) the second order Eulerian scheme, with the refinedgrid [scheme (ii)], produces better resolution then the first order one [scheme (i)], and converges to the MEL solution[scheme (iv)]. However, the second order Eulerian scheme results with numerical dispersion in zones close to the pumpingwells where advection is dominant. The EL scheme [scheme (iii)] and scheme (i) produce similar numerical resolutions.

Thereafter we compared between schemes (i) and (ii) for further investigation of their numerical resolutions.In the case of a heterogeneous field characterized by regular random distribution of dispersivities and hydraulic conduc-

tivity, we note that scheme (iv) demonstrates (as expected) concentration contours focusing towards the pumping wellwhile the extent of the salt intrusion is significantly less then that predicted by scheme (i).

Considering the same heterogeneous characteristics and for the case that the flow PDE is decoupled from the transportPDE, the numerical predictions of schemes (i) and (iv) demonstrate significant less deviation when compared to those ob-tained for the coupled set of PDE’s. Yet, migration of solute when considering scheme (i) is to a somewhat greater distanceand more dispersed in comparison to the resolution of scheme (iv). The superiority of the MEL scheme over that of the EL oneis particularly pronounced in the case of high heterogeneity (up to discontinuity) characteristics of the medium, addressedby the particles apparent velocity. In that regard we find suitable to refer to LeVeque [11] who argues (Section 8.5, ‘‘order ofaccuracy Isn’t everything’’) that: ‘‘The quality of a numerical method is often summarized by a single number, its order ofaccuracy. This is indeed an important consideration, but it can be a mistake to put too much emphasis on this one attribute.It is not always true that a method with a higher order of accuracy is more accurate on a particular grid or for a particularproblem.’’

We thus conclude that the MEL scheme produced the most reliable results and is better fitted to simulate coupled systemsof balance equations.

Appendix A. Convergence of MEL schemes

Convergence study of difference schemes approximating PDE’s can be found, e.g., in Anderson et al. [2], Ganzha and Vor-ozhtsov [9], Morton and Mayers [13], Samarskii [20]. In what follows the convergence of MEL schemes will be proved on thebasis of Samarskii’s [20] approach.

Consider a diffusion PDE in its Eulerian form

@U@t¼ @

@xDðxÞ @U

@x

� �; 0 < x < 1; DðxÞP c0 > 0 ðA- 1Þ

Uðx;0Þ ¼ 0; Uð0; tÞ ¼ U0; Uð1; tÞ ¼ U1 ðA-2Þ

The MEL formulation of (A-1) reads

DVx

DtðUÞ ¼ DðxÞ @

2U@x2 ;

DVx

DtðxÞ ¼ Vx � �

@D@x

ðA-3Þ

Without loss of generality we may assume that the coefficient of diffusion D, being a sufficiently smooth Samarskii[20] func-tion at any point x – n, has a discontinuity at a single point x = n, i.e., DL � D(n � 0) – DR � D(n + 0). In such a case the analyticfunction U(x, t) is not differentiable at x = n, however Samarskii [20] in reference to its jump, [ ], conditions we write

½U� � Uðnþ 0Þ � Uðn� 0Þ ¼ 0 and D@U@x

� ¼ 0 if x ¼ n ðA-4Þ

Consider the grids Xs = {tk,k = 0,1, . . .}, Xx = {xi, i = 0,1, . . . ,N}, X = Xs �Xx.

3098 S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100

Let us assign h (=const) to be the spatial increments and s (=const) for the time intervals, and let xi � ih; xiþ0:5 � ðiþ 0:5Þh;tk � ks; u � uðxÞ � Uðx; tkþ1Þ; �u � �uðxÞ � Uðx; tkÞ; ui � Uðxi; tkþ1Þ; �ui � Uðxi; tkÞ; ux � ux;i � ðuiþ1 � uiÞ=h;u�x � u�x;i � ðui � ui�1Þ=h;u�xx � u�xx;i � u�x;iþ1 � u�x;i

�=h � ðuiþ1 � 2ui þ ui�1Þ=h2. Assuming that the point of discontinuity, x = n, is situated within the

interval (xn,xn+1), i.e., we write

n ¼ xn þ hh; 0 6 hðhÞ 6 1; 0 < n < N ðA-5Þ

To solve (A-3) numerically, we will use the backward particle tracking method. Assuming that at the time level tk + s a par-ticle is situated at the location x, we find solving the pathline equation in (A-3) that at the time level tk the particle was at thebackward location bxð� xþ

