An operator splitting method for nonlinear convection-diffusion equations

AN OPERATOR SPLITTING METHOD FORNONLINEAR CONVECTION-DIFFUSION EQUATIONSKenneth Hvistendahl Karlsen, Nils Henrik RisebroAbstract. We present a semi-discrete method for constructing approximate solutions to the initial value problemfor the m-dimensional convection-di�usion equation ut +r � f(u) = "�u. The method is based on the use of operatorsplitting to isolate the convection part and the di�usion part of the equation. In the casem > 1, dimensional splittingis used to reduce the m-dimensional convection problem to a series of one-dimensional problems. We show that themethod produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fullydiscrete method is analyzed, and demonstrated in the case of one and two space dimensions.1. Introduction. In this paper we present an operator splitting method for the scalar convection-di�usionequation(1.1) ut + mXi=1 fi(u)xi = " mXi=1 uxixi ; u (x1; : : : ; xm; 0) = u0 (x1; : : : ; xm) :For brevity of notation, we shall write this equation asut +r � f (u) = "�u; u(x; 0) = u0(x);where x = (x1; : : : ; xm), f (u) = (f1 (u) ; : : : ; fm (u)), r = (@=@x1; : : : ; @=@xm), and � =Pmi=1 @2=@x2i .Equations such as (1.1) arise in a variety of applications, ranging from models of turbulence [1], via tra�c ow [17], to two phase ow in porous media [19]. Equation (1.1) can also be viewed as a model problem for asystem of convection-di�usion equations, such as three phase ow in porous media [23], or the Navier-Stokesequations.Of particular importance is the case where convection dominates di�usion, i.e., " is small compared with otherscales in (1:1). This is often the case in models of two phase ow in oil reservoirs. Accurate numerical simulationof such models is consequently often complicated by both unphysical oscillations and numerical di�usion. Theoperator splitting approach presented here is especially well suited to the case where "� 1.The quasi-linear parabolic equation (1.1) was �rst properly analyzed by Oleinik in [18], where she provedexistence of a unique classical solution, and also that weak solutions to (1.1) coincide with classical solutions.Operator splitting, or fractional steps, methods have been extensively used in connection with conservationlaws, starting with Godunov [10], who used this method to study gas dynamics. Operator splitting (dimensionalsplitting) for a scalar conservation law in several space dimensions was studied by Crandall and Majda in [2],where they analyzed both a semi-discrete and a discrete method, and showed the convergence of the method forseveral numerical schemes. Holden and Risebro [12] studied dimensional splitting coupled with front tracking,and convergence rates for dimensional splitting methods were obtained in [22] and [13].If " � 1, then (1.1) is \almost hyperbolic", and it is natural to exploit this when constructing numericalmethods. This approach has indeed been taken by several authors, we only mention Douglas and Russell [5],[21], [8], Espedal and Ewing [6], [7], and more recently Dahle [4]. In [21] a characteristic element method isused to solve the hyperbolic part of (1.1). In [5], error estimates are obtained for a linear version of (1.1).The splitting method analyzed in this paper can be summarized as follows: Let v(x; t) = S(t)v0 be the uniqueentropy satisfying solution to vt +r � f (v) = 0; v(x; 0) = v0(x);and let w(x; t) = H(t)w0 be the solution ofwt = "�w; w(x; 0) = w0(x):1991 Mathematics Subject Classi�cation. 35L65.Key words and phrases. Convection-di�usion equations, operator splitting, front tracking, initial value problem.The research of the �rst author has been supported by VISTA, a research cooperation between the Norwegian Academy ofScience and Letters and Den norske stats oljeselskap a.s. (Statoil). Typeset by AMS-TEX1

