A Nonconventional Eulerian-LagrangianSingle-Node Collocation Method withHermite Polynomials for Unsteady-StateAdvection-Diffusion EquationsLi Wu,1 Hong Wang,2 George F. Pinder3

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

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

3Department of Civil and Environmental Engineering, University of Vermont,Burlington, Vermont 05405

Received 5 July 2002; accepted 5 September 2002

DOI 10.1002/num.10047

We developed a nonconventional Eulerian-Lagrangian single-node collocation method (ELSCM) withpiecewise-cubic Hermite polynomials as basis functions for the numerical simulation to unsteady-stateadvection-diffusion transport partial differential equations. This method greatly reduces the number ofunknowns in the conventional collocation method, and generates accurate numerical solutions even if verylarge time steps are taken. The method is relatively easy to formulate. Numerical experiments are presentedto show the strong potential of this method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial DifferentialEq 19: 271–283, 2003

Keywords: advection-diffusion equations; characteristic methods; collocation method; convection-diffu-sion equations; Eulerian-Lagrangian methods; Hermite polynomials

I. INTRODUCTION

Advection-diffusion partial differential equations describe transports of solute in groundwaterand surface water, processes of pollution spreading, displacements of oil by invading fluid in

Correspondence to: Li Wu, Department of Mathematics, University of Rhode Island, 9 Greenhouse Road, Suite 3,Kingston, RI 02881-0816 (e-mail: liwu@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 InitiativeGrantContract grant sponsor: National Science Foundation; contract grant number: DMS 0079549

© 2003 Wiley Periodicals, Inc.

petroleum recovery, and miscible fluid flow precesses in many other important applications [1,2]. These equations admit solutions with moving steep fronts and complicated structures, whereimportant physical and chemical phenomena take place. In practice, most reasonable numericalmethod performs well when physical diffusion dominates the transport process. However, manydifficult problems arise from the numerical simulation of these equations when advectiondominates the transport process. Space- and time-centered methods tend to generate numericalsolutions with nonphysical oscillations. Although some methods, such as spatially upwindedschemes, can suppress oscillations dramatically, they often generate solutions with excessivenumerical diffusion and severe damping of the moving steep fronts. Hence important informa-tion in physics could not be revealed by the numerical simulations. The development of decentnumerical methods for seeking accurate and efficient numerical solutions of advection-domi-nated transport equations has been a challenging task.

Collocation methods were first investigated in [3, 4] for the one-dimensional parabolicequations. These methods attempt to minimize the residual in the variational formulation or theweighted residual methods for the governing differential equations by forcing the residual to bezero at a finite number of discrete or collocation points within the physical domain. Thesemethods are attractive because they are theoretically simple and exhibit a small discretizationerror. Consequently, collocation methods were extended subsequently in various ways tonumerically simulate a wide variety of problems [1, 5–8]. Recently, Wu and Pinder [9]introduced a single-node strategy to simplify the original two-degree collocation methods thatuse Hermite polynomials as basis functions. This modification involves the replacement of thefirst-order space derivative in the trial functions by its finite-difference quotient approximations,the averaging of the discrete equations at consecutive nodes, and the upwinded approximationto the advection terms. Computational experiments have shown the potential of the single-nodecollocation methods.

Most collocation methods fall into a larger class of Eulerian methods. Eulerian methods usea fixed spatial grid and the standard temporal discretization in the time direction. They attemptto minimize the error in approximating spatial derivatives in advection-diffusion equations andoften yield an upstream bias in the resulting numerical schemes. Thus, they are easily influencedby the time truncation errors that cause numerical diffusion and restrictions on the Courantnumber. In other words, Eulerian methods have difficulties in simulating all of the waveinteractions that take place if important information propagates more than one cell per time stepeither due to the reason of stability or accuracy. Consequently, they tend to be ineffective fortransient advection-dominated problems so as to they require small time steps in numericalsimulations, so that the CFL condition is satisfied. The Courant number in these simulationsoften has to be much smaller than one [10, 11]. Finally, in the context of advection-diffusionequations, those discrete algebraic systems resulting from the Eulerian methods often lose thecoercivity (unless the grids are extremely refined which is impossible in practice). This explainsfrom another point of view why many numerical methods that work very well for elliptic andparabolic problems tend to perform poorly for advection-diffusion equations.

