Numerische Mathematik

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1 Numer. Math. (997) 77: Numerische Mathematik c Springer-Verlag 997 Electronic Edition An operator splitting method for nonlinear convection-diffusion equations Kenneth Hvistendahl Karlsen, Nils Henrik Risebro Department of Mathematics, University of Bergen, N 58 Bergen, Norway; kennethk@mi.uib.no Department of Mathematics, University of Oslo, N 36 Oslo, Norway; nilshr@math.uio.no Received February, 996 / Revised version received June 4, 996 Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the m-dimensional convection-diffusion equation u t + f(u)=ε u. The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case m >, dimensional splitting is used to reduce the m-dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. Mathematics Subject Classification (99): 35L65. Introduction In this paper we present an operator splitting method for the scalar convectiondiffusion equation (.) u t + m m f i (u) xi = ε u xi x i, u (x,...,x m,)=u (x,...,x m ). i= For brevity of notation, we shall write this equation as i= u t + f(u)=ε u, u(x, ) = u (x), where x = (x,...,x m ),f(u)=(f (u),...,f m (u)), = ( ) / x,..., / x m, and = m i= / xi. The research of the first author has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil) Correspondence to: N.H. Risebro page 365 of Numer. Math. (997) 77:

2 366 K.H. Karlsen, N.H. Risebro Equations such as (.) arise in a variety of applications, ranging from models of turbulence [], via traffic flow [7], to two phase flow in porous media [9]. Equation (.) can also be viewed as a model problem for a system of convectiondiffusion equations, such as three phase flow in porous media [3], or the Navier- Stokes equations. Of particular importance is the case where convection dominates diffusion, i.e., ε is small compared with other scales in (.). This is often the case in models of two phase flow in oil reservoirs. Accurate numerical simulation of such models are consequently often complicated by both unphysical oscillations and numerical diffusion. The operator splitting approach presented here is especially well suited to the case where ε. The quasi-linear parabolic equation (.) was first properly analyzed by Oleinik in [8], where she proved existence of a unique classical solution, and also that weak solutions to (.) coincide with classical solutions. Operator splitting, or fractional steps, methods have been extensively used in connection with conservation laws, starting with Godunov [], who used this method to study gas dynamics. Operator splitting (dimensional splitting) for a scalar conservation law in several space dimensions was studied by Crandall and Majda in [], where they analyzed both a semi-discrete and a discrete method, and showed the convergence of the method for several numerical schemes. Holden and Risebro [] studied dimensional splitting coupled with front tracking, and convergence rates for dimensional splitting methods were obtained in [] and [4]. If ε, then (.) is almost hyperbolic, and it is natural to exploit this when constructing numerical methods. This approach has indeed been taken by several authors, we only mention Douglas and Russell [5,, 8], Espedal and Ewing [6, 7], and more recently Dahle [4]. In [] a characteristic element method is used to solve the hyperbolic part of (.). In [5], error estimates are obtained for a linear version of (.). The splitting method analyzed in this paper can be summarized as follows: Let v(x, t) =S (t)v be the unique entropy satisfying solution to v t + f(v)=, v(x,) = v (x), and let w(x, t) =H(t)w be the solution of w t = ε w, w(x, ) = w (x). Operator splitting is based on the following approximation (.) u(x, n t) [ H ( t)s ( t) ] n u (x). Here we study the convergence properties as the time step t. In applications, the exact solution operators S and H are replaced by approximations. We use one-dimensional front tracking as defined by Dafermos [3], see also [], as an approximate solution operator for the hyperbolic part if m =.Ifm> page 366 of Numer. Math. (997) 77:

3 An operator splitting method 367 we use the dimensional splitting method described in [] as an approximate solution operator for the hyperbolic part. In both one and several space dimensions a finite element method is used as an approximate operator for the diffusion part. We also establish convergence of the approximate solutions in the fully discrete case. The rest of this paper is organized as follows: In Sect. we obtain compactness of the sequence of approximate solutions generated by (.), and show that any convergent subsequence converges to the unique classical solution of (.). In Sect. 3, we show the same if the exact solution operators are replaced by the approximations mentioned above. In Sect. 4 we study applications of the method in one and two space dimensions, and present tentative convergence rates based on the numerical examples.. The semi-discrete method In this section we shall describe the operator splitting of (.), that is, we will obtain the solution of (.) through a composition of the solution operator to a hyperbolic equation and to a parabolic equation. We study (.) for x = (x,...,x m ) R m, where m. Let therefore f(u) =(f (u),...,f m (u)), where u is a scalar. Furthermore, let S (t) bethe operator which takes an initial function v (x) to the entropy weak solution at time t of (.) v t + f(v)=v t + m f i (v) xi =, v(x,) = v (x), i= that is, we write the (weak) solution v(x, t) ass (t)v (x). We shall also need the solution operator taking the initial data v (x) to the entropy weak solution at time t of the one dimensional conservation law (.) v t + f i (v) xi =, v(x,) = v (x). This operator we denote S fi (t). Note that in (.), x j, j /= i, act only as parameters. Following Kružkov [5], we know that (.) has a unique entropy weak solution if f is Lipschitz continuous, and the initial data are in L B.V.. Here, B.V. denotes the space of functions with bounded total variation. Similarly, let H (t) be the solution operator (at time t) for the parabolic equation m (.3) w t = ε w = ε w xi x i, i= w(x,) = w (x). Now fix T > and t >, and let N be such that N t = T. Let u (x) =u (x), and let (.4) u n+ (x) = [ H( t)s fm ( t)...s f ( t) ] u n (x) page 367 of Numer. Math. (997) 77:

4 368 K.H. Karlsen, N.H. Risebro for n =,...,N. Observe that by the results in [], S ( t) [ S fm ( t)... S f ( t) ]. We will now show that the sequence {u n } is compact in L loc. To accomplish this we use a technique introduced by Oleinik [8]. This approach is now standard in the theory of conservation laws, and involves proving a priori uniform L, B.V. and L (x) Lipschitz in time, bounds on the sequence. More concretely, we show that the sequence {u n } is uniformly (in time) bounded in L and T.V. norms, with bounds independent of the discretization parameters. Then we appeal to Helly s theorem to conclude the existence of a subsequence converging in L loc (x) to some function in B.V. (x) for each fixed t. By a diagonalization argument, we obtain convergence for all t in some dense countable subset of [, T ]. Convergence in L loc (x) for all t is finally obtained by showing that each member of the convergent subsequence is L Hölder continuous in time, i.e., u m (x) u n (x) dx L ( (m n) t ) /. R m To simplify the notation we define S f (t) =S fm (t)...s f (t). We also note that if w (x) is bounded, then the solution of (.3) is given by convolution with the heat kernel K (x, t), where [ ] x K (x, t) = exp = k (x (4πεt) m/, t)...k(x m,t) 4εt with k(x, t) given by k(x, t) = 4πεt exp [ x 4εt ], i.e., we can write (.5) H (t)w (x) = K (x y, t) w (y) dy. R m Lemma.. For n =,...,N, (.6) u n u. Proof. This follows from the fact that both S f (t) and H (t) obey a maximum principle similar to (.6). Lemma.. For n =,...,N, (.7) T.V. (u n ) T.V. (u ). page 368 of Numer. Math. (997) 77:

5 An operator splitting method 369 Proof. Recall that for a function g(x) its total variation is defined by g(x + h) g(x) T.V. (g) = lim sup dx, h R h and for a function g(x) of several variables m T.V. (g) = T.V. xi (g) d m x, i= R m where T.V. xi (g) denotes the total variation of g with respect to the ith variable. We have that T.V. ( ) S fi (t)v T.V. (v ) and that These two facts imply, see [], that S fi (t)v S fi (t)ˆv v ˆv. T.V. ( S f (t)v ) T.V. (v ). The lemma will follow by induction if we can show that (.8) T.V. ( H (t)w ) T.V. (w ). Let e i denote the unit vector in the ith direction. We calculate w (x + he i, t) w(x, t) lim sup dx i h R h = lim sup K (y, t) w (x + he i y) w (x y) dy dx i h R R h m K (y, t)t.v. xi (w ) dy = T.V. xi (w ). R m Hence, (.8) holds. Lemma.3. There is a constant C, independent of k, l and t, such that (.9) u k u l C (k l) t for all l k N. Proof. We begin by showing that the splitting solutions are weakly Lipschitz continuous in the time variable, that is, (.) φ(x) ( u k (x) u l (x) ) dx C (k l) t ( ) φ + max φ xi, φ C. R m i Using finite speed of propagation, we have that S fi (t) (and thus S f (t)) obey a stronger condition than (.), namely S fi (t + t)v (x) S fi (t)v (x) dx i C t. R page 369 of Numer. Math. (997) 77:

6 37 K.H. Karlsen, N.H. Risebro From this it follows that (.) φ(x) ( S f (t + t)v (x) S f (t)v (x) ) dx C 3 t φ. R m Using the differential equation for w(x, t) =H(t)w (x) and integration by parts, we get the bound φ(x) ( H (t + t)w (x) H (t)w (x) ) dx R m m ( ) t+ t = φ(x) εw(x,τ) xix i dτ dx i= R m t (.) C 4 tmax φ xi, i where we mention that C 4 depends on T.V. (w ). Using (.) and (.), we now readily compute φ(x) ( u k (x) u l (x) ) k dx φ(x) ( u i+ (x) u i (x) ) dx R m i=l R m C (k l) t ( ) φ + max φ xi i for some proper choice of constant C. Finally, we claim that the estimate (.9) follows from an approximation argument. To see this, let ω h (x) be a standard m-dimensional C -mollifier with smoothing radius h. Let d(x) =u k (x) u l (x), and define β(x) = sign (d(x)) for x r hand β(x) = for x > r h, where r > h. Moreover, define β h = ω h β and note that β h C with support in { x r}. By choosing φ = β h in (.) and recalling several elementary properties of the mollifier function, e.g. ( β h) Const./h, it follows that x i u k (x) u l (x) dx d(x) β h (x)d(x) dx x r x r + β h (x)d(x) dx x r C 5 h + C 6 ((k l) t) /h. Choosing h = (k l) t and letting r, we obtain (.). This concludes the proof of the lemma. In order to investigate the convergence of the sequence {u n }, we need to work with functions that are not only defined for each t = n t, but in the entire interval [, T ]. To do this we define α j,n = n + j m + page 37 of Numer. Math. (997) 77:

7 An operator splitting method 37 for j =,...,m +, and u i,n (x) =S fi ( t)...s f ( t)u n (x) for i =,...,m, where u,n = u n. Let now the sequence {u t } be defined by ( ( S fi (m+) t αi,n t )) u i,n (x), for t [ α i,n t,α i,n t, u t (x, t) = H ( (m +) ( t α m,n t )) u m,n (x), for t [ α m,n t,α m+,n t. This method of extending {u n } to a function defined for all t was first used by Crandall and Majda in []. Regarding {u t } we have the following lemma. Lemma.4. For any sequence { t}, such that t, there exists a subsequence { t j } and a function u such that as j,u tj uinl loc (R m [, T ]). Proof. By Lemma. and. the sequence {u t } is uniformly bounded and has uniformly bounded total variation. We can therefore use Helly s theorem to conclude that a subsequence { u tj } converges in L on bounded boxes [ r, r] m for each fixed t. Since r is arbitrary, we can apply this argument a countable number of times to conclude that there is a further subsequence, again denoted by { u tj }, such that { u tj (, t) } converges in L loc (R m ) for each fixed t. Bya diagonalization we can have such a convergence for a dense countable subset {t l } in [, T ]. Now, for some t {t l }, let {t k } {t l }such that t k t. We compute u ti (x, t) u tj (x, t) dx u ti (x, t) u ti (x, t k ) dx [ r,r] m [ r,r] m + u ti (x, t k ) u tj (x, t k ) dx [ r,r] m + u tj (x, t k ) u tj (x, t) dx. [ r,r] m By Lemma.3, we can make the first and third term arbitrarily small by making k large, and since {u tj (, t k )} is a Cauchy sequence in L loc (R m ), the middle term can be made arbitrarily small by making i and j large. Thus { u tj (, t) } is a Cauchy sequence in L loc (R m ) for all t [, T ]. We denote the limit by u. This concludes the proof of the lemma. Next, we justify the term approximate solution by showing: Theorem.5. Assume that u is in L B.V., and that f(u) is Lipschitz continuous. Then, for any sequence { t} tending to zero, there exists a subsequence { t j } such that the corresponding subsequence { u tj } converges to a solution of the initial value problem u t + f(u)=ε u, u(x, ) = u (x). page 37 of Numer. Math. (997) 77:

8 37 K.H. Karlsen, N.H. Risebro Remark. With a solution of (.) we understand a function u(x, t) which is twice continuously differentiable in x and once in t such that the differential equation is satisfied in the classical sense for t >. Moreover, the initial data should be assumed in the weak sense. It is well-known that such a solution exists, cf. [8, 6, 5]. Proof. From Lemma.4 we have convergence of the sequence {u j } ( = { }) u tj to u. We continue by showing that u is indeed a weak solution to (.). To this end define the functional T L φ (u) = (uφ t +f(u) φ+εu φ) dt dx + u (x)φ(x, ) dx, R m R m ( for φ C R m,t] ).IfL φ (u) = for all such φ, then u is a weak solution. But we have more, for following [8], we have that a weak solution to (.) is, in fact, a classical solution possessing the necessary smoothness for t >. Therefore it suffices to show that L φ (u) =, where u = lim j u j, for a proper set of test functions φ. Let now v i,n j (t) =S fi (t)u i,n j for t [ ], t j. We also define a new test function φ by ( t ) φ(x, t) =φ x,. m+ Then we have αi,n t j (( ) ( ) (.) )u j φ t +f i uj φxi m + R m = = α i,n t j m + R m tj dt dx ( v i,n j (x,τ) φ τ ( x,τ +(m+)αi,n t j ) +f i (v i,n ( ) j (x,τ)) φ ) xi x,τ +(m+)αi,n t j dτdx ( ( uj x,αi,n t ) φ ( ) x,α i,n t j m+ R ( m u j x,αi,n t ) φ ( )) x,α i,n t j dx, by a change of variable; τ =(m+) ( ) t α i,n t j. In order to deduce the second equality of (.) from the first, we have used that v i,n j (t) is a weak solution in the interval [ ] ( ) ( ), t j with initial and final values uj x,αi,n t j and u j x,αi,n t j, respectively. Similarly, we have αm+,n t j (( ) ) (.3) u j φ t +εu j φ dt dx R m α m,n t j m + ( ( ) ( ) = uj x,αm+,n t j φ x,αm+,n t j m + R ( m ) ( )) u j x,αm,n t j φ x,αm,n t j dx. page 37 of Numer. Math. (997) 77:

9 An operator splitting method 373 If we sum (.) over i =,...,m and add the result to (.3), and finally sum over n =,...,N j, where N j t j = T, the resulting equality takes the form ( T ( ) ) m ( ) u j φ t + χ i f i uj φxi + εχ m+ u j φ dt dx R m m + i= + (.4) u j (x,)φ(x,) dx =, m + R m where χ i (t) is defined as { for t [ ] α i,n t j,α i,n t j for some n, χ i (t) = otherwise, for i =,...,m +.As t j, χ i tends weakly to /(m +)inl. Consequently, taking the limit in (.4), we obtain that L φ (u) = lim L ( ) φ uj =, j where u(x, t) = lim j u j (x, t). This concludes the proof of Theorem.5. Remark. Since the solution of (.) is unique, it follows that the whole sequence {u t } converges, and not just some subsequence { u tj }. 3. A discrete method In this section we present a numerical method implementing the ideas of the operator splitting described earlier. Here the solution operator for the hyperbolic part of the equation is replaced by a solution generated by front tracking, and the solution operator for the diffusion part is replaced by a finite element method. Front tracking is inherently a numerical method for one-dimensional conservation laws, in several dimensions we use the dimensional splitting described in [] as the approximation to S. Let S η denote the approximate solution operator associated with (.), i.e., S η (t)v (x) is the result of using front tracking, possibly coupled with dimensional splitting, for a single timestep of length t, on the piecewise constant function v (x). We have that S η (t)v (x) is a piecewise constant function of x (on a fixed grid if m > ). Here η is a parameter measuring the discretization used; η =(δ, x) =(δ, x,..., x m ), where x i denotes the grid spacing in the ith direction. For a detailed description of the front tracking method for scalar conservation laws in one dimension, we refer the reader to [], and for a description of the dimensional splitting method to []. We shall use a finite element method for the solution of (.3), with elements determined by the discontinuities in S η (t)v (x). Let H x (t) denote the operator which takes an initial function page 373 of Numer. Math. (997) 77:

10 374 K.H. Karlsen, N.H. Risebro w (x) = M a i ϕ i (x) i= to the approximate solution of (.3) obtained by the element method using basis functions ϕ i (x), i =,...,M. We assume that these basis functions are associated with x such that M as x. The approximate solution is then written as M H x (t)w (x) = a i (t)ϕ(x), where a i (t) is the solution of the following system of ordinary differential equations (3.) M i= da i (t) dt i= ( ) M ( ) ϕi,ϕ j +ε ϕi, ϕ j ai (t) = i= for j =,...,M, where (, ) denotes the inner product in L. For a description of element methods for problems such as (.3), see [3]. Standard theory for parabolic equations, see e.g. [, Ch. ], says that H x ( )w is a continuous function of t taking values in L. If we use a suitable numerical scheme when integrating (3.), and if w is bounded, then also H x (t)w will be a bounded function for all t, independently of M. We restrict ourselves to some bounded (rectangular) domain Ω in R m, therefore H x ( )w is a continuous function of t taking values in L, and we have the following estimate (3.) H x (t)w w h(t), where h(t) is some continuous function with h() =. As mentioned above, we shall restrict ourselves to a bounded rectangular domain Ω in R m, x i [a i, b i ]. To do this, we ought to impose boundary values, but for ease of presentation we shall make these as simple as possible. If m =, we shall require the solution to satisfy (3.a) u(a, t) =u a, u(b,t)=u b, where u a and u b are constants. If m >, we require the solution to satisfy (3.b) u Ω = c, where c is some constant, and we observe that there is no loss of generality in choosing c =. Now we can describe the numerical approximation. Let δ and x i, i =,...,m, be small numbers, and let u η(x) be a function which is constant in each grid block, {x =(x,...,x m ):a i +n i x i x i <a i +(n i +) x i,i =,...,m} for n i =,...,M i, where a i + M i x i = b i. We choose this function such that page 374 of Numer. Math. (997) 77:

11 An operator splitting method 375 (3.3) lim η u η u =. For t = n t we define u n η(x) as (3.4) u n η(x) = [ H x ( t)s η ( t) ] n u η (x). Here, the case m = needs special mention. In this case S η does not give a result that is constant on a time independent grid. Therefore, to ensure convergence of H x, we have to add grid points whenever the spacing between two discontinuities become larger than C x for some fixed constant C.Inthe computations presented in Sect. 4, we use C =. By mimicking the proofs of Lemma. to Lemma.3, and using the remarks above, it is straightforward to prove: Lemma 3.. Let u n η(x) be defined by (3.4), then for all m, n =,...,N, (3.5) uη n M, T.V. ( uη) n M, uη m uη n M h( t(m n) ), where M is some number independent of n, m, η and t, and h(s) is a continuous function with h()=. An immediate consequence of this lemma is that our numerical method, (3.4), converges as η and t tend to zero. Furthermore, we can also mimic the arguments used in proving Theorem.5 to show: Theorem 3.. Assume that u is in L B.V., and that f(u) is Lipschitz continuous. Let u η, t (x, t) be given by (3.4) for t = n t, and the analog of u t (x, t) (the semi-discrete method) for t /= n t. Then u(x, t) = lim u η, t(x, t) η, t is the unique solution to the convection-diffusion problem u t + f(u)=ε u, u(x, ) = u (x), with boundary data (3.a) if m =, and (3.b) if m >. Remark. In the implementation presented in Sect. 4, we have used Strang splitting u n η(x) = [ H x ( t/ ) Sη ( t)h x ( t/) ] n u η (x) instead of (3.4). It is clear that convergence and Theorem 3. 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 the sense that the time step is not limited by the space discretization. page 375 of Numer. Math. (997) 77:

12 376 K.H. Karlsen, N.H. Risebro 4. Some numerical experiments In this section we shall present an implementation of the operator splitting algorithm presented in the last section in the case of one and two space dimensions. In particular, we will try to numerically determine a convergence rate of this method. To do this we shall measure the error at a fixed time T (in the experiments below we use T = ). Since the convergence results above were formulated in L, we measure the error in a relative L norm, i.e., E = un η u (, T ) u (, T ), where u is some reference solution. If we have no exact solution available, then we have used a solution calculated with a very fine 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 of the form E C [ x α + t β], 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 fix a small t and observe the convergence order α with respect to x. The other is to test the temporal convergence rate with respect to t. Here we fix a small x and observe β. We use least-squares fit to obtain the exponents α and β. In the two-dimensional case we shall assume that the error is of the form E C x α with y = x, and relate the time step t to space discretization x through the CFL number; ( t/ x ) max ( ) f, g. The reason for doing so is that we want to explore, in addition to the convergence rate, large time step behavior of the method. Our method is unconditionally stable in the sense that there is no CFL number restriction on the discretization parameters in order to obtain convergence, therefore it is interesting to see whether the error increases if larger time steps are taken. We will present L errors and convergence rates for three different CFL numbers, namely,, and 4. In one dimension, front tracking for a scalar conservation law has two discretization parameters; the distance between the (linear) interpolation points used when approximating the flux function, and the initial discretization used when approximating the initial function. Furthermore, the operator splitting needs two additional parameters, namely the timestep t and the size of the support of the basis functions. In one dimension we have used piecewise linear basis functions of the sort if x x j, ϕ j (x) = if x = x j, if x x j+. page 376 of Numer. Math. (997) 77:

13 An operator splitting method 377 Here, the numbers {x j } are given as the position of the j th front from the left, or, if the spacing between adjacent fronts is greater than x, then x j + = x j + x, where x is the dizcretization used when approximating the initial function u. That is, we use (4.) u η(x) =u (a+(i+/) x) for x [a + i x, a +(i+) x. We set the distance between the interpolation points in the flux function to. for all computations in one dimension and. for the computations in two dimensions. In two dimensions, the approximation, uη(x, y), of the initial data is given by the two-dimensional analog of (4.). Furthermore, the finite element method uses a standard uniform triangulation of Ω, and piecewise linear basis functions given by { if i = j, ϕ j (N i )= if i /= j, where N i is some node in the triangulation. When integrating (3.) numerically, for stability reasons, we have set the matrix {( )} φ i,φ j equal to the identity matrix, a technique known as lumping, see [9]. The integration of (3.) is done by Euler s backward method. This might seem as a restriction on the order of the splitting method, but as the numerical convergence orders obtained here are all well below, we believe this not to be so. Example (one-dimensional test case). Our first example will be the solution of Burgers equation in one space dimension, ( ) (4.) u t + u = εu xx. x This equation has a time independent solution given by ( x (4.) u(x, t) = tanh, ε) so that it is well suited as a test case. In Fig. we show all discontinuities in the approximated solution in the (x, t) plane for t in the interval [, ]. This computation was done with ε =., x =.5 and t =.. Note how the discontinuities converge towards the steepest place in the exact solution at x =. In the left column of Fig. we show the numerical convergence rates obtained with x = 8, with computational domain [, ], and boundary data given by (4.). These rates were obtained using t =,..., 8. The figure shows the logarithm of the error versus the logarithm of t. The straight lines are obtained from standard linear regression. In the right column we show numerical convergence rates in x. Here t = 8, and x =,..., 8. The three rows of the figure correspond to ε =., ε =. and ε =.. The computations above correspond to the case where the conservation law would have a shock solution, we can do a similar analysis in the case where page 377 of Numer. Math. (997) 77:

14 378 K.H. Karlsen, N.H. Risebro Fig.. Discontinuities in the (x, t) plane 5 - epsilon=., order=.7 epsilon=., order= epsilon=., order=.8 epsilon=., order= epsilon=., order=.5 epsilon=., order= Fig.. Numerical convergence rates in t (left) and x (right), with u given by (4.) the conservation law has a rarefaction wave solution. Also in this case, via the Hopf-Cole transform, we have an explicit solution. Now we have initial data { for x, (4.3) u (x) = for x >, and the solution is where g is given by u(x, t) = g( x,t) g(x,t) g( x,t)+g(x,t), ( ) g(x, t) =e t+x t + x 4ε erfc. 4εt The computational domain is the interval [, ], and the boundary values are g(, t). In Fig. 3 we show the numerical convergence rates in t and x for this rarefaction case. The most notable feature of these computations is the page 378 of Numer. Math. (997) 77:

15 An operator splitting method 379 poor convergence rate in t. Comparing the vertical scale of the left and right columns of Fig. 3, we see that even for large values of t, the error in the left column is roughly the same as the smallest 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 significantly more sensitive to the choice of t epsilon=., order=. epsilon=., order= epsilon=., order=. epsilon=., order= epsilon=., order= epsilon=., order= Fig. 3. Example ; Numerical convergence rates in t (left) and x (right), with u given by (4.3) Fig. 4. Exact solutions (dotted line) versus approximate solutions (piecewise constant) from Example. The shock case (left) and the rarefaction case (right) are both calculated with x =., t =.5, and ε =. Example (two-dimensional test case). We consider the equation u t +(u+(u.5) 3 ) x (u + u ) y = ε(u xx + u yy ), with initial data given by { for (x.5) +(y.5) <.5, u (x, y) = otherwise. page 379 of Numer. Math. (997) 77:

16 38 K.H. Karlsen, N.H. Risebro The computational domain is [, 5] [, 5] and we calculate approximate solutions for x =7,...,7 8. Boundary values are set to zero, which is consistent with the initial data u. Fig. 5 shows log-log plots of the error and numerical convergence rates in x for ε =.,. and CFL numbers,, and 4. See Fig. 7 for a picture of an approximate solution. We observe that the error and the convergence rates are more or less independent of the choice of the CFL number, and, consequently, large time steps can be taken without loosing accuracy. epsilon=., CFL=, order=.9 epsilon=., CFL=, order= epsilon=., CFL=, order=. epsilon=., CFL=, order= epsilon=., CFL=4, order=. epsilon=., CFL=4, order= Fig. 5. Example ; Numerical convergence rates in x for different values of the CFL number Example 3 (two-dimensional test case). We now present an example where we generate approximate solutions to the equation u t + f (u) x + g(u) y = ε(u xx + u yy ) with initial data and u (x, y) = { for x + y <.5, otherwise, g(u) = u u +( u), f(u) = g(u)( 5( u) ). The flux functions f and g are both S-shaped with f () = g() = and f () = g() =. This problem is motivated from two-phase flow in porous media with a gravitation pull in the x-direction. The computational domain is [ 3, 3] [ 3, 3] and approximate solutions are computed for x =6,...,6 8. Boundary values are again put equal to zero. We use the same values as in the previous example for the viscosity constant ε and the CFL number. The results are presented in Fig. 6 (see also Fig. 7), and we observe again that the accuracy is uncorrelated to the CFL number. Finally, let us also mention (without page 38 of Numer. Math. (997) 77:

17 An operator splitting method 38 log-log plots ) that the convergence rate in t for both the two-dimensional examples seems to be well below, which is consistent with the observations in Example. epsilon=., CFL=, order=. epsilon=., CFL=, order= epsilon=., CFL=, order=. epsilon=., CFL=, order= epsilon=., CFL=4, order=. epsilon=., CFL=4, order= Fig. 6. Example 3; Numerical convergence rates in x for different values of the CFL number y axis x axis 3 4 y axis 3 x axis 3 Fig. 7. Left: Approximate solution and initial data taken from Example. Right: Approximate solution from Example 3. The solutions are both computed on a grid, time steps, and ε =. In closing, we would like to mention that we plan to continue the study of convergence rates, this time in a more theoretical setting, in a forthcoming paper. Acknowledgement. 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. References. Burgers, J.M. (974): The nonlinear diffusion equation. Reidel, Dordrecht. Crandall, M., Majda, A. (98): The method of fractional steps for conservation laws, Numer. Math. 34, Dafermos, C.M. (97): Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38, 33 4 page 38 of Numer. Math. (997) 77:

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