2 KARLSEN, RISEBROOperator splitting is based on the following approximation(1.2) u(x; n�t) � [H(�t)S(�t)]n u0(x):Here we study the convergence properties as the time step �t! 0. In applications, the exact solution operatorsS and H are replaced by approximations. We use one-dimensional front tracking as de�ned by Dafermos [3], seealso [11], as an approximate solution operator for the hyperbolic part if m = 1. If m > 1 we use the dimensionalsplitting method described in [12] as an approximate solution operator for the hyperbolic part. In both one andseveral space dimensions a �nite element method is used as an approximate operator for the di�usion part. Wealso establish convergence of the approximate solutions in the fully discrete case.The rest of this paper is organized as follows: In section 2 we obtain compactness of the sequence of approx-imate solutions generated by (1.2), and show that any convergent subsequence converges to the unique classicalsolution of (1.1). In section 3, we show the same if the exact solution operators are replaced by the approxima-tions mentioned above. In section 4 we study applications of the method in one and two space dimensions, andpresent tentative convergence rates based on the numerical examples.2. The semi-discrete method. In this section we shall describe the operator splitting of (1.1), that is, wewill obtain the solution of (1.1) through a composition of the solution operator to a hyperbolic equation and toa parabolic equation.We study (1.1) for x = (x1; : : : ; xm) 2 Rm, where m � 1. Let therefore f (u) = (f1 (u) ; : : : ; fm (u)), whereu is a scalar. Furthermore, let S(t) be the operator which takes an initial function v0 (x) to the entropy weaksolution at time t of(2.1) vt +r � f (v) = vt + mXi=1 fi(v)xi = 0; v(x; 0) = v0(x);that is, we write the (weak) solution v(x; t) as S(t)v0 (x). We shall also need the solution operator taking theinitial data v0(x) to the entropy weak solution at time t of the one dimensional conservation law(2.2) vt + fi(v)xi = 0; v(x; 0) = v0(x):This operator we denote Sfi (t). Note that in (2.2), xj, j 6= i, act only as parameters. Following Kru�zkov [15],we know that (2.1) has a unique entropy weak solution if f is Lipschitz continuous, and the initial data are inL1TB:V:. Here, B:V: denotes the space of functions with bounded total variation. Similarly, let H(t) be thesolution operator (at time t) for the parabolic equation(2.3) wt = "�w = " mXi=1 wxixi ; w(x; 0) = w0(x):Now �x T > 0 and �t > 0, and let N be such that N�t = T . Let u0(x) = u0(x), and let(2.4) un+1(x) = �H(�t)Sfm (�t) : : :Sf1(�t)�un(x)for n = 0; : : : ; N � 1. Observe that by the results in [2], S(�t) � �Sfm (�t) : : :Sf1 (�t)�.We will now show that the sequence fung is compact in Lloc1 . To accomplish this we use a techniqueintroduced by Oleinik [18]. This approach is now standard in the theory of conservation laws, and involvesproving a priori uniform L1, B:V: and L1(x) continuity in time, bounds on the sequence. More concretely, weshow that the sequence fung is uniformly (in time) bounded in L1 and T:V: norms, with bounds independentof the discretization parameters. Then we appeal to Helly's theorem to conclude the existence of a subsequenceconverging in Lloc1 (x) to some function in B:V: (x) for each �xed t. By a diagonalization argument, we obtainconvergence for all t in some dense countable subset of [0; T ]. Convergence in Lloc1 (x) for all t is �nally obtainedby showing that each member of the convergent subsequence is L1 H�older continuous in time, i.e.,ZRm jum(x) � un(x)j dx � L (j(m � n)�tj)1=2 :To simplify the notation we de�ne Sf (t) = Sfm (t) : : :Sf1 (t):

AN OPERATOR SPLITTING METHOD 3We also note that if w0(x) is bounded, then the solution of (2.3) is given by convolution with the \heat kernel"K(x; t), where K(x; t) = 1(4�"t)m=2 exp ��jxj24"t � = k (x1; t) : : :k (xm; t)with k(x; t) given by k(x; t) = 1p4�"t exp ��x24"t � ;i.e., we can write(2.5) H(t)w0 (x) = ZRmK (x� y; t)w0(y) dy:Lemma 2.1. For n = 0; : : : ; N ,(2.6) jjunjj1 � jju0jj1:Proof. This follows from the fact that both Sf (t) and H(t) obey a maximum principle similar to (2.6). �Lemma 2.2. For n = 0; : : : ; N ,(2.7) T:V: (un) � T:V: (u0) :Proof. Recall that for a function g(x) its total variation is de�ned byT:V: (g) = lim suph!0 ZRjg(x+ h)� g(x)jh dx;and for a function g(x) of several variablesT:V: (g) = mXi=1 ZRm�1 T:V:xi (g) dm�1x;where T:V:xi(g) denotes the total variation of g with respect to the ith variable. We have that T:V: �Sfi (t)v0� �T:V: (v0) and that jjSfi(t)v0 � Sfi (t)v̂0jj1 � jjv0 � v̂0jj1:These two facts imply, see [12], that T:V: �Sf (t)v0� � T:V: (v0) :The lemma will follow by induction if we can show that(2.8) T:V: (H(t)w0) � T:V: (w0) :Let ei denote the unit vector in the ith direction. We calculatelim suph!0 ZRjw (x+ hei; t)� w(x; t)jh dxi = lim suph!0 ZRZRmK(y; t) jw0 (x+ hei � y)� w0(x � y)jh dy dxi� ZRmK(y; t)T:V:xi (w0) dy = T:V:xi (w0) :Hence, (2.8) holds. �

4 KARLSEN, RISEBROLemma 2.3. There is a constant C, independent of k, l and �t, such that(2.9) jjuk � uljj1 � Cp(k � l)�tfor all 0 � l � k � N .Proof. We begin by showing that the splitting solutions are weakly Lipschitz continuous in the time variable,that is,(2.10) ��ZRm �(x)�uk(x) � ul(x)� dx�� C1(k � l)�t�jj�jj1+maxi jj�xijj1�; 8� 2 C10 :Using �nite speed of propagation, we have that Sfi(t) (and thus Sf (t)) obeys a stronger condition than (2.10),namely ZR��Sfi (t+�t)v0(x) � Sfi (t)v0(x)�� dxi � C2�t:From this it follows that(2.11) ��ZRm �(x) �Sf (t +�t)v0(x)� Sf (t)v0(x)� dx�� C3�tk�k1:Using the di�erential equation for w(x; t) = H(t)w0(x) and integration by parts, we get the bound(2.12) ��ZRm �(x)�H(t+�t)w0(x) �H(t)w0(x)� dx�� = �� mXi=1 ZRm �(x) Z t+�tt "w(x; � )xixi d�! dx�� C4�tmaxi jj�xijj1;where we mention that C4 depends on T:V: (w0). Using (2.11) and (2.12), we now readily compute��ZRm �(x)�uk(x) � ul(x)� dx�� k�1Xi=l ��ZRm �(x)�ui+1(x) � ui(x)� dx�� C1(k � l)�t�jj�jj1 +maxi jj�xijj1�for some proper choice of constant C1.Finally, we claim that the estimate (2.9) follows from an approximation argument. To see this, let !h(x)be a standard m-dimensional C10 -molli�er with smoothing radius h. Let d(x) = uk(x) � ul(x), and de�ne�(x) = sign (d(x)) for jxj � r� h and �(x) = 0 for jxj > r� h, where r > h. Moreover, de�ne �h = !h � � andnote that �h 2 C1 with support in fjxj � rg. By choosing � = �h in (2.10) and recalling several elementaryproperties of the molli�er function, e.g. j ��h�xi j � Const.=h, it follows thatZjxj�r ��uk(x)� ul(x) dx�� Zjxj�r ��jd(x)j � �h(x)d(x)�� dx+ ��Zjxj�r �h(x)d(x) dx�� C5h+ C6 ((k � l)�t) =h:Choosing h =p(k � l)�t and letting r!1, we obtain (2.10). This concludes the proof of the lemma. �In order to investigate the convergence of the sequence fung, we need to work with functions that are notonly de�ned for each t = n�t, but in the entire interval [0; T ]. To do this we de�ne�j;n = n+ jm+ 1for j = 0; : : : ;m+ 1, and ui;n(x) = Sfi (�t) : : :Sf1 (�t)un(x)for i = 1; : : : ;m, where u0;n = un. Let now the sequence fu�tg be de�ned byu�t(x; t) = � Sfi ((m + 1) (t� �i�1;n�t))ui�1;n(x); for t 2 [�i�1;n�t; �i;n�ti;H ((m + 1) (t � �m;n�t))um;n(x); for t 2 [�m;n�t; �m+1;n�ti:This method of extending fung to a function de�ned for all t was �rst used by Crandall and Majda in [2].Regarding fu�tg we have the following lemma.