By using a characteristic tracking, Eulerian-Lagrangian methods follow the movement ofinformation or particles as well as their interactions along the characteristics. Algorithmically,Eulerian-Lagrangian methods solve the advection term by a characteristic tracking algorithmand treat the diffusion term on a fixed Eulerian grid separately. Because the solutions of thegoverning differential equations are much smoother along the characteristics than they are in thetime direction, these methods have significantly reduced truncation errors compared with theEulerian treatments, have generated accurate numerical solutions even if large time steps areemployed, and have relaxed restriction on the Courant number in Eulerian methods. Although

272 WU ET AL.

Eulerian-Lagrangian methods have many advantages, such as the accuracy and the efficiency,they usually require extra implementational effort and raise many implementational and ana-lytical issues that need to be addressed.

In this paper, based on the single-node collocation method in [9] and the Eulerian-Lagrangianconcept, we developed a nonconventional Eulerian-Lagrangian single-node collocation method(ELSCM) for unsteady-state advection-diffusion transport partial differential equations. The restof this paper is organized as follows: in section II, we present a Eulerian-Lagrangian semi-discretized approximation. In section III, we derive the nonconventional Eulerian-Lagrangiansingle-node collocation method with the piecewise-cubic Hermite polynomials. In section IV,we address several numerical issues and concerns. In section V, we carry out several numericalexperiments to observe the performance of the ELSCM scheme. In the last section, section VI,we summarize the results in this paper and discuss directions of the future work.

II. A EULERIAN-LAGRANGIAN SEMI-DISCRETIZED APPROXIMATION TOADVECTION-DIFFUSION EQUATIONS

In sections II–IV of this article, we focus on the following initial-boundary value problem forthe one-dimensional unsteady-state advection-diffusion equation in the development of theELSCM scheme:

R�x, t��u

�t� V�x, t�

�u

�x�

�

�x �D�x, t��u

�x� � f�x, t�, �x, t� � �a, b� � �0, T�,

u�a, t� � g1�t�, u�b, t� � g2�t�, t � �0, T�, (2.1)

u�x, 0� � u0�x�, x � �a, b�.

In Eq. (2.1), the nomenclature is such that R(x, t) is a retardation coefficient; V(x, t) is the fluidvelocity; D(x, t) is a diffusion coefficient; f(x, t) is a given source or sink function; and u(x, t),the unknown function, represents a measure of the concentration of a dissolved substance insubsurface flow. u0(x), g1(t), and g2(t), are initial and boundary data that are needed to close thesystem.

The Eulerian-Lagrangian method uses a time-marching algorithm. Let N be a positiveinteger. We define a partition on the 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 [12] andthe Eulerian-Lagrangian localized adjoint method [10, 11, 13, 14] to combine the advection termand the time derivative in the governing Eq. (2.1) into a directional derivative term as

R�x, t��u

�t�x, t� � V�x, t�

�u

�x�x, t� � �R2�x, t� � V2�x, t�

�u

���x, t�. (2.3)

along the characteristic x � r(�; x�, t�) defined by

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 273

dx

d��

V�x, t�

�R2�x, t� � V2�x, t�,

dt

d��

R�x, t�

�R2�x, t� � V2�x, t�,

x���t � r�t�; x� , t�� � x� . (2.4)

By (2.3) and (2.4), we rewrite the governing advection-diffusion equation in (2.1) as aparabolic equation along the characteristics:

�R2�x, t� � V2�x, t��u

���x, t� �

�

�x �D�x, t��u

�x�x, t�� � f�x, t�. (2.5)

To derive a numerical scheme, we approximate the characteristic curve x � r(�; x, tn) passingthrough (x, tn) by a Euler quadrature in the time-stepping procedure. Let t*n(x) � tn�1 if r(�; x,tn) does not backtrack to the boundary during the time period [tn�1, tn], or t*n � t*n(x) � [tn�1,t] is the time instant when r(�; x, tn) intersects the boundary otherwise. We also let

�t*n�x� � tn � t*n�x�, x* � x �V�x, tn�

R�x, tn��t*n�x�. (2.6)

Figure 1 illustrates the connection between two consecutive time steps.We approximate the directional derivative (�u/��)(x, tn) at time tn by the backward difference

quotient along the approximate characteristic [10, 12, 14] as

�R2�x, tn� � V2�x, tn��u

���x, tn� � �R2�x, tn� � V2�x, tn�

u�x, tn� � u�x*, t*n�

���t*n�x��2 � �x � x*�2

FIG. 1. The connection between two consecutive time steps.

