Pointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang

Transcription

1 Pointwise convergence rate for nonlinear conservation laws Eitan Tadmor and Tao Tang Abstract. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and nite dierence approximations of scalar conservation laws with piecewise smooth solutions. This method can deal with nitely many shocks with possible collisions. The key ingredient in our approach is an interpolation inequality between the L 1 and Lip + -bounds, which enables us to convert a global result into a (non-optimal) local estimate. A bootstrap argument yields optimal pointwise error bound for both the vanishing viscosity and nite dierence approximations. 1. Introduction We study solutions to the single hyperbolic conservation laws with small viscosity of the form subect to the initial condition u t + f(u ) x = u xx ; x R; t>;> (1) u (x) =u (x): () We are interested in the relation between its solution, u, and the solution u of the corresponding conservation laws without viscosity The initial condition for (3) is given by u t + f(u) x =; x R; t>: (3) u(x; ) = u (x): (4) We will investigate in this paper the pointwise error estimates between u and u, when u has nitely many shocks. It is well-known that u (;t) converges strongly in L 1 to u(;t), where u(;t) is the unique, so-called entropy solution of (3)-(4). It is shown in [8] that if the ux f is strictly convex, f >; (5)

2 94 Eitan Tadmor and Tao Tang then the L 1 convergence rate in this case is upper bounded by ku (;t), u(;t)k L 1 const.: (6) It is understood that the L 1 error estimate is a global one, while in many practical cases we are interested in the local behavior of u(x; t). Consequently, when the error is measured by the L 1 -norm, there is a loss of information due to the poor resolution of shock waves in u(x; t). In this work, we will provide the optimal pointwise convergence rate for the viscosity approximation. The previous results for the optimal order one convergence rates, in both L 1 and L 1 spaces, are all based on a matching method and traveling wave solutions, see e.g. [1,, 8]. In this work, however, we will not use the traveling wave solutions; instead our arguments are based on energy-like estimates. The proof of our results is based upon two ingredients: (1): Lip + -boundedness along [4] which enables us to \convert" a global result into a local estimate, (): A weighted quantity of the error satisfying a transport inequality such that the maximum principal applies. Unlike previous work on pointwise estimates [1, ], this framework can deal with nitely many shocks with possible collisions. The extensions to the general case can be found in [6]. Moreover, although we only consider the Lax-Friedrichs scheme the idea can be used to obtain the same results for monotone schemes [7]. The paper is organized as follows. In x we consider the viscosity methods when there are nitely many shocks. In x3 we discuss the extensions to the Lax- Friedrichs scheme.. Viscosity methods To begin with, we let kk Lip + denote the Lip + -seminorm w(x), w(y) + kwk Lip + := ess sup ; x6=y x, y where [w] + = H(w)w, with H() the Heaviside function. To convert global L 1 -error bounds (for Lip + bounded solutions) into local pointwise error estimates, the following lemma is at the heart of matter. T Lemma.1. Assume that v L 1 Lip + (I), and w Cloc 1 (x, ;x+ ) for an interior x such that (x, ;x+ ) I. Then the following estimate holds: v(x), w(x) Const 1 kv, wk L 1 + n kvk Lip + (x,;x+) + w C 1 loc (x,;x+) o : In particular, if the size of the smoothness neighborhood for w can be chosen so that kv, wk 1=,1= 1 kvk L 1 (I) Lip + + w C 1 loc I then the following estimate holds: v(x), w(x) Const kv, wk 1= L 1 (I) hkvk Lip + + w C 1 loc(x,;x+)i 1= : (7)