R tþst Vxdh, for which Vx � � @D

@x ¼ dðx� nÞ the Dirac delta function is a valid possibility) wherebu(x) � u(bx) the backward value is considered. In view of (A-3) we thus write

uðxÞ � buðxÞs ¼ DðxÞ @

2u@x2 ; ðA-6Þ

or, what is the same,

sDu00 � u ¼ �bu; u00 � @2u@x2 ðA-7Þ

The difference scheme for (A-7) can be written in the form

sDiy�xx � yi ¼ �byi; i ¼ 1;2; . . . ;N � 1; y0 ¼ U0; yN ¼ U1; ðA-8Þ

where byi is an estimate of bu(xi), evaluated by piecewise linear interpolation.By virtue of Samarskii [20] that a scheme with discontinuous coefficients will be convergent only if it is conservative, we

will demonstrate that there exists such a s0 that for any s such that 0 < s 6 s0 the MEL difference scheme (A-8) is conserva-tive, i.e. its operator is self-adjoint. Consider the (A-8) scheme so that xn 6 n 6 xn+1, Di+1 > Di, "i and let s0 be the time duringwhich the particle situated at x = xn+1 will be shifted to the x = xn position. In view of (A-3), we approximate the particlevelocity Vn by

Vn ¼Dnþ1 � Dn

h; ðA-9Þ

and use linear interpolation to estimate bun at the backward position

bun ¼ �un þ�unþ1 � �un

hVns ðA-10Þ

Considering that xn+1 = xn + Vns0 and by virtue of (A-9) we obtain

s0 ¼h

Vn¼ h2

Dnþ1 � DnðA-11Þ

Addressing (A-1)–(A-3), we can estimate Un(t) and Un+1(t) using Taylor’s sequences namely

�un ¼ un � s @@t

Unðt þ sÞ þ Oðs2Þ; �unþ1 ¼ unþ1 � s @@t

Unþ1ðt þ sÞ þ Oðs2Þ ðA-12Þ

By virtue of (A-11) and (A-12) we obtain

�unþ1 � �un

hVns ¼

unþ1 � un

h2 ðDnþ1 � DnÞs� ðDnþ1 � DnÞs2 @

@tðUnþ1 � UnÞ þ

sh2 Oðs2Þ ðA-13Þ

Following (A-5), the estimate for bun is obtained by an expansion in the vicinity of x = n

@

@tUnþ1 ¼

@

@tUðn; t þ sÞ þ ð1� hÞh @

@tU0R þ Oðh2Þ; U0R �

@

@xUðnþ 0; t þ sÞ; ðA- 14Þ

@

@tUn ¼

@

@tUðn; t þ sÞ þ hh

@

@tU0L þ Oðh2Þ; U0L �

@

@xUðn� 0; t þ sÞ ðA-15Þ

By virtue of (A-13)–(A-15), we obtain from (A-10)

bun ¼ �un þunþ1 � un

h2 ðDnþ1 � DnÞsþ Oðs2Þ þ sOðhÞ ðA-16Þ

In view of (A-16), the MEL scheme (A-8) can be approximated, with an approximation error of O(s) + O(h), by

ui � �ui

s� uiþ1 � ui

h2 ðDiþ1 � DiÞ ¼ Diu�xx; i ¼ 1; . . . N � 1; ðA-17Þ

leading to

S. Sorek, V. Borisov / Journal of Computational Physics 231 (2012) 3083–3100 3099

Di

h2 ui�1 �1sþ Di þ Diþ1

h2

� �ui þ

Diþ1

h2 uiþ1 ¼ ��ui

s; i ¼ 1; . . . N � 1 ðA-18Þ

We conclude from (A-18) that the scheme operator is self-adjoint, i.e. the scheme is conservative. Moreover, rewriting (A-18)in the form

ui � �ui

s¼ Diu�xð Þx �

Diþ1ðuiþ1 � uiÞ � Diðui � ui�1Þh2 ; ðA-19Þ

we note that in Samarskii [20] the (A-19) scheme is proven to be stable and that it provides a convergent (hence, consistent)approximation for (A-1). Hence, the (A-8) scheme, being consistent with (A-19), provides a consistent approximation for (A-1). To prove the stability of (A-8), we choose a general irregular grid Xbx = {bxi, i = 1, . . . ,N � 1}. Following the notion of theboundary maximum principle (e.g. Borisov and Sorek [7], Samarskii [20]), any grid node of Xbx is considered as a boundarypoint where the grid function is equal to a prescribed value namely

byi ¼ �uðbxiÞ ðA-20Þ

Rewriting the scheme (A-8) in the form

yi ¼sDi

h2 þ 2sDi

ðyi�1 þ yiþ1Þ þh2

h2 þ 2sDi

byi; ðA-21Þ

we note that the sum of the absolute values of the coefficients multiplying the nodal arguments in the RHS of (A-21) is equalto a unit, i.e.

2sDi

h2 þ 2sDi

þ h2

h2 þ 2sDi

¼ 1 ðA-22Þ

Hence, by virtue of Theorem 2.2 (Borisov and Sorek [7]) concerning sufficient conditions for the monotonicity of differenceschemes, (A-8) is Samarskii-monotone and hence it is stable with respect to Chebyshev norm. Thus in view of Lax equiva-lence theorem, Richtmyer and Morton [19], the MEL scheme provides a convergent approximation for the boundary valueproblem (A-1) even for the case of discontinuous coefficients.

In perfect analogy, following (A-19)–(A-22), it can be proven that the (A-18) scheme is conservative stable and conver-gent, when Di+1 < Di for some i < N.

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