274 WU ET AL.

� E1�u�x, tn��

� R�x, tn�u�x, tn� � u�x*, t*n�

�t*n�x�� E1�u�x, tn��, (2.7)

where E1(u(x, tn)) is the resulting local truncation error. We then obtain a semi-discretizedapproximate equation to (2.5) as

R�x, tn�u�x, tn� � u�x*, t*n�

�t*n�x��

�

�x �D�x, tn��u

�x�x, tn�� � f�x, tn�. (2.8)

III. AN NONCONVENTIONAL EULERIAN-LAGRANGIAN SINGLE-NODECOLLOCATION METHOD USING THE PIECEWISE-CUBIC HERMITEPOLYNOMIALS

We notice that the semi-discretized approximate Eq. (2.8) is in the form of a parabolic equationalong the approximate characteristics and so it retains the coercivity. Hence, any method thatperforms well for an elliptic or parabolic equation should potentially work well for (2.8). Let Ibe a positive integer. We define a spatial partition on [a, b] by

a � x0 � x1 � x2 � · · · � xi � · · · � xI�1 � xI � b. (3.1)

A. A Brief Overview of a Conventional Collocation Method

A conventional collocation method for second-order problems uses piecewise-cubic Hermitepolynomials as basis functions, leading the trial function U(x, tn) to the form

U�x, tn� � �i�0

I �U�xi, tn��i�x� ��U

�x�xi, tn�i�x�� , (3.2)

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

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

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

with ij � (xi � xj) being the Dirac- function.If we incorporate Eq. (3.2) into Eq. (2.8) and enforce the resulting equation at some

collocation points X � [xi�1, xi] for i � 1, 2, . . . , I, we would obtain a Eulerian-Lagrangiancollocation scheme. At the time step tn, this scheme would look like a standard collocationscheme for a parabolic problem and would have both U(xi, tn), (�U/�x)(xi, tn) as unknowns ateach node. Thus, after using the boundary conditions in (2.1), this scheme would require 2Icollocation points on interval (a, b) to close the system at each time level. Similarly, atwo-dimensional analogue of the collocation scheme would need four collocation points withineach cell [xi�1, xi] [yj�1, yj] to determine the function values U(xi, yj, tn), (�U/�x)(xi, yj, tn),(�U/�y)(xi, yj, tn), and (�2U/�x�y)(xi, yj, tn) at each interior node. And a corresponding scheme

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 275

for three-dimensional problems would require employing eight collocation points on each cubiccell.

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

To further improve the computational efficiency of the Eulerian-Lagrangian collocation scheme,we apply a single-node collocation method which uses only nodal values as unknowns, Hermitepolynomials as basis functions, and perserves 4th order convergence rate as described in [9].Still using piecewise-cubic Hermite polynomials, in this section, we develop a nonconventionalEulerian-Lagrangian single-node collocation method. For simplicity of expression, we assumea uniform spatial partition in (3.1).

In order to eliminate the first-order spatial derivative (�U/�x)(xi, tn) as a unknown at eachnode in the trial function (3.2), we replace it by its fourth-order finite difference quotient:

�U

�x�xi, tn� � xU�xi, tn�

U�xi�2, tn� � 8U�xi�1, tn� � 8U�xi�1, tn� � U�xi�2, tn�

12�x. (3.4)

Consequently, we obtain the trial function of the form:

U�x, tn� � �i�0

I

�U�xi, tn��i�x� � xU�xi, tn�i�x��. (3.5)