3 Pointwise convergence rate 95 Thus, (7) tells us that if the global L 1 -error kv, wk L 1 is small, then the pointwise error v(x), w(x) is also small whenever w x is bounded. This does not require the C 1 -boundedness of v; the weaker one-sided Lip + bound will suce. The detailed proof of the above lemma can be found in [6]. In this section, we rst assume that the entropy solution of (3)-(4) has only one shock discontinuity. The shock curve x = X(t) satises the Rankine-Hugoniot and the Lax conditions: X [f(u(x; t)] = ; (8) [u(x; t)] f (u(x(t),;t)) >X (t) >f (u(x(t)+;t)) : (9) Owing to the convexity of the ux f, the viscosity solutions of (1) satisfy a Lip + -stability condition, similar to the familiar Oleinik's E-condition, which asserts an a priori upper bound for the Lip + -seminorm of the viscosity solution 1 ku (;t)k Lip + ku k,1 + t ; (1) Lip + where u is the solution of (1)-(), is the convexity constant of the ux f given by (5). The above result suggests that if the initial data do not contain non- Lipschitzian increasing discontinuities then the viscosity solution of (1) will keep the same property. The same is true for entropy solution of (3)-(4). Equipped with (7), together with the global error bound (6) and the Lip + -boundedness (1), we obtain the following pointwise error bound: u (x; t), u(x; t)c p ; for dist(x; S(t)) p : (11) The basic idea of the pointwise error estimate in this section is as follows: Step #1: Set E(x; t) :=(u (x; t), u(x; t))(x; t); (1) where is a suitably dened distance function to the shock curve x = X(t). We will also choose a suitable domain of smoothness, D, such that the following dierential equation holds: E t + h(x; t)e x, E xx = p(x; t)e + q(x; t); (x; t) D: (13) Here h; p and q are smooth functions in D. Step #: The functions p and q in (13) can be (uniformly) upper bounded and bounded, respectively: p(x; t) Const:; q(x; t) Const:; for all (x; t) D: (14) Step #3: denote the usual boundary for this domain of smoothness, it will be shown that max E(x; t)c: (15) (x;t)@d

4 96 Eitan Tadmor and Tao Tang The inequality (15), together with the maximum principal for (13)-(14), yield E(x; t) C, for all (x; t) D, which in turn implies the pointwise estimate u (x; t), u(x; t) C, for (x; t) away from the shock curve x = X(t). In Step #1 mentioned above, the function E is a weighted error function which is continuous for (x; t) R(;T]. The key point in this step is to introduce the distance function, which satises! asdist(x; S(t))! and O(1) when dist(x; S(t)) O(1). The proof for Step # is based upon the interpolation between the global L 1 -error estimate and the Lip + -stability that leads to a local pointwise estimate. The proper use of the Lax entropy condition (9) is also crucial in this step. The third step is dependent on the choice of the weighted distance function,. We rst consider the pointwise error estimate in the region x>x(t). Let e(x; t) :=u, u and set the weighted error E(x; t) =e(x; t) (x, X(t)) : Here, (x, X(t)) is a weighted distance to the shock set. The function (x) C ([; 1)) satises x (x) ; if x 1 (16) 1; if x 1; with 1 to be determined later. More precisely, the function satises () = ; (x) > ; (x) x ; for x>; (17) x (x) (x); for x ; (18) (k) (x) Const; x ; (19) e.g., (x) =(1, e,x ). Roughly speaking, the weighted function behaves like (x) min(x ; 1). Direct calculations using the denition of E give us E t + f (u )E x, E xx = e t + f (u )e x, e xx +, X (t)+f (u ) {z } I 1 e {z } I, ex, e: () For ease of notation, denotes (x, X(t)) in the remaining of this section. It follows from the viscosity equation (1) and the limit equation (3) that I 1 =, f (u )u x + f (u)u x + u xx =,f ()(u, u)u x + u xx =,f ()u x E + u xx (1) where (and below) denotes some intermediate value between,ku k 1 and ku k 1. Let u (t) =u(x(t) ;t) and let I 3 (t) =,X (t)+f (u + ):

5 Pointwise convergence rate 97 Observing that u, u + = u x ( 1 )(x, X(t)), where 1 is an intermediate value between x and X(t), we obtain I = I 3, f (u + )+f (u), f (u)+f (u ) e = ei 3 + ef ()(u, u + )+ f ()e = I 3 + f ()e E + f (x, X(t)) ()u x ( 1 ) E; () where in the last step we have used the fact E = e. It is noted that e x = (E x, e)=. This, together with ()-(), yield the rst desired result, (13): E t + h(x; t)e x, E xx = p(x; t)e + q(x; t) ; where the coecient of the convection term is given by h(x; t) =f (u )+ ; (3) and the functions p := p 1 + p and q are given by p 1 (x; t) :=I 3 + f ()e + ; (4) p (x; t) :=,f () u x + f (x, X(t)) ()u x ( 1 ) ; (5) q(x; t) :=u xx, e: (6) We have then nished the Step #1. Next we move to Step #, verifying the boundedness of the coecients p and q inside a suitable domain. We nowchoose a proper domain of smoothness, D, inside the region x>x(t). Let D := n (x; t) x X(t)+ 1= ; t T o : (7) Using Lax geometrical entropy condition (9), u + (t) u, (t), and the convexity of f, it follows that I 3 is nonpositive I 3 (t) =,X (t)+f (u + )= = Z 1 Z 1 f ()(1, )d (u +, u, ) : h i f (u + ), f (u + +(1, )u, ) d For (x; t) D; x > X(t) + p, and hence by the property (18) of the weighted distance function we have C x, X(t) C,1= ; for (x; t) D: The last two upper bounds, together with (11), lead to the following estimate for p 1 p1 +C 1=,1= + C,1 C; for (x; t) D: (8)

6 98 Eitan Tadmor and Tao Tang By the property of, (x, X(t)) (x, X(t))=(x, X(t)) Const and the regularity ofu, u x Const for all (x; t) D, we obtain that p is also upper bounded. Again, due to the C -smoothness assumption on u, q is bounded in the domain of smoothness, D. This completes Step #. Finally, we need to verify Step #3, upper bounding E We rst check that the maximum value for E on the left boundary is bounded by O(). On the left boundary, wehave x, X(t) = 1= ; hence by x, and by e(x; t) = O( 1= ), we have E(x; t) = e(x; t)c = 1= : Choosing =1,wehave E(x; t) =O() on the left boundary of the domain D. On the right and the bottom of D, E(x; t) vanishes. This completes Step #3. Hence, the maximum principal gives E(x; t)c; for (x; t) D: This implies that the weighted error u (x; t), u(x; t) (x, X(t)) is bounded by O(), in particular for (x; t) bounded away from the shock curve x = X(t) we have O() pointwise error bound. Similarly, we can show that the same is true when (x; t) is on the left side of the shock. The argument can be extended when there are nitely many shocks. We summarize what we have shown by stating the following: Theorem.. Let u (x; t) be the viscosity solutions of (1)-(). Let u(x; t) be the entropy solution of (3)-(4), and assume it has nitely many shock discontinuities, then the following error estimates hold: For a weighted distance function, (x) min(x; 1), (u, u)(x; t) (x, X(t)) =O(): (9) In particular, if (x; t) is bounded away from the singular support of u, then (u, u)(x; t) C(h); for dist(x; S(t)) h>: (3) Since the weighted function (x) x, it follows from (9) that (u, u)(x; t) dist(x; S(t)),1 : (31) This implies that the thickness of the shock layer is of order O(). 3. The Lax-Friedrichs scheme The Lax-Friedrichs (LxF) scheme is of the following form: v n+1 = 1 v n +1 + vn,1, f(v n+1 ), f(vn,1 ) : (3)

7 Pointwise convergence rate 99 It is well known that the truncation error of the LxF scheme is O(t ). This, and the assumption that u xx is uniformly bounded in the region x>x(t), imply that U n+1, 1 U n +1 + U n,1 + f(u n+1 ), f(u n,1 ) = O(t ); (33) where U n := u(x ;t n );x = x; t n = nt. We require that := t=x satises the standard CFL condition For a xed integer n, we dene J(n) = min maxf (v n ) < 1: (34) n x X(t n )+t 1=4 o : (35) The following lemmas are useful in obtaining our error bounds. However, due to the limitation of space we will omit the detail proofs. They can be found in [7]. Lemma 3.1. For any given T> and given integer m>, there exists a positive constant C(m; T ) > such that v n, U n C(m; T )t1=4 ; J(n), m; n T=t; (36) where U n := u(x ;t n ), J(n) is dened by (35). The above lemma is established by using Lemma.1, the interpolation between L 1 and Lip + estimates. The uniform Lip + -bounds are obtained by Nessyahu and Tadmor [5]. In the discrete case, the optimal L 1 error bounds for the case with nitely many shocks are not available. The best result was obtained by Kuznetsov [3]: kv n, U n k L 1 := X xv n, U n Const t 1= : (37) With the above non-optimal L 1 -error bound, the order of the pointwise error bound (36) is even less than 1=. However, it will suce to derive the optimal error bound by a bootstrap argument. Lemma 3.. Let e n = vn, U n and n = Z 1 f (v n +1 +(1, )v n,1)d : In the smooth region of u, the following result holds: e n+1, 1 e n +1 + en,1 + n e n, +1 en,1 =tt n + O(t ) ; J(n) ; (38) where the term T n can be bounded by T n C, e n +1 + e n,1 : (39)

8 3 Eitan Tadmor and Tao Tang The above lemma is established by using Taylor expansions. The main part of our error analysis is to estimate E n+1, 1 E n +1 + E n,1 + n E n +1, E n,1 ; J(n); (4) where E n = (x,x(t n ))e n, is given in x (with = 3). Let n = (x,x(t n )) The following facts will be used frequently 8 < : The following Taylor expansion together with Lemma 3., lead to E n+1, 1 E n +1 + E n,1 = n 1 = n x( ) n + O(x ) ; n +1, n + n,1 =( ) n x + O(x 3 ) : n+1 = n, tx (t n )( ) n + O(x ) : + n E n +1, E n,1 n, tx (t n )( ) n + O(x ) 1 (en +tt n + O(t ), 1 E n +1 + E n,1 + n (41) +1 + e n,1), n (e n +1, e n,1) E n +1, E n,1 = n, n +1 n E n +1 + n, n,1 +1 n E n,1,,1 n n (e n +1, e n,1) (4) + n E n, +1 En,1, tx (t n )( ) n 1 (en + +1 en ),1 + tx (t n )( ) n n (en +1, en,1 )+Tn O(t)+O(t ) : It follows from (41) that, n n (en +1, en,1 )+ n + n = t n ( ) n E n, +1 En,1 =, n n (en +1, en,1 ) he n+1 (n +x( ) n ), en,1 (n, x( ) n ) i + O(x ) 1 (en +1 + en,1 )+O(t ) : where in the last step we have used the fact x =t. This result, together with (4), lead to E n+1, 1 E n +1 + E n,1 + n E n +1, E n,1 = I 1 + J 1 + I + J + I 3, J 3 + T n O(t)+O(t ) (43)

9 Pointwise convergence rate 31 for J(n), where I 1 = n, n +1 n ; J 1 = n, n,1 ; +1 n,1 I = t ( ) n n +1, X (t n )+ n ; J = t ( ) n n,1, X (t n )+ n ; I 3 = tx (t n ) ( ) n n n ; +1 It follows from (43) that J 3 = tx (t n ) ( ) n n n :,1 E n+1 = 1, n + I 1 + I + I 3 E n n + J 1 +J, J 3 E n,1 + Tn O(t)+O(t ) ; (44) for J(n). We can further show that for J(n), 8 < : I 1 + J 1 Ct 3= ; I Ct; J Ct; I 3, J 3 Ct 3= ; I k O(t 3=4 ) ; J k O(t 3=4 ) ; k =1; ; 3: It follows from (34) and the fact f > that (45) max n < 1: (46) This and the last two inequalities of (45) imply that the coecients for E n in 1 (44) are nonnegative, provided that t is suciently small. This result implies that from (44) and the rst three inequalities in (45) we have E n+1 1+Ct 1+Ct max J(n),1 max J(n),1 In other words, we have proved that max J(n) E n+1 1+Ct E n + Ct(e n +1 + e n,1)+ct E n + Ct for J(n): (47) max J(n),1 E n + Ct : (48) The above inequality is not an exact Gronwall type inequality. By using the information on the numercal boundary = J(n), it can be improved to the standard Gronwall type inequality which yields max E n Ct: (49) J(n);nN This result implies that v n, u(x ;t n ) can be bounded by Ct, if(t ;t n )ison the right side of the shock and is of any O(1) distance away from the shock curve. Similarly, we can show that the same is true if (x ;t n ) is on the left side of the shock curve. The argument can be extended when there are nitely many shocks. We summarize what we have shown by stating the following:

10 3 Eitan Tadmor and Tao Tang Theorem 3.3. Let fv n g be the solution of the Lax-Friedrichs scheme (3). Let u(x; t) be the entropy solution of (3)-(4), and assume it has nitely many shock discontinuities, then the following error estimate holds: For a weighted distance function, (x) min(x 3 ; 1), v n, u(x ;t n ) (x, X(t n )) =O(t) : (5) In particular, if (x ;t n ) is bounded away from the singular support, then v n, u(x ;t n )C(h)t; dist(x ;S(t n )) h>: (51) Acknowledgment. Research was supported in part by ONR grant N14-91-J- 176, NSF grant DMS and NSERC Canada grant OGP References [1] B. Engquist and S.-H. Yu, Convergence of nite dierence schemes for piecewise smooth solutions with shocks, Preprint (1997). [] J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 11 (199), pp [3] N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a rst-order quasi-linear equation, USSR Comput. Math. and Math. Phys., 16 (1976), pp [4] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., 8 (1991), pp [5] H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal., 9 (199), pp [6] E. Tadmor and T. Tang, Pointwise error estimates for scalar conservation laws with piecewise smooth solutions. UCLA CAM Report 98-3, (1998). [7] E. Tadmor and T. Tang, In preperation. [8] T. Tang and Z.-H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp., 66 (1997), pp Department of Mathematics, UCLA, Los Angelas, California 995; School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Iseral. address: tadmor@math.ucla.edu Depratment of Mathematics, Simon Fraser University, Vancouver, B.C., Canada. address: tangtao@cs.sfu.ca