AN OPERATOR SPLITTING METHOD 5Lemma 2.4. For any sequence f�tg, such that �t ! 0, there exists a subsequence f�tjg and a function usuch that as j !1, u�tj ! u in Lloc1 (Rm� [0; T ]).Proof. By Lemma 2.1 and 2.2 the sequence fu�tg is uniformly bounded and has uniformly bounded totalvariation. We can therefore use Helly's theorem to conclude that a subsequence �u�tj converges in L1 onbounded boxes [�r; r]m for each �xed t. Since r is arbitrary, we can apply this argument a countable number oftimes to conclude that there is a further subsequence, again denoted by �u�tj, such that �u�tj (�; t) convergesin Lloc1 (Rm) for each �xed t. By a diagonalization we can have such a convergence for a dense countable subsetftlg in [0; T ]. Now, for some t 62 ftlg, let ftkg � ftlg such that tk ! t. We computeZ[�r;r]m ��u�ti(x; t)� u�tj (x; t)�� dx � Z[�r;r]m ju�ti(x; t) � u�ti (x; tk)j dx+ Z[�r;r]m ��u�ti (x; tk) � u�tj (x; tk)�� dx+ Z[�r;r]m ��u�tj (x; tk)� u�tj (x; t)�� dx:By Lemma 2.3, we can make the �rst and third term arbitrarily small by making k large, and since fu�tj(�; tk)gis a Cauchy sequence in Lloc1 (Rm), the middle term can be made arbitrarily small by making i and j large.Thus �u�tj (�; t) is a Cauchy sequence in Lloc1 (Rm) for all t 2 [0; T ]. We denote the limit by u. This concludesthe proof of the lemma. �Next, we justify the term \approximate solution" by showing:Theorem 2.5. Assume that u0 is in L1 \B:V:, and that f (u) is Lipschitz continuous. Then, for any sequencef�tg tending to zero, there exists a subsequence f�tjg such that the corresponding subsequence �u�tj convergesto a solution of the initial value problemut +r � f (u) = "�u; u(x; 0) = u0(x):Remark. With a solution of (1.1) we understand a function u(x; t) which is twice continuously di�erentiablein x and once in t such that the di�erential equation is satis�ed in the classical sense for t > 0. Moreover, theinitial data should be assumed in the weak sense. It is well-known that such a solution exists, cf. [18], [16].Proof. From Lemma 2.4 we have convergence of the sequence fujg �= �u�tj� to u. We continue by showingthat u is indeed a weak solution to (1.1). To this end de�ne the functionalL�(u) = ZRm Z T0 (u�t + f (u) � r�+ "u��) dt dx+ ZRm u0(x)�(x; 0) dx;for � 2 C20 (Rm� h0; T ]). If L�(u) = 0 for all such �, then u is a weak solution. But we have more, for following[18], we have that a weak solution to (1.1) is, in fact, a classical solution possessing the necessary smoothnessfor t > 0. Therefore it su�ces to show that L�(u) = 0, where u = limj uj , for a proper set of test functions �.Let now vi;nj (t) = Sfi(t)ui�1;nj for t 2 [0;�tj]:We also de�ne a new test function ~� by ~�(x; t) = ��x; tm + 1� :Then we have(2.13) ZRm Z �i;n�tj�i�1;n�tj �� 1m+ 1�uj�t + fi (uj)�xi� dt dx =1m+ 1 ZRm Z �tj0 �vi;nj (x; � )~�� (x; � + (m + 1)�i�1;n�tj) + fi(vi;nj (x; � ))~�xi (x; � + (m+ 1)�i�1;n�tj)� d� dx= 1m + 1 ZRm (uj (x; �i;n�t)� (x; �i;n�tj) � uj (x; �i�1;n�t)� (x; �i�1;n�tj)) dx;

6 KARLSEN, RISEBROby a change of variable; � = (m+1) (t � �i�1;n�tj). In order to deduce the third line of (2.12) from the second,we have used that vi;nj (t) is a weak solution in the interval [0;�tj] with initial and �nal values uj (x; �i�1;n�tj)and uj (x; �i;n�tj), respectively. Similarly, we have(2.14) ZRm Z �m+1;n�tj�m;n�tj �� 1m + 1�uj�t + "uj�� dt dx =1m + 1 ZRm (uj (x; �m+1;n�tj)� (x; �m+1;n�tj) � uj (x; �m;n�tj)� (x; �m;n�tj)) dx:If we sum (2.12) over i = 1; : : : ;m and add the result to (2.13), and �nally sum over n = 0; : : : ; Nj � 1, whereNj�tj = T , the resulting equality takes the form(2.15) ZRm Z T0 � 1m + 1�uj�t + mXi=1 �ifi (uj)�xi + "�m+1uj��! dt dx+ 1m + 1 ZRm uj (x; 0)�(x; 0) dx = 0;where �i(t) is de�ned as �i(t) = � 1 for t 2 [�i�1;n�tj; �i;n�tj] for some n;0 otherwise,for i = 1; : : : ;m+1. As �tj ! 0, �i tends weakly to 1=(m+1) in L2. Consequently, taking the limit in (2.14),we obtain that L� (u) = limj!1L� (uj) = 0;where u(x; t) = limj uj(x; t). This concludes the proof of Theorem 2.5. �Remark. Since the solution of (1.1) is unique, it follows that the whole sequence fu�tg converges, and notjust some subsequence �u�tj.3. A discrete method. In this section we present a numerical method implementing the ideas of the operatorsplitting described earlier. Here the solution operator for the hyperbolic part of the equation is replaced bya solution generated by front tracking, and the solution operator for the di�usion part is replaced by a �niteelement method.Front tracking is inherently a numerical method for one-dimensional conservation laws, in several dimensionswe use the dimensional splitting described in [12] as the approximation to S. Let S� denote the approximatesolution operator associated with (2.1), i.e., S�(t)v0(x)is the result of using front tracking, possibly coupled with dimensional splitting, for a single timestep of lengtht, on the piecewise constant function v0(x). We have that S�(t)v0(x) is a piecewise constant function of x (on a�xed grid if m > 1). Here � is a parameter measuring the discretization used; � = (�;�x) = (�;�x1; : : : ;�xm),where �xi denotes the grid spacing in the ith direction. For a detailed description of the front tracking methodfor scalar conservation laws in one dimension, we refer the reader to [11], and for a description of the dimensionalsplitting method to [12].We shall use a �nite element method for the solution of (2.3), with elements determined by the discontinuitiesin S�(t)v0(x). Let H�x(t) denote the operator which takes an initial functionw0(x) = MXi=1 ai'i(x)to the approximate solution of (2.3) obtained by the element method using basis functions 'i(x), i = 1; : : : ;M .We assume that these basis functions are associated with �x such that M !1 as �x! 0. The approximatesolution is then written as H�x(t)w0(x) = MXi=1 ai(t)'(x);where ai(t) is the solution of the following system of ordinary di�erential equations(3.0) MXi=1 dai(t)dt ('i; 'j) + " MXi=1 (r'i;r'j) ai(t) = 0