The expression (3.5) involves only the function values U(xi, tn) for i � 0, 1, . . . , I and thusallows the use of one collocation point on each subinterval to match the number of unknowns.Hence, on elements [xi�1, xi], we choose Xi � (xi�1 � xi)/2 as the collocation point for i � 1,2, . . . , I. We incorporate the expression (3.5) into Eq. (2.8) and collocate the resulting equationat each collocation point Xi for i � 1, 2, . . . , I. Using the notation [ f ]X � �a

b f(y)(y � X)dy �f(X), we obtain a collocation scheme:

�R�x, tn�U�x, tn� � U�x*, t*n�

�t*n�x��

�

�x �D�x, tn��U

�x�x, tn���

Xi

� �f�x, tn��Xi(3.6)

for i � 1, 2, . . . , I.However, the expression (3.5) involves I � 1 unknowns U(xi, tn) for i � 0, 1, . . . , I. After

applying the boundary conditions in (2.1), only I � 1 equations are needed to close the system.Thus, the scheme (3.6) is an over-determined system. To overcome this difficulty, we averagethe collocation Eq. (3.6) at every two consecutive collocation points Xi and Xi�1 to obtain ascheme with I � 1 equations:

1

2 �R�x, tn�U�x, tn� � U�x*, t*n�

�t*n�x��

�

�x �D�x, tn��U

�x�x, tn���

Xi

� �R�x, tn�U�x, tn� � U�x*, t*n�

�t*n�x��

�

�x �D�x, tn��U

�x�x, tn���

Xi�1

�

276 WU ET AL.

�1

2 �f�x, tn��Xi

� �f�x, tn��Xi�1�, i � 1, 2, . . . , I. (3.7)

IV. COMPUTATIONAL ISSUES

In this section, we address some computational issues and concerns arising in the numericalimplementation of the Eulerian-Lagrangian single-node collocation scheme (3.7).

A. Characteristic Tracking

In section II, we considered a first-order Euler tracking algorithm:

r��; x� , t�� � x� � �� � t��V�x� , t��

R�x� , t��(4.1)

in the derivation of Eq. (2.8) for simplicity. It is well known that the accuracy of a trackingalgorithm has a very significant impact on the accuracy of a Eulerian-Lagrangian scheme [10,11]. To improve the accuracy of the characteristic tracking, we can use higher-order trackingalgorithms. For example, we can use a second-order Runge-Kutta tracking algorithm:

r��; x� , t�� � x� ��� � t��

2 �V�x� , t��

R�x� , t���

V�r1��, x� , t��, ��

R�r1��, x� , t��, ��� , (4.2)

with r1(�; x�, t�) given by (4.1). Alternatively, we can use a fourth-order Runge-Kutta algorithm:

r��; x� , t�� � x� �16

�K1��; x� , t��, � 2K2��; x� , t�� � 2K3��; x� , t�� � K4��; x� , t���,

(4.3)

with

K1��; x� , t�� � �� � t��V�x� , t��

R�x� , t��

K2��; x� , t�� � �� � t��V�x� �

12

K1��; x� , t��, t� �12

�� � t���

R�x� �12

K1��; x� , t��, t� �12

�� � t���

K3��; x� , t�� � �� � t��V�x� �

12

K2��; x� , t��, t� �12

�� � t���

R�x� �12

K2��; x� , t��, t� �12

�� � t���

K4��; x� , t�� � �� � t��V�x� � K3��; x� , t��, ��

R�x� � K3��; x� , t��, ��. (4.4)

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 277

We can also combine these algorithms with the use of micro time steps in the characteristictracking to further improve the accuracy of a tracking algorithm [10].

B. Evaluation of U(x*, tn�1)

In this subsection, we address the issue of the evaluation of U(x*, tn�1). It has been shown thatthe evaluation of the term involving U(x*, tn�1) in a Eulerian-Lagrangian method is problematicin practice [15]. Without careful implementation, a nonexact integration of the term involvingU(x*, tn�1) in the context of finite element or finite volume setting may cause numericaloscillations and other problems. In a Eulerian-Lagrangian collocation method, the situationcould be even worse partly because of the following reasons: (i) a Eulerian-Lagrangiancollocation method uses only the value U(X*i, tn�1) as a representative value for the elementintegral with U(x*, tn�1) arising in the finite element or finite volume methods; (ii) the foot X*iat time tn�1 (see Fig. 1), which is obtained via a characteristic tracking from Xi at time tn, couldhappen to be any point at time tn�1.

