Research Collection. On the definition of nonlinear stability for numerical methods. Working Paper. ETH Library. Author(s): Pirovino, Magnus

Size: px

Start display at page:

Download "Research Collection. On the definition of nonlinear stability for numerical methods. Working Paper. ETH Library. Author(s): Pirovino, Magnus"

Josephine Thomas
6 years ago
Views:

Research Collection Working Paper On the definition of nonlinear stability for numerical methods Author(s): Pirovino, Magnus Publication Date: 1991 Permanent Link: https://doi.org/10.

1 Research Collection Working Paper On the definition of nonlinear stability for numerical methods Author(s): Pirovino, Magnus Publication Date: 1991 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 On the Denition of Nonlinear Stability for Numerical Methods Magnus Pirovino Research Report No October 1991 Seminar fur Angewandte Mathematik Eidgenossische Technische Hochschule CH-8092 Zurich Switzerland

3 On the Denition of Nonlinear Stability for Numerical Methods Magnus Pirovino Seminar fur Angewandte Mathematik Eidgenossische Technische Hochschule CH-8092 Zurich Switzerland Research Report No October 1991 Abstract In this paper a new denition of nonlinear stability for the general nonlinear problem F (u) = 0 and the corresponding family of discretized problems Fh(uh) = 0 is given. The notion of nonlinear stability introduced by Keller [Kel75] and later by Lopez-Marcos and Sanz-Serna [LS88a] have the disadvantage that the Lipschitz constant of the derivative offh(uh) has to be known which, in many applications, is not practicable. The modication proposed here allows us to use linearized stability in a ball containing the solution uh to get nonlinear stability. The usual result remains true: nonlinear stability together with consistency implies convergence. Keywords: nonlinear stability, linearized stability Mathematics Subject Classication: 65H10, 65L20, 65M12, 65N12

4 1. Introduction and preliminaries. Let us consider the general nonlinear problem F (u) =0 (1) where F : D F X! Y is a dierentiable mapping between the Banach spaces X Y with domain D F. We only consider isolated solutions u of (1), i.e. the Frechet derivative d u F : X! Y shall be boundedly invertible. A numerical method may be applied in order to solve equation (1). This leads in general to a family of equations F h (u h )=0 (2) where the F h : D Fh X h! Y h are mappings between nite dimensional Banach spaces. Here the domain D Fh is an open set and F h () is continuous on D Fh. The subscript h indicates the dependence of the discretization on a small parameter such as the mesh size. Let us assume that h takes values inaset H of positive parameters with inf H = 0 and sup H = h 0 < 1.In order to dene convergence of the approximating solution u h of (2) to u, we require that there exist linear mappings obeying the condition R h X : D R h X X! X h and R h Y : D R h Y Y! Y h lim kr h (x)k = kxk x 2 D R h h!0 2fX Y g y : Here the D R h are required to be dense in.for example, it may be reasonable to work with a function space X,say L 2 (), and with a restriction represented by a grid function U(i j) =u(ih jh) (i j) 2 Z 2. The domain of this restriction is a set containing all continuous