AN OPERATOR SPLITTING METHOD 7for j = 1; : : : ;M , where (�; �) denotes the inner product in L2. For a description of element methods for problemssuch as (2.3), see [14]. Standard theory for parabolic equations, see e.g. [20, ch 10], says that H�x(�)w0 is acontinuous function of t taking values in L2. If we use a suitable numerical scheme when integrating (3.0), andif w0 is bounded, then also H�x(t)w0 will be a bounded function for all t, independently of M . We restrictourselves to some bounded (rectangular) domain in Rm, therefore H�x(�)w0 is a continuous function of ttaking values in L1, and we have the following estimate(3.1) jjH�x(t)w0 � w0jj1 � h(t);where h(t) is some continuous function with h(0) = 0.As mentioned above, we shall restrict ourselves to a bounded rectangular domain in Rm, xi 2 [ai; bi]. Todo this, we ought to impose boundary values, but for ease of presentation we shall make these as simple aspossible. If m = 1, we shall require the solution to satisfy(3.2a) u(a; t) = ua; u(b; t) = ub;where ua and ub are constants. If m > 1, we require the solution to satisfy(3.2b) uj@ = c;where c is some constant, and we observe that there is no loss of generality in choosing c = 0.Now we can describe the numerical approximation. Let � and �xi, i = 1; : : : ;m, be small numbers, and letu0�(x) be a function which is constant in each grid block,fx = (x1; : : : ; xm) : ai + ni�xi � xi < ai + (ni + 1)�xi, i = 1; : : : ;mgfor ni = 0; : : : ;Mi � 1, where ai +Mi�xi = bi. We choose this function such that(3.3) lim�!0 jju0� � u0jj1 = 0:For t = n�t we de�ne un� (x) as(3.4) un� (x) = [H�x(�t)S�(�t)]n u0�(x):Here, the case m = 1 needs special mention. In this case S� does not give a result that is constant on a timeindependent grid. Therefore, to ensure convergence of H�x, we have to add grid points whenever the spacingbetween two discontinuities become larger than C�x for some �xed constant C. In the computations presentedin section 4, we use C = 1.By mimicking the proofs of Lemma 2.1 to Lemma 2.3, and using the remarks above, it is straightforward toprove:Lemma 3.1. Let un� (x) be de�ned by (3.4), then for all m;n = 0; : : : ; N ,(3.5) jjun� jj1 �M;T:V: �un�� M;jjum� � un� jj1 �M~h(j�t(m� n)j);where M is some number independent of n, m, � and �t, and ~h(s) is a continuous function with ~h(0) = 0.An immediate consequence of this lemma is that our numerical method, (3.4), converges as � and �t tendto zero. Furthermore, we can also mimic the arguments used in proving Theorem 2.5 to show:Theorem 3.2. Assume that u0 is in L1 \B:V:, and that f (u) is Lipschitz continuous. Let u�;�t(x; t) be givenby (3.4) for t = n�t, and the analogue of u�t(x; t) (the semi-discrete method) for t 6= n�t. Thenu(x; t) = lim�;�t!0u�;�t(x; t)is the unique solution to the convection-di�usion problemut +r � f (u) = "�u; u(x; 0) = u0(x);with boundary data (3.2a) if m = 1, and (3.2b) if m > 1.Remark. In the implementation presented in section 4, we have used Strang splittingun� (x) = [H�x (�t=2)S�(�t)H�x(�t=2)]n u0�(x)instead of (3.4). It is clear that convergence and Theorem 3.2 hold for this construction as well. Furthermore,it should be noted that if H�x is chosen properly, our discrete method (3.4) is unconditionally stable in thesense that the time step is not limited by the space discretization.

8 KARLSEN, RISEBRO4. Some numerical experiments. In this section we shall present an implementation of the operator splittingalgorithm presented in the last section in the case of one and two space dimensions. In particular, we will tryto numerically determine a convergence rate of this method. To do this we shall measure the error at a �xedtime T (in the experiments below we use T = 1). Since the convergence results above were formulated in L1,we measure the error in a relative L1 norm, i.e.,E = jjuN� � u�(�; T )jj1jju�(�; T )jj1 ;where u� is some reference solution. If we have no exact solution available, then we have used a solutioncalculated with a very �ne discretization. We shall in the following adopt the notation x = (x; y), f = (f; g),and �x = (�x;�y).When trying to determine the convergence rate in the one-dimensional case we assume that the error is ofthe form E � C ��x� +�t�� ;for some constant C independent of �x and �t. In our experiments we perform two kinds of computations:One is to test the spatial convergence rate, where we �x a small �t and observe the convergence order � withrespect to �x. The other is to test the temporal convergence rate with respect to �t. Here we �x a small �xand observe �. We use least-squares �t to obtain the exponents � and �.In the two-dimensional case we shall assume that the error is of the form E � C�x� with �y = �x, andrelate the time step �t to space discretization �x through the CFL number; (�t=�x)max(jjf 0jj1; jjg0jj1).The reason for doing so is that we want to explore, in addition to the convergence rate, large time step behaviorof the method. Our method is unconditionally stable in the sense that there is no CFL number restriction onthe discretization parameters in order to obtain convergence, therefore it is interesting to see whether the errorincreases if larger time steps are taken. We will present L1 errors and convergence rates for three di�erent CFLnumbers, namely, 1, 2 and 4.In one dimension, front tracking for a scalar conservation law has two discretization parameters; the distancebetween the (linear) interpolation points used when approximating the ux function, and the initial discretiza-tion used when approximating the initial function. Furthermore, the operator splitting needs two additionalparameters, namely the timestep �t and the size of the support of the basis functions. In one dimension wehave used piecewise linear basis functions of the sort'j(x) = 8><>: 0 if x � xj�1,1 if x = xj,0 if x � xj+1.Here, the numbers fxjg are given as the position of the jth front from the left, or, if the spacing between adjacentfronts is greater than �x, then xj+1 = xj +�x, where �x is the dizcretization used when approximating theinitial function u0. That is, we use(4.0) u0�(x) = u0(a+ (i + 1=2)�x) for x 2 [a+ i�x; a+ (i + 1)�xi.We set the distance between the interpolation points in the ux function to 0:1 for all computations in onedimension and 0:01 for the computations in two dimensions. In two dimensions, the approximation, u0�(x; y),of the initial data is given by the two-dimensional analogue of (4.0). Furthermore, the �nite element methoduses a standard uniform triangulation of , and piecewise linear basis functions given by'j(Ni) = � 1 if i = j,0 if i 6= j,where Ni is some node in the triangulation.When integrating (3.0) numerically, for stability reasons, we have set the matrix f(�i; �j)g equal to theidentity matrix, a technique known as lumping , see [9]. The integration of (3.0) is done by Euler's backwardmethod. This might seem as a restriction on the order of the splitting method, but as the numerical convergenceorders obtained here are all well below 1, we believe this not to be so.Example 1 (one-dimensional test case). Our �rst example will be the solution of Burgers' equation in onespace dimension,(4.1) ut +�12u2�x = "uxx:

AN OPERATOR SPLITTING METHOD 9This equation has a time independent solution given by(4.2) u(x; t) = � tanh� x2"� ;so that it is well suited as a test case.In Figure 4.1. we show all discontinuities in the approximated solution in the (x; t) plane for t in the interval[0; 1]. This computation was done with " = 0:1, �x = 0:05 and �t = 0:1. Note how the discontinuities\converge" towards the steepest place in the exact solution at x = 0. 0.00

0.33

0.67

1.00

-1.00 -0.33 0.33 1.00Figure 4.1. Discontinuities in the (x; t) plane.In the left column of Figure 4.2 we show the numerical convergence rates obtained with �x = 2�8, withcomputational domain [�1; 1], and boundary data given by (4.2). These rates were obtained using �t =2�1; : : : ; 2�8. The �gure shows the logarithm of the error versus the logarithm of �t. The straight lines areobtained from standard linear regression. In the right column we show numerical convergence rates in �x. Here�t = 2�8, and �x = 2�1; : : : ; 2�8. The three rows of the �gure correspond to " = 0:1, " = 0:01 and " = 0:001.-6 -5 -4 -3 -2 -1 0

-4

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epsilon=0.01, order=0.8

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epsilon=0.001, order=1.0Figure 4.2. Numerical convergence rates in �t (left) and �x (right), with u0 given by (4.2).The computations above correspond to the case where the conservation law would have a shock solution, wecan do a similar analysis in the case where the conservation law has a rarefaction wave solution. Also in thiscase, via the Hopf-Cole transform, we have an explicit solution. Now we have initial data(4.3) u0(x) = � �1 for x � 0,1 for x > 0,

10 KARLSEN, RISEBROand the solution is u(x; t) = g(�x; t)� g(x; t)g(�x; t) + g(x; t) ;where g is given by g(x; t) = e t+2x4" erfc� t+ xp4"t� :The computational domain is the interval [�1; 1], and the boundary values are g(�1; t). In Figure 4.3 we showthe numerical convergence rates in �t and �x for this \rarefaction case". The most notable feature of thesecomputations is the poor convergence rate in �t. Comparing the vertical scale of the left and right columnsof Figure 4.3, we see that even for large values of �t, the error in the left column is roughly the same as thesmallest error in the right column. Another way to look at this is that the error is largely independent of �t,i.e., one can take large time steps without much loss in accuracy in this \rarefaction case", as opposed to the\shock case" where the error is signi�cantly more sensitive to the choice of �t.-6 -5 -4 -3 -2 -1 0

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-1

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epsilon=0.001, order=0.8Figure 4.3. Example 1; Numerical convergence rates in �t (left) and �x (right), with u0 given by (4.3) -1.10

-0.37

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1.10

-1.00 -0.33 0.33 1.00 -1.00 -0.33 0.33 1.00

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Figure 4.4. Exact solutions (dotted line) versus approximate solutions (piecewise constant) from Example 1. The shock case (left)and the rarefaction case (right) are both calculated with �x = 0:01, �t = 0:5, and " = 0:01.Example 2 (two-dimensional test case). We consider the equationut + (u+ (u� 0:25)3)x � (u+ u2)y = "(uxx + uyy);

AN OPERATOR SPLITTING METHOD 11with initial data given by u0(x; y) = � 1 for (x� 0:25)2 + (y � 2:25)2 < 0:5,0 otherwise.The computational domain is [�2; 5]�[�2; 5] and we calculate approximate solutions for �x = 7�2�4; : : : ; 7�2�8.Boundary values are set to zero, which is consistent with the initial data u0. Figure 4.5 shows \log-log plots"of the error and numerical convergence rates in �x for " = 0:1, 0:01 and CFL numbers 1, 2, and 4. See Figure4.7 for a picture of an approximate solution. We observe that the error and the convergence rates are more orless independent of the choice of the CFL number, and, consequently, large time steps can be taken withoutloosing accuracy.−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

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0epsilon=0.1, CFL=2, order=1.1

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0epsilon=0.01, CFL=4, order=1.1Figure 4.5. Example 2; Numerical convergence rates in �x for di�erent values of the CFL number.Example 3 (two-dimensional test case). We now present an example where we generate approximatesolutions to the equation ut + f(u)x + g(u)y = "(uxx + uyy)with initial data u0(x; y) = � 1 for x2 + y2 < 0:5,0 otherwise,and g(u) = u2u2 + (1� u)2 ;f(u) = g(u)(1 � 5(1� u)2):The ux functions f and g are both \S-shaped" with f(0) = g(0) = 0 and f(1) = g(1) = 1. This problem ismotivated from two-phase ow in porous media with a gravitation pull in the x-direction. The computationaldomain is [�3; 3]� [�3; 3] and approximate solutions are computed for �x = 6 � 2�4; : : : ; 6 � 2�8. Boundaryvalues are again put equal to zero. We use the same values as in the previous example for the viscosity constant" and the CFL number. The results are presented in Figure 4.6 (see also Figure 4.7), and we observe again thatthe accuracy is uncorrelated to the CFL number. Finally, let us also mention (without \log-log plots") thatthe convergence rate in �t for both the two-dimensional examples seems to be well below 1, which is consistentwith the observations in Example 1.In closing, we would like to mention that we plan to continue the study of convergence rates, this time in amore theoretical setting, in a forthcoming paper.Acknowledgment. The authors thank Knut-Andreas Lie and Vidar Haugse for the use of computer code.The authors would also like to thank Helge K. Dahle and Magne S. Espedal for valuable discussions.

12 KARLSEN, RISEBRO−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

−4

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−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−4

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−1epsilon=0.1, CFL=2, order=1.0

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−2

0epsilon=0.01, CFL=4, order=1.0Figure 4.6. Example 3; Numerical convergence rates in �x for di�erent values of the CFL number.

−2−1

01

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x−axisy−axis −3−2

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x−axisy−axisFigure 4.7. Left: Approximate solution and initial data taken from Example 2. Right: Approximate solution from Example 3. Thesolutions are both computed on a 45� 45 grid, 10 time steps, and " = 0:01.References1. J. M. Burgers, The nonlinear di�usion equation, Reidel, Dordrecht, 1974.2. M. Crandall, A. Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), 285{314.3. C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal.Appl. 38 (1972), 33{41.4. H. K. Dahle, ELLAM-based operator splitting for nonlinear advection-di�usion equations, UiB report 98 (1995).5. J. Douglas, T. F. Russell, Numerical methods for convection dominated di�usion problems based on combining the method ofcharacteristiscs with �nite element or �nite di�erence procedures, SIAM J. Numer. Anal. 19 (1982), 871{885.6. M. Espedal, R. E. Ewing, Characteristic Petrov{Galerkin subdomain methods for two-phase immiscible ow, Comput. Meth-ods Appl. Mech. Engrg. 64 (1987), 113{135.7. R. E. Ewing, Operator splitting and Eulerian-Lagrangian localized adjoint methods for multiphase ow, The Mathematics ofFinite Elements and Applications VII MAFELAP (J. Whiteman, eds.), Academic Press, San Diego, 1990, pp. 215{232.8. R. E. Ewing, T. F. Russell, E�cient time stepping methods for miscible displacement problems in porous media, SIAM J.Numer. Anal. 19 (1982), 1{67.9. I. Fried, Numerical integration of partial di�erential equations, Academic, New York, 1979.10. S. K. Godunov, Finite di�erence methods for numerical computations of discontinuous solutions of the equations of uiddynamics, Mat. Sbornik. 47 (1959), 271{295.11. H. Holden, L. Holden, R. H�egh-Krohn, A numerical method for �rst order nonlinear scalar conservation laws in one{dimension, Comput. Math. Applic. 15 (1988), 595{602.12. H. Holden, N. H. Risebro, A method of fractional steps for scalar conservation laws without the CFL condition, Math. Comp.60 (1993), 221{232.13. K. Hvistendahl Karlsen, On the accuracy of a dimensional splitting method for scalar conservation laws, Master thesis (1994).14. C. Johnson, Numerisk l�osning av partiella di�erentialekvationer med �nita element metoden, Studentlitteratur, Lund, 1980.15. S. N. Kru�zkov, First order quasilinear equations in several independent variables, Mat. Sbornik 10 (1970), 217{243.16. S. N. Kru�zkov, O. Oleinik,Quasi-linear second order equations with several indendent variables, Uspehi. Mat. Nauk 16 (1961),115{155.

AN OPERATOR SPLITTING METHOD 1317. M. J. Lighthill, G. B. Whitham, On kinematic waves. II. Theory of tra�c ow on long crowded roads, Proc. Roy. Soc. 229A(1955), 317{345.18. O. Oleinik, Discontinuous solutions of non-linear di�erential equations, Amer. Math. Soc. Transl. 26 (1963), 95{172.19. D.W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier, Amsterdam, 1977.20. M. Renardy, R. C. Rodgers, An introduction to partial di�erential equations, Springer, New York, 1993.21. T. F. Russell, Galerkin time stepping along characteristics for Burgers' equation, Scienti�c computing (R. Stepleman, eds.),IMACS, North-Holland, 1983, pp. 183{192.22. Z. H. Teng, On the accuracy of fractional step methods for conservation laws in two dimensions, SIAM J. Numer. Anal. 31(1994), 43{63.23. J. A. Trangenstein, Mathematical structure of the black-oil model for petroleum reservoir simulation, SIAM J. Appl. Math.49 (1989), 749{783.Department of MathematicsUniversity of BergenJohs. Brunsgt. 12N{5008 Bergen, Norway(Hvistendahl Karlsen)E-mail address: [email protected] of MathematicsUniversity of OsloP.O. Box 1053, BlindernN{0316 Oslo, Norway(Risebro)E-mail address: [email protected]

An operator splitting method for nonlinear convection-diffusion equations

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