An Eulerian-Lagrangian Single-Node CollocationMethod for Transient Advection-DiffusionEquations in Multiple Space DimensionsLi Wu,1 Hong Wang2

1Department of Mathematics, University of Rhode Island,Kingston, Rhode Island 02881

2Department of Mathematics, University of South Carolina,Columbia, South Carolina 29208

Received 5 September 2002; accepted 20 August 2003

DOI 10.1002/num.10094

We developed a nonconventional Eulerian-Lagrangian single-node collocation method for transientadvection-diffusion transport partial differential equations in multiple space dimensions. This methodgreatly reduces the number of unknowns in conventional collocation method, generates accurate numericalsolutions, and allows large time steps to be used in numerical simulations. We perform numericalexperiments to show the strong potential of the method. © 2003 Wiley Periodicals, Inc. Numer Methods PartialDifferential Eq 20: 284–301, 2004

Keywords: advection-diffusion equations; characteristic methods; collocation method; advection-diffusionequations; Eulerian-Lagrangian methods; Hermite polynomials

I. INTRODUCTION

Advection-diffusion partial differential equations arise in mass and heat transfer problems, inwhich both advection and diffusion contribute to the transport. These equations model transportof solute in groundwater and surface water in environmental science, displacements of oil byinvading fluid in petroleum recovery, and miscible fluid flow precesses in many other importantapplications [1, 2]. These equations admit solutions with moving sharp fronts and complicatedstructures and present serious mathematical and numerical difficulties. Classical second-ordermethods tend to generate numerical solutions with spurious oscillations. Upstream-weighting

Correspondence to: Li Wu, Department of Mathematics, Tyler 211, University of Rhode Island, Kingston, RI 02881(e-mail: liwu@math.uri.edu)Contract grant sponsor: Mobil Technology Company and ExxonMobil Upstream Research Company (to H.W.)Contract grant sponsor: South Carolina State Commission of Higher Education: South Carolina Research InitiativeGrant (to H.W.)

© 2003 Wiley Periodicals, Inc.

methods can suppress these oscillations dramatically, but they often generate solutions withexcessive numerical diffusion and grid orientation effect [3, 4].

Many improved numerical methods were developed, aiming at seeking accurate and efficientnumerical solutions of advection-diffusion equations. Collocation methods attempt to minimizethe residual in the variational formulation or the weighted residual methods for the governingdifferential equations by forcing the residual to be zero at a finite number of discrete orcollocation points within the physical domain. Because they are relatively easy to formulate andto implement, collocation methods have been extended in various ways to numerically simulatea wide variety of problems [1, 5–8]. Recently, Wu and Pinder [9] introduced a single-nodestrategy to simplify the standard collocation methods that use Hermite polynomials as basisfunctions, primarily by replacing the first-order space derivatives in the trial functions byupstream-weighted, finite-difference quotient approximations. Computational experiments haveshown the potential of the single-node collocation method.

Like many other Eulerian methods, collocation methods typically use a fixed spatial grid andan Eulerian temporal discretization in the time direction. Thus, they are easily influenced by thetime truncation errors that cause numerical diffusion and suffer from the Courant-Friedrich-Levy (CFL) restriction. Eulerian-Lagrangian methods follow the movement of information orparticles as well as their interactions along the characteristics. Because solutions of transientadvection-diffusion equations are much smoother along the characteristics than they are in thetime direction, so these methods greatly reduce truncation errors compared with the Euleriantreatments, generate accurate numerical solutions even if large time steps are employed, andrelaxed the CFL restriction.

In a previous article [10], Wu and collaborators developed an Eulerian-Lagrangiansingle-node collocation method (ELSCM) for one-dimensional transient advection-diffu-sion equations by combining the single-node collocation method in [9] with the Eulerian-Lagrangian concept. The ELSCM method generates very accurate numerical solutions forone-dimensional transport equations, despite that the method uses large time steps andgreatly reduce number of unknown variables. Motivated by the success of the ELSCMmethod for one-dimensional problems [10], We develop a nonconventional Eulerian-Lagrangian single-node collocation method for transient advection-diffusion equations inmultiple space dimensions, by overcoming new difficulties that occur due to higher spacedimensionality. The rest of this article is organized as follows: In section II, we present anEulerian-Lagrangian semi-discretized approximation. In section III, we derive a noncon-ventional Eulerian-Lagrangian single-node collocation method with piecewise-cubic Her-mite polynomials in two space dimensions. In section IV, we address several numericalissues and concerns. In section V, we carry out numerical experiments to observe theperformance of the ELSCM scheme. In section VI, we extend this scheme to three-dimensional space with some numerical tests, and at the end, section VII, we summarize theresults in this article and discuss the directions of future work.