To overcome this difficulty, we follow the idea of the Godunov scheme [16] to use cellaverage values to define an approximation U(X*i, tn�1) to U(X*i, tn�1), which minimizes oreliminates the possible oscillations presented in U(X*i, tn�1). More precisely, suppose that X*i fallsinto the interval [Xk�1, Xk], we define the cell-averages

U�Xk�1, tn�1� �1

�x �Xk�1�

12�x

Xk�1�12�x

U�y, tn�1�dy,

U�Xk, tn�1� �1

�x �Xk�

12�x

Xk�12�x

U�y, tn�1�dy. (4.5)

Then, we define U(X*i, tn�1) to be the linear interpolation of U(Xk�1, tn�1) and U(Xk, tn�1). Inthe Eulerian-Lagrangian single-node collocation scheme (3.7), we replace U(X*i, tn�1) by U(X*i,tn�1).

To improve the accuracy of U(X*i, tn�1), we can apply the slope limiters in the total variationdiminishing (TVD) methods to define a piecewise-linear approximation U(X*k�1, tn�1) andU(X*k, tn�1), as in the MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws)and Minmod methods [17–20]. We can also apply the techniques in the essentially nonoscil-latory (ENO) schemes and weighted ENO schemes [21–24] to define higher-order approxima-tions to U(X*i, tn�1) in the ELSCM scheme (3.7).

C. Boundary Conditions and the Matrix Structure

In this section, we point out the strategy for handling boundary conditions.In the trial function U(x, tn) defined in (3.5), xU(xi, tn) is used to replace (�U/�x)(xi, tn) in

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

278 WU ET AL.

�U

�x�x0, tn� � xU�x0, tn�

�25U�x0, tn� � 48U�x1, tn� � 36U�x2, tn� � 16U�x3, tn� � 3U�x4, tn�

12�x(4.6)

�U

�x�x1, tn� � xU�x1, tn�

�3U�x0, tn� � 10U�x1, tn� � 18U�x2, tn� � 6U�x3, tn� � U�x4, tn�

12�x(4.7)

Both quotients in (4.6) and (4.7) are 4th order approximations to corresponding derivatives.Lower-order quotient approximations can also be applied for derivatives. Similar approxima-tions can be derived for nodes close to right hand boundary of the domain.

Introducing xU(xi, tn) to the numerical scheme reduces number of unknowns, but at nodexi, its neighboring nodes xi�2, xi�1, xi�1, and xi�2 are used. Meanwhile, averaging consecutiveequations as in (3.7) recruits more nodes around xi. The matrix we derived from (3.7) is a sparse,diagonally dominated matrix of bandwidth 7. This is the computational cost to pay in exchangefor the reduction of number of unknowns, which is acceptable for one-dimensional problems.However, this could bring disadvantages for multidimensional problems. In this case, furtherwork has to be done on recruiting fewer nodes in each equation to reduce the bandwidth of thematrix. This will be addressed elsewhere.

V. NUMERICAL EXPERIMENTS

In this section, we carry out two numerical experiments to observe the performance of thenonconventional Eulerian-Lagrangian single-node collocation method developed in this paper.For each example, we consider an advection-dominated case and a pure advection case. In eachcase, we present the numerical solutions at different time steps.

A. A Shifting Gaussian Pulse

The first example considers the transport of a Gaussian Hill over the spatial domain [0, 1]. Theinitial condition is given by

u0�x� � exp���x � x0�

2

2�2 �, (5.1)

where the centered deviation x0 � 0.25 and the standard deviation � � 0.0447, whichaccordingly determines the steepness of the Gaussian hill and gives 2�2 � 0.004. When thecoefficient D is a constant, V(x, t) � 0.5(1 � 0.1x), the source function f(x, t) can be chosen suchthat the analytical solution corresponding to (5.1) has the form:

u�x, t� � � 2�2

2�2 � 4Dtexp��

�x � x0 � V�x, t�t�2

2�2 � 4Dt �. (5.2)