II. AN EULERIAN-LAGRANGIAN SEMI-DISCRETIZED APPROXIMATION TO TWO-DIMENSIONAL ADVECTION-DIFFUSION EQUATIONS

We consider the following initial-boundary value problem for two-dimensional transient ad-vection-diffusion equations in the development of the ELSCM scheme:

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 285

R�x, y, t��u

�t� V1�x, y, t�

�u

�x� V2�x, y, t�

�u

�y�

�

�x �D�x, y, t��u

�x� ��

�y �D�x, y, t��u

�y� � f �x, y, t�,

�x, y� � �, t � �0, T�,u�x, y, 0� � u0�x, y�, �x, y� � �,u�x, y, t� � g�x, y, t�, �x, y� � ��, t � �0, T�.

(2.1)

Here � � (a, b) � (c, d ) is a rectangular domain in �2. R(x, y, t) is a retardation coefficient;V1(x, y, t) and V2(x, y, t) are the x� and y� components of the velocity field; D(x, y, t) is thediffusion coefficient; f (x, y, t) is a given source or sink function; u(x, y, t) is the unknownconcentration of a dissolved substance in subsurface flow. u0(x, y) and g(x, y, t) are prescribedinitial and boundary data that needed to close the system.

Eulerian-Lagrangian methods use a time-marching algorithm. Let N be a positive integer. Wedefine a partition of time interval [0, T] by

0 � t0 � t1 � t2 � · · · � tn � · · · � tN�1 � tN � T. (2.2)

We follow the Eulerian-Lagrangian treatment in the modified method of characteristics [11] andthe Eulerian-Lagrangian localized adjoint method [12, 16]. Let the characteristic curves

x � r1��; x� , y� , t��,

y � r2��; x� , y� , t��,

t � r3��; x� , y� , t�� (2.3)

be the solution of the differential equation

dx

d��

V1�x, y, t�

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t�,

dy

d��

V2�x, y, t�

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t�,

dt

d��

R�x, y, t�

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t�, (2.4)

which is subject to the initial condition

x���t� � r1�t�; x� , y� , t�� � x� ,

y���t� � r2�t�; x� , y� , t�� � y� ,

t���t� � r3�t�; x� , y� , t�� � t�. (2.5)

286 WU AND WANG

We combine the advection term and the time derivative term in the governing equation in(2.1) into a directional derivative term

�u

���

R�x, y, t�

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t�

�u

�t

�V1�x, y, t�

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t�

�u

�x�

V2�x, y, t�

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t�

�u

�y.

(2.6)

Incorporating equation (2.6) into Equation (2.1), we symmetrize the governing Equation (2.1)and rewrite it as a parabolic equation along the characteristics (2.3):

�R2�x, y, t� � V12�x, y, t� � V2

2�x, y, t��u

���

�

�x �D�x, y, t��u

�x� ��

�y �D�x, y, t��u

�y�� f �x, y, t�, �x, y� � �, t � �0, T�. (2.7)

Note that the initial-value problem of the differential Equations (2.4) and (2.5) cannot besolved analytically for a general velocity field. Hence, to derive an Eulerian-Lagrangiannumerical scheme, we need to approximate the characteristic curves (2.3) by, e.g., the Eulerquadrature in the time stepping procedure. Let t*n(x, y) � tn�1 if the (approximate) characteristiccurve r(�; x, y, tn) � (r1(�; x, y, tn), r2(�; x, y, tn), r3(�; x, y, tn)) does not backtrack to theboundary during the time period [tn�1, tn], or t*n � t*n(x, y) � [tn�1, tn] be the time instant whenr(�; x, y, tn) intersects the boundary otherwise. We also let

t*n�x, y� � tn � t*n�x, y�,

x* � x �V1�x, y, tn�

R�x, y, tn�t*n�x, y�,

y* � y �V2�x, y, tn�

R�x, y, tn�t*n�x, y�. (2.8)

We then approximate the characteristic directional derivative (�u/��)(x, y, tn) at time tn by thebackward difference quotient along the approximate characteristic [11, 12]:

�R2�x, y, tn� � V12�x, y, tn� � V2

2�x, y, tn��u

���x, y, tn�

� �R2�x, y, tn� � V12�x, y, tn� � V2

2�x, y, tn� �u�x, y, tn� � u�x*, y*, t*n�

��t*n�x, y��2 � �x � x*�2 � �y � y*�2

� E1�u�x, y, tn�� � R�x, y, tn�u�x, y, tn� � u�x*, y*, t*n�

t*n�x, y�� E1�u�x, y, tn��, (2.9)

where E1(u(x, y, tn)) is the resulting local truncation error [13].

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 287

Incorporating Equation (2.9) into Equation (2.7) and dropping the local truncation error term,we obtain a semi-discretized Eulerian-Lagrangian approximation scheme

R�x, y, tn�u�x, y, tn� � u�x*, y*, t*n�

t*n�x, y��

�

�x �D�x, y, tn��u

�x�x, y, tn��

��

�y �D�x, y, tn��u

�y�x, y, tn�� � f �x, y, tn�. (2.10)

III. A EULERIAN-LAGRANGIAN SINGLE-NODE COLLOCATION METHOD USINGPIECEWISE-BICUBIC HERMITE POLYNOMIALS

We notice that the semi-discretized approximate Equation (2.10) is in the parabolic equationform without the advection term along the approximate characteristics. Thus it retains thecoercivity. Hence, any method that performs well for an elliptic or parabolic equation shouldpotentially work well for (2.10). Let I, J be positive integers. We define a spatial partition of [a,b] � [c, d] by

a � x0 � x1 � x2 � · · · � xi � · · · � xI�1 � xI � b,

c � y0 � y1 � y2 � · · · � yj � · · · � yJ�1 � yJ � d. (3.1)

A. A Second Look at a Conventional Collocation Method

A conventional collocation method for second-order elliptic or parabolic equations uses piece-wise-bicubic Hermite polynomials as basis functions. It employs the trial function U(x, y, tn) ofthe form [1, 14, 15]

U�x, y, tn� � �j�0

J �i�0

I �U�xi, yj, tn��i�x��j�y� ��U

�x�xi, yj, tn�i�x��j�y�

��U

�y�xi, yj, tn��i�x�j�y� �

�2U

�x�y�xi, yj, tn�i�x�j�y�� , (3.2)

where �i() and i() are the Hermite nodal basis functions satisfying

�i�j� � �ij, �i�j� � 0, i, j � 0, 1, . . . , I,

i�j� � 0, i�j� � �ij, i, j � 0, 1, . . . , I, (3.3)

with �ij � �(i � j) being the Dirac-� function.If we incorporate the expression (3.2) into Equation (2.10) and enforce the resulting equation

at some collocation points (X, Y ) � [xi�1, xi] � [yj�1, yj] for i � 1, 2, . . . , I and j � 1, 2, . . . ,J, we would obtain an Eulerian-Lagrangian collocation scheme. At time step tn, this schemewould look like a conventional collocation scheme for a parabolic equation without advectionterm and would involve U(xi, yj, tn), (�U/�x)(xi, yj, tn), (�U/�y)(xi, yj, tn), and (�2U/�x�y)(xi,

288 WU AND WANG

yj, tn) as unknowns. Thus, after incorporating the boundary conditions in (2.1), this schemewould require four collocation points within each cell [xi�1, xi] � [yj�1, yj], and 4(I � J ) totalcollocation points on the region (a, b) � (c, d ) to close the system at time tn. Similarly, acorresponding scheme for three-dimensional problems would require eight collocation pointsper cell.

B. A Nonconventional Eulerian-Lagrangian Single-Node Collocation Method

To further improve the computational efficiency of the Eulerian-Lagrangian collocation scheme,we follow the idea in [9] to develop a nonconventional Eulerian-Lagrangian single-nodecollocation method using piecewise-bicubic Hermite polynomials.

In order to eliminate the first-order spatial derivative (�U/�x)(xi, yj, tn), (�U/�y)(xi, yj, tn), andthe mixed derivative (�2U/�x�y)(xi, yj, tn) in the trial function (3.2), we replace them by theirfinite-difference quotient approximations:

U�x, y, tn� � �j�0

J �i�0

I

�Ui,jn �i�x��j�y� � �xUi,j

n i�x��j�y� �

�yUi,jn �i�x�j�y� � �x�yUi,j

n i�x�j�y��. (3.4)

Here Ui, jn � U(xi, yj, tn); and �xUi, j

n and �yUi, jn , and �x�yUi, j

n are the divided 4th orderfinite-difference approximations to the first-order spatial derivatives (�U/�x)(xi, yj, tn) and(�U/�y)(xi, yj, tn), and the mixed derivative (�2U/�x�y)(xi, yj, tn), respectively. For simplicityof expression, we assume a uniform spatial partition in (3.1) below. This leads to the followingfinite-difference approximations �xUi, j

n and �yUi, jn to the first-order spatial derivatives �U/�x and

�U/�y at (xi, yj, tn):

�xUi,jn �

Ui�2,jn � 8Ui�1,j

n � 8Ui�1,jn � Ui�2,j

n

12x,

�yUi,jn �

Ui,j�2n � 8Ui,j�1

n � 8Ui,j�1n � Ui,j�2

n

12y. (3.5)

Then, the product of �x and �y gives the following approximation to the mixed spatial derivative(�2U/�x�y)(xi, yj, tn):

�x�yUi,jn �

1

144xy �Ui�2,j�2

n � 8Ui�1,j�2n � 8Ui�1,j�2

n � Ui�2,j�2n � � 8�Ui�2,j�1

n � 8Ui�1,j�1n

� 8Ui�1,j�1n � Ui�2,j�1

n � � 8�Ui�2,j�1n � 8Ui�1,j�1

n � 8Ui�1,j�1n � Ui�2,j�1

n �

� �Ui�2,j�2n � 8Ui�1,j�2

n � 8Ui�1,j�2n � Ui�2,j�2

n ��. (3.6)

Expression (3.4) involves only the function values U(xi, yj, tn) for i � 2, 3, . . . , I � 2 andj � 2, 3, . . . , J � 2. For nodes near the boundaries, we use one-sided finite-differenceapproximations for �x and �y [10]. Then overall, this makes it possible to use one collocationpoint only on each cell to match the number of unknowns. Hence, we choose the collocationpoint to be the cell center

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 289

Xi �xi�1 � xi

2, Yj �

yj�1 � yj

2, (3.7)

for each cell [xi�1, xi] � [yj�1, yj], i � 1, 2, . . . and j � 1, 2, . . . , J. We incorporate theexpression (3.4) into Equation (2.10) and collocate the resulting equation at each collocationpoint (Xi, Yj) for i � 1, 2, . . . , I and j � 1, 2, . . . , J. Using the notation

�w��X,Y � � �a

b �c

d

w �x, y���x � X ���y � Y �dxdy � w �X, Y �, (3.8)

we obtain a collocation scheme

�R�x, y, tn�U�x, y, tn� � U�x*, y*, t*n�

t*n�x, y��

�

�x �D�x, y, tn��U

�x�x, y, tn��

��

�y �D�x, y, tn��U

�y�x, y, tn���

�Xi,Yj�

� �f �x, y, tn���Xi,Yj�. (3.9)

After incorporating the boundary condition in (2.1), expression (3.4) involves only (I � 1)(J � 1) unknowns. However, Equation (3.9) defines (I, J ) equations, leading to an overdeter-mined system. To circumvent this difficulty, we average the collocation equation (3.9) aroundevery four adjacent collocation points to define an equation at the common vertex. In this way,we obtain an Eulerian-Lagrangian single-node collocation method for the two-dimensionaladvection-diffusion Equation (2.1):

1

4 �k�i

i�1 �l�j

j�1 �R�x, y, tn�U�x, y, tn� � U�x*, y*, t*n�

t*n�x, y��

�

�x �D�x, y, tn��U

�x�x, y, tn��

��

�y �D�x, y, tn��U

�y�x, y, tn���

�Xk,Yl�

�1

4 �k�i

i�1 �l�j

j�1

�f �x, y, tn���Xk,Yl�,

1 � i � I � 1, 1 � j � J � 1. (3.10)

IV. COMPUTATIONAL CONSIDERATIONS

In this section, we address some computational issues in the implementation of the Eulerian-Lagrangian single-node collocation scheme (3.10).

A. Reduction of the Bandwidth of the Coefficient Matrix

In the numerical implementation, we rewrite equation (3.10) in the form

290 WU AND WANG

�k�i

i�1 �l�j

j�1 �R�x, y, tn�U�x, y, tn� � t*n�x, y��

�x �D�x, y, tn��U

�x�x, y, tn��

� t*n�x, y��

�y �D�x, y, tn��U

�y�x, y, tn���

�Xk,Yl�

� �k�i

i�1 �l�j

j�1

�R�x, y, tn�U�x*, y*, t*n�

� t*n�x, y�f �x, y, tn���Xk,Yl�, 1 � i � I � 1, 1 � j � J � 1. (4.1)

The average of equation (3.9) at four collocation points around the node (i, j ) in the Eulerian-Lagrangian single-node collocation scheme (3.10) recruits more nonzero entries in the coeffi-cient matrix. In solving the discrete algebraic system, we arrange the unknowns from left to rightand from bottom to top.

Expressions (3.4)–(3.6) show that the finite difference approximation �xUi, jn uses the function

values U(xi�2, yj, tn), U(xi�1, yj, tn), U(xi�1, yj, tn), and U(xi�2, yj, tn) at the four neighboringnodes of (xi, yj, tn) in the x-direction. Furthermore, the average in the x-direction in Equation(4.1) involves �xUi�1, j

n , �xUi, jn , and �xUi�1, j

n . Therefore, Equation (4.1) involves 7 neighboringnodes in the x-direction, and 7 nodes in the y-direction. Furthermore, Equation (3.6) couples allthese nodes. Consequently, the coefficient matrix of the Eulerian-Lagrangian single-nodecollocation scheme (3.10) is a 49-band matrix, with most of the coupling from the finite-difference approximation �x�yUi, j

n in (3.6) to the mixed spatial derivative (�2U/�x�y)(xi, yj, tn).To reduce the bandwidth of the coefficient matrix, we replace (3.6) by the following

2nd-order finite-difference approximation:

�x�yUi,jn �

1

4xy�Ui�1,j�1

n � Ui�1,j�1n � Ui�1,j�1

n � Ui�1,j�1n �. (4.2)

With this approximation, the Eulerian-Lagrangian single-node collocation scheme (3.10) re-duces the coupling to 4 nodes each at xi�1 and xi�1 as well as yj�1 and yj�1, in addition to thecoupling of 7 each at xi and yj. In other words, for each row of the coefficient matrix, thistreatment reduces the nonzero entries from 49 in (3.6) to 25 in (4.2).

For the linear matrix system, we use a preconditioned iterative method to solve it. In short,at each time level, we use the three diagonals to generate the preconditioned solution, thenmodify the solution by off-diagonal terms.

B. Characteristic Tracking

In sections II and III, we developed the Eulerian-Lagrangian single-node collocation scheme(3.10) based on an Euler tracking algorithm to the characteristic curves (2.3)–(2.5) for simplicityof presentation. It is well known that the accuracy of a tracking algorithm has a very significantimpact on the accuracy of an Eulerian-Lagrangian numerical scheme [12, 16]. To improve theaccuracy of the tracking algorithm, we can use higher-order numerical quadratures, such assecond-order Runge-Kutta methods to improve the accuracy of the tracking algorithm. We canalso combine these algorithms with the use of micro time steps in the characteristic tracking tofurther improve the accuracy of a tracking algorithm [17].

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 291

C. Treatment of Boundary Conditions

At the nodes that are close to the outflow boundary (identified by v � n � 0, where v � (V1(x,y, t), V2(x, y, t)) and n is the unit outward normal to the boundary), we could use a one-sidedfinite difference approximation in the upstream direction to approximate �xU and �yU.

In the trial function U(x, y, tn) defined in (3.4), �xUi, jn is used to replace (�U/�x)(xi, yj, tn) in

(3.2) for most interior nodes (xi, yj) i � 2, 3, . . . , I � 2. When the flux is known at oneboundary, it can be directly applied into (3.2); otherwise, we need to employ some interior nodesto approximate (�U/�x)(xi, yj, tn) for i � 0, 1, I � 1 and I. For example, we can approximate(�U/�x)(x0, yj, tn) and (�U/�x)(x1, yj, tn) as the following, respectively,

�U

�x�x0, yj, tn� �xU�x0, yj, tn�

�25U�x0, yj, tn� � 48U�x1, yj, tn� � 36U�x2, yj, tn�

12x

�16U�x3, yj, tn� � 3U�x4, yj, tn�

12x. (4.3)

�U

�x�x1, yj, tn� �xU�x1, yj, tn�

�3U�x0, yj, tn� � 10U�x1, yj, tn� � 18U�x2, yj, tn�

12x

�6U�x3, yj, tn� � U�x4, yj, tn�

12x. (4.4)

Both quotients in (4.3) and (4.4) are 4th order approximations to corresponding derivatives. Andwe can also derive the one-sided difference quotients for (�U/�x)(xI�1, yj, tn) and (�U/�x)(xI,yj, tn). Similarly, these can be applied to (�U/�y)(xi, yk, tn) for k � 0, 1, J � 1 and k � J. Lowerorder quotient approximations can also be the options for replacing derivatives.

However, such treatment does not work near the inflow boundary (identified by v � n � 0),because the one-sided finite difference approximation would become a downstream approxi-mation. In this case, we could combine an extrapolation or extension with a backtracking of thecharacteristics from a node near the inflow boundary to the boundary to define the numericalscheme near inflow boundary [12].

V. NUMERICAL EXPERIMENTS

In this section, we carry out two numerical example runs for problem (2.1) to observe theperformance of the Eulerian-Lagrangian single-node collocation method. The first example is apure transport problem. The second example is a purely advective or advective-diffusivetransport of a two-dimensional Gaussian pulse.

A. An Incoming Pulse

In this numerical example, we apply the ELSCM method to simulate transport of a pulse thatmoves into the domain � � [0, 1] � [0, 1] from the boundary of x � 0. The initial conditionof this incoming pulse is

u0�x, y� � �exp���y � 0.5�2

0.004 �, if x � 0.15,

exp���x � 0.15�2 � �y � 0.5�2

0.004 �, otherwise. (5.1)

292 WU AND WANG

The contour and surface plots of the initial condition (t � 0.0) are shown in Fig. 1.The velocity field is (V1, V2) � (0.5, 0.0). In the numerical simulation, we use a mesh size

of x � y � 164

, a time step of t � 110

. We present the numerical result at t � 0.5 in Fig. 2.

B. Transport of a Two-Dimensional Gaussian Pulse

In the second numerical example run, we apply the ELSCM to simulate transport of a Gaussianpulse. This problem provides an example for a homogeneous two-dimensional advection-diffusion equation with a variable velocity field and a known analytical solution. Moreover, thisproblem changes from the advection dominance in most of the domain to the diffusiondominance in the region near the center. These types of problems often arise in many importantapplications and are more difficult to simulate compared with purely advection-dominatedproblems. This example has been used widely to test for numerical artifacts of different

FIG. 1. Contour and mesh plots for the initial condition (t � 0.0).

FIG. 2. Contour and mesh plots for numerical solution at t � 0.5. x � y � 1

64, t � 1

10, V� � (0.5, 0.0),

diffusion � 0.0.

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 293

schemes, such as numerical stability and numerical diffusion, spurious oscillations, and phaseerrors.

The spatial domain is [0, 1] � [0, 1]; the rotating velocity field is imposed as V1(x, y, t)� sin(4t) and V2(x, y, t) � �cos(4t); the time interval is [0, T] � [0, /2], which is the timeperiod required for a full revolution. The initial condition is given as

u0�x, y� � exp���x � 0.25�2 � �y � 0.5�2

2�2 �. (5.2)

The corresponding analytical solution with a constant diffusion coefficient D and f � 0 has theform

u�x, y, t� �2�2

2�2 � 4Dtexp��

�x � 0.5 � 0.25 cos 4t�2 � �y � 0.5 � 0.25 sin 4t�2

2�2 � 4Dt �. (5.3)

In the numerical experiments, the remaining parameters are chosen as follows: � � 0.0447,which gives 2�2 � 0.004; x � y � 1/64; t � /40; and D � 0.0001 and 0.0, respectively.We present the surface and contour plots of the initial condition (5.2), which is also theanalytical solution of the pure advection problem after one complete revolution, and theanalytical solution (5.3) for D � 0.0001 in Fig. 3(a–d). We present the corresponding numericalsolutions for D � 0 and D � 0.0001 at t � /2 in Fig. 4(a–d).

At t � / 2, the pulse finishes one full revolution. When no diffusion is presented, thatis diffusion � 0.0, the computed solution gives its maximum value as 1.0029, comparingwith the analytical solution having maximum as 1.0; when diffusion � 0.0001, thecomputed solution presents its maximum value as 0.8647, whereas the analytical solutionhas its maximum value as 0.8642. Apparently, the numerical solutions reveal the physicsvery well in both cases.

In Table I, we provide the comparison of maximum values for the gaussian pulse computedby different numerical schemes with the analytical solution at t � /2. For all schemes here,x � y � 1

64, D � 0.0001. ELLAM-Er and ELLAM-RK are ELLAM methods with Euler and

Runge-Kutta tracking algorithms. Same as ELSCM, they allow taking large time steps and stillgenerate good simulations; the Crank-Nicolson Cubic Petrov-Galerkin (CN-CPG) finite elementmethod and the Crank-Nicolson Finite-Difference method (CN-FDM) are restricted by Courantnumber such that only small time steps can be employed to generate reasonable results. Theytake more CPU time and the solutions have some deformation. Some figures and morecomparison can be found in [12].

VI. THE EXTENSION OF ELSCM TO THREE-DIMENSIONAL ADVECTION-DIFFUSION EQUATIONS

In this section, we focus on the extension of ELSCM to unsteady-state advection-diffusionequations in the three-dimensional spaces.

294 WU AND WANG

A. The Eulerian-Lagrangian Semi-Discretization

In a three-dimensional space, the problem of interest has the form

R�x, t��u

�t� V � �u � � � �D�x, t��u� � f �x, t�, x � �, t � �0, T�,

u�x, 0� � u0�x�, x � �,

u�x, t� � g�x, t�, x � ��, t � �0, T�. (6.1)

Here � � �3 is a three-dimensional rectangular prism. V � (V1(x, t), V2(x, t), V3(x, t)) is thevelocity field; Other terms are defined as the same as those in (2.1).

We define the characteristic curves x � r(�; x� , t�) � (r1(�; x� , t�), r2(�; x� , t�), r3(�; x� , t�)), t(�; x� ,t�) � rt(�; x� , t�), in terms of the variable � by the ordinary differential equations

drd�

�V�x, t�

�,

dt

d��

R�x, t�

�, (6.2)

with � � �R2 � V12 � V2

2 � V32, which are subject to the initial condition

FIG. 3. 2-D Gaussian pulse: (a, b) Initial condition, t � 0, max � 1. Diffusion � 0.0 or 0.0001. (c, d)Analytical solution with diffusion � 0.0001 at t � /2, max � 0.8642.

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 295

r���t� � r�t�; x� , t�� � x� , t���t� � rt�t�; x� , t�� � t�. (6.3)

After combining the advection term and the time derivative term in the governing equation in(6.1) into a directional derivative term

�u

���

R�x, t�

�

�u

�t�

V�x, t�

�� �u, (6.4)

we can follow the steps as in (2.7)–(2.10) to obtain the following semi-discretized Eulerian-Lagrangian approximation scheme in three dimensional space

FIG. 4. 2-D Gaussian pulse: (a, b) Computed solution with diffusion � 0.0 at t � /2, max � 1.0029.(c, d) Computed solution with diffusion � 0.0001 at t � /2, max � 0.8647.

TABLE I. Comparison of maximum values for different methods.

Method t Maximum Figure Method t Maximum Figure

Analytical N/A 0.8642 3(c,d) ELSCM /40 0.8647 4(c,d)ELLAM-Er /40 0.8302 ELLAM-RK /30 0.8630CN-FDM /240 0.8699 CN-CPG /40 0.8555

296 WU AND WANG

R�x, tn�u�x, tn� � u�x*, t*n�

t*n�x�� � � �D�u�x, tn�� � f �x, tn�. (6.5)

B. An Eulerian-Lagrangian Single-Node Collocation Method in Three-DimensionalSpace

We define an orthogonal partition on the spatial domain � as

x0 � x1 � · · · � xi � · · · � xI�1 � xI,

y0 � y1 � · · · � yj � · · · � yJ�1 � yJ,

z0 � z1 � · · · � zk � · · · � zK�1 � zK. (6.6)

A conventional collocation method for second-order elliptic or parabolic problems usespiecewise-cubic Hermite polynomials as basis functions, leading to the trial function U(x, tn) ofthe form [1, 14, 15]

U�x, tn� � �i,j,k

�Ui,j,kn �i,j,k�x� �

�U

�x �i,j,k

n

�i,j,kx �x� �

�U

�y �i,j,k

n

�i,j,ky �x� �

�U

�z �i,j,k

n

�i,j,kz �x� �

�2U

�x�y�i,j,k

n

�i,j,kxy �x�

��2U

�y�z�i,j,k

n

�i,j,kyz �x� �

�2U

�x�z�i,j,k

n

�i,j,kxz �x� �

�3U

�x�y�z�i,j,k

n

�i,j,kxyz �x�� (6.7)

Here Ui, j,kn � U(xi, yj, zk, tn), �i, j,k(x) � �i(x)�j(y)�k(z), �i, j,k

x (x) � i(x)�j(y)�k(z), and�i, j,k

yz (x) � �i(x)j(y)k(z). Others are defined similarly. In U(x, tn), Ui, j,kn , �U/�x�i, j,k

n , �U/�y�i, j,kn ,

�U/�z�i, j,kn , �2U/�x�y�i, j,k

n , �2U/�y�z�i, j,kn , �2U/�x�z�i, j,k

n , and �3U/�x�y�z�i, j,kn are unknowns. Thus,

after incorporating the boundary conditions in (6.1), this scheme would require eight collocationpoints within each cell Ei, j,k, and 8(I � J � K ) total collocation points on the entire domain �to close the system at time tn.

In order to eliminate all the derivatives as unknowns in the trial function (6.7), we replace thestandard piecewise tri-cubic Hermite polynomial trial function by the following expression:

U�x, tn� � �i,j,k

�Ui,j,kn �i,j,k�x� � �xUi,j,k

n �i,j,kx �x� � �yUi,j,k

n �i,j,ky �x� � �zUi,j,k

n �i,j,kz �x� � �xyUi,j,k

n �i,j,kxy �x�

� �yzUi,j,kn �i,j,k

yz �x� � �xzUi,j,kn �i,j,k

xz �x� � �xyzUi,j,kn �i,j,k

xyz �x��. (6.8)

Here �x is a fourth-order finite-difference approximations to the corresponding first-order spatialderivatives given by

�xUi,j,kn �

1

12x�Ui�2,j,k

n � 8Ui�1,j,kn � 8Ui�1,j,k

n � Ui�2,j,kn �, (6.9)

�y and �z have similar expressions. �xy is a simplified version of the product of �x and �y,

�xyUi,j,kn �

1

4xy Ui�1,j�1,k

n � Ui�1,j�1,kn � Ui�1,j�1,k

n � Ui�1,j�1,kn �, (6.10)

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 297

so are �yz and �xz. A practical version of the triple produce �xyz in use has the form

�x�y�zU�xi, yj, zk, tn� �1

8xyz

� U�xi�1, yj�1, zk�1, tn� � U�xi�1, yj�1, zk�1, tn�� U�xi�1, yj�1, zk�1, tn� � U�xi�1, yj�1, zk�1, tn�� U�xi�1, yj�1, zk�1, tn� � U�xi�1, yj�1, zk�1, tn�� U�xi�1, yj�1, zk�1, tn� � U�xi�1, yj�1, zk�1, tn��. (6.11)

The expression (6.8) involves only function values Ui, j,kn as unknowns, thus on each cell, only

one collocation point is needed, which is the center point on each cell. After incorporating theboundary conditions in (6.1), we average every eight adjacent equations where we set up at thosecollocation points that share a common vertex to get rid of the over-determinant system, weobtain the following Eulerian-Lagrangian single-node collocation scheme for the three-dimen-sional advection-diffusion equation (6.1),

1

8 �m�i

i�1 �q�j

j�1 �l�k

k�1 �R�x, tn�U�x, tn� � U�x*, t*n�

t*n�x�� � � �D�x, tn��U�x, tn���

xm,q,l

�1

8 �m�i

i�1 �q�j

j�1 �l�k

k�1

�f �x, tn��xm,q,l, 1 � i � I � 1, 1 � j � J � 1, 1 � k � K � 1. (6.12)

(6.12) is the general form of the ELSCM method. For nodes close to the boundaries, itrequires to handle the difference quotients, such as (6.9) and (6.10), by different treatments aspresented in section 4.

C. Numerical Experiments

In this part, we carry out three-dimensional numerical experiments to observe the performanceof ELSCM. This numerical example is similar to the second testing problem in section 5, exceptthat the pulse is a three-dimensional Gaussian pulse, its moving path is a spiral, in xy-direction,it moves along a circle, in z direction, it moves at a constant speed.

The spatial domain of interest is [0, 1] � [0, 1] � [0, 1]. We move the pulse along the spiralpath: x � 1

2� 1

4cos(4t), y � 1

2� 1

4sin(4t), z � 1

4� (5/8 )t, then the rotating field is imposed as

V1(x, t) � sin(4t), V2(x, t) � �cos(4t), V3 � Vz � 5/8 ; the time interval is [0, T] � [0, /2],which is the time period required for a full revolution only on the xy-plane. The initial conditionis given as

u0�x� � exp���x �

14�2 � �y �

12�2 � �z �

14�2

2�2 �. (6.13)

When the diffusion term D(x, t) � 0 and the source term f (x, t) � 0, the analytical solutionis just a translation of the initial condition. In the numerical experiments, the remainingparameters are chosen as follows: � � 0.0447, which gives 2�2 � 0.004; x � y � z � 1/64;z0 � 0.25, t � /40, and T � /2. The contour plots of this pulse at different times are

298 WU AND WANG

presented in Fig. 5. Although we put these four plots together in the same figure, they are, infact, located on four different z-planes, and each pulse keeps its maximum as or very close to1.0.

We also run this test with small diffusion term, D(x, t) � 0.0001. Other parameters remainthe same as before. The contour plots and the maximum values of the pulse at t � T/4, T/2, 3T/4,T are presented in Fig. 6. Still these four contour plots are for the pulses on different z-planes,according to the time changes.

FIG. 5. Contour plots with diffusion � 0.0 at (t, z) � (T/4, z0 � VzT/4), (T/2, z0 � VzT/2), (3T/4, z0

� 3VzT/4), (T, z0 � VzT ), where T � /2, z0 � 0.25, Vz � 5/8 , x � y � z � 1

64, t � /40 � T/20.

FIG. 6. Contour plots with diffusion � 0.0001 at (t, z) � (T/4, z0 � VzT/4), (T/2, z0 � VzT/2), (3T/4, z0

� 3VzT/4), (T, z0 � VzT ), where T � /2, z0 � 0.25, Vz � 5/8 , x � y � z � 1

64, t � /40 � T/20.

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 299

VII. CONCLUSION AND FUTURE WORK

In this article, we have developed a nonconventional Eulerian-Lagrangian single-node colloca-tion method with piecewise Hermite polynomials for the numerical simulation of transientadvection-diffusion transport equations in two- and three-dimensional spaces. The ELSCMgreatly reduces the number of unknown variables in conventional collocation methods andallows the use of large time steps in numerical simulations to generate accurate numericalsolutions, to maintain the stability of the numerical method, and to avoid severe nonphysicaloscillation or excessive numerical diffusion. The ELSCM scheme is relatively easy to formulate.Numerical experiments conducted in this article show that the ELSCM scheme generatesaccurate numerical solutions without noticeable numerical diffusion or severe nonphysicaloscillations.

Future work includes the study on how to handle other types of boundary conditions andinvestigation on applying the ELSCM scheme to nonlinear problems and coupled pressure andtransport systems.

The authors thank Dr. George F. Pinder for his contribution and influence in the developmentof the single-node collocation method.

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