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 279

Other parameters are: mesh size �x � 0.01, the time step �t � 1/12, and the diffusioncoefficient D(x, t) � 0.0001 or 0. In Fig. 2(a,b), we present the numerical solutions (dashedlines) against the analytical solutions (solid lines) at t � 0.5 and t � 1.0, along with the initialcondition at t � 0, with D � 0.0001, and D � 0, respectively. Figure 2(a) shows that whendiffusion is present, the Gaussian pulse moves to the right while diffusing slightly. Figure 2(b)shows that the initial pulse moves horizontally to the right without noticeable numericaldiffusion. In both cases, the shapes of the Gaussian pulse at different times are well preservedby the numerical solutions.

In particular, if V(x, t) is a constant, u(x, t) is the analytical solution for (2.1) with f � 0. Inthe numerical experiments in Fig. 3, we use a mesh size of �x � 0.01 and a time step size of�t � 1/8, and a constant velocity of V(x, t) � 0.5. This gives a non-integer Courant number of6.25. We present the numerical solutions for D � 0 and D � 0.0001 in Figure 3(a) and 3(b),respectively. The numerical solutions in Figure 3(a) corresponds to a diffusion coefficient ofD � 0.0. We observe that during the entire transport process, the numerical solution to the pulsepreserves the maximum value, does not generate any noticeable nonphysical oscillation ornumerical diffusion. When a diffusion of D � 0.0001 is present, the maximum values of theexact solutions are 0.9759 at t � 0.5 and 0.9535 at t � 1, respectively. The computed solutionsgive the maximum � 0.9763 at t � 0.5, and 0.9542 at t � 1.0, as shown in Fig. 3(b). We noticethat the numerical solutions in Fig. 2 introduce some errors. We believe that these errors are due

FIG. 2. Gaussian pulse: (a) Diffusion � 0.0001. (b) Diffusion � 0.0.

FIG. 3. Gaussian pulse: (a) Diffusion � 0.0 (b) Diffusion � 0.0001.

280 WU ET AL.

those in the characteristic tracking, which show the impact of an accurate tracking of charac-teristics on numerical solutions.

B. A Moving Sharp Front

To observe the impact of the boundary conditions on the numerical solutions generated by theEulerian-Lagrangian single-node collocation method, we consider the transport of a movingsharp front. The data in this example is chosen as follows: the velocity is V(x, t) � (�/4)cos[(�/2)t]; the diffusion term D(x, t) � 0.0001 or 0; the space domain is � � [0, 1] with a spatial gridsize of �x � 0.01; and the time step is �t � 1/6. This again yields a noninteger Courant number.The initial condition and boundary conditions are given by

u0�x� � 1, if x � �0, 0.1�,0, otherwise.

g1�t� � 1, g2�t� � 0. (5.3)

The numerical solutions are plotted at t � 0.5 and t � 1.0, along with the initial curve (t � 0),in Fig. 4(a,b), for D � 0.0001 and D � 0, respectively. We observe that the numerical solutionsretain the moving steep fronts without introducing noticeable nonphysical oscillations ornumerical diffusion.

VI. CONCLUSION AND FUTURE WORK

In this article, we have developed a nonconventional Eulerian-Lagrangian single-node colloca-tion method with the piecewise-cubic Hermite polynomials for the numerical simulation tounsteady-state advection-diffusion transport equations. The method greatly reduces the numberof unknown variables in the conventional collocation method and allows the use of large timesteps in numerical simulations to generate accurate numerical solutions, to maintain the stabilityof the numerical method, and to avoid severe nonphysical oscillation or excessive numericaldiffusion. The ELSCM scheme is relatively easy to formulate and is very competitive with manywell perceived numerical methods. Numerical experiments conducted in this article show that

FIG. 4. Sharp front: (a) Diffusion � 0.0001; (b) diffusion � 0.0.

A NEW EULERIAN-LAGRANGIAN COLLOCATION METHOD 281

the ELSCM scheme generates accurate numerical solutions without noticeable numericaldiffusion or severe nonphysical oscillations.

Future work includes the study on extension to multidimensional problems, how to handleother types of boundary conditions, and investigation on applying the ELSCM scheme inrealistic applications for the numerical simulation of coupled pressure and transport systems.

References

1. M. B. Allen and C. L. Murphy, A finite-element collocation method for variably saturated flow in twospace dimensions, Water Resour Res 22 (1986), 1537–1542.

2. J. Bear, Hydraulics of Groundwater, New York, McGraw-Hill, 1979.

3. J. H. Cerutti and S. V. Parter, Collocation methods for parabolic partial differential equations in onespace dimension, Numer Math 26 (1976), 227–254.

4. J. Douglas and T. Dupont, A finite element collocation method for quasilinear parabolic equations,Math Comp 27 (1973), 17–28.

5. B. Bialecki and G. Fairweather, Orthogonal spline collocation methods for partial differential equa-tions, Numerical Analysis 2000, Vol. VII, Partial differential equations, J Comput Appl Math 128(2001), 55–82.

6. M. A. Celia and G. F. Pinder, Generalized alternating-direction collocation methods for parabolicequations. II. Transport equations with application to seawater intrusion problems, Numer MethodsPartial Differential Eq 6 (1990), 215–230.

7. C. E. Greenwell-Yanik and G. Fairweather, Analyses of spline collocation methods for parabolic andhyperbolic problems in two space variables, SIAM J Numer Anal 23 (1986), 282–296.

8. P. Percell and M. F. Wheeler, An C1 finite element collocation method for elliptic equations, SIAMJ Numer Anal 17 (1980), 605–622.

9. L. Wu and G. F. Pinder, Single-degree freedom collocation method using Hermite polynomials,Contemp Math 295 (2002), 489–499.

10. H. Wang, H. K. Dahle, R. E. Ewing, M. S. Espedal, R. C. Sharpley, and S. Man, An ELLAM schemefor advection-diffusion equations in two dimensions, SIAM J Sci Comput 20 (1999), 2160–2194.

11. H. Wang, R. E. Ewing, G. Qin, S. L. Lyons, M. Al-Lawatia, and S. Man, A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations, J ComputPhys 152 (1999), 120–163.

12. J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problemsbased on combining the method of characteristics with finite element or finite difference procedures,SIAM J Numer Anal 19 (1982), 871–885.

13. M. A. Celia, T. F. Russell, I. Herrera, and R. E. Ewing, An Eulerian-Lagrangian localized adjointmethod for the advection-diffusion equation, Adv Water Res 13 (1990), 187–206.

14. H. Wang, R. E. Ewing, and T. F. Russell, Eulerian-Lagrangian localized methods for convection-diffusion equations and their convergence analysis, IMA J Numer Anal 15 (1995), 405–459.

15. K. W. Morton, A. Priestley, and E. Suli, Stability of the Lagrangian-Galerkin method with nonexactintegration, RAIRO M2 AN 22 (1988), 123–151.

16. S. K. Godunov, A difference scheme for numerical computation of discontinuous solutions of fluiddynamics, Mat Sb 47 (1959), 271–306.

17. P. Colella, A direct Eulerian MUSCL scheme for gas dynamics, SIAM J Sci Stat Comput 6 (1985),104–117.

18. J. B. Goodman and R. J. LeVeque, A geometric approach to high resolution TVD schemes, SIAM JNumer Anal 25 (1988), 268–284.

282 WU ET AL.

19. B. van Leer, Towards the ultimate conservative difference scheme II. Monotonicity and conservationcombined in a second order scheme, J Comput Phys 14 (1974), 361–370.

20. B. van Leer, On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher,and Roe, SIAM J Sci Stat Comput 5 (1984), 1–20.

21. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order accurate essentiallynonoscillatory schemes, III, J Comput Phys 71 (1987), 231–241.

22. A. Harten and S. Osher, Uniformly high-order accurate non-oscillatory schemes, I, SIAM J NumerAnal 24 (1987), 279–309.

23. X.-D. Liu, S. Osher, and T. Chan, Weighted essentially nonoscillatory schemes, J Comput Phys 115(1994), 200–212.

24. C.-W. Shu, Essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO)schemes for hyperbolic conservation laws. A. Quarteroni, editor, Advanced Numerical Approximationof Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, New York, Springer-Verlag, Vol1697, 1997, pp 325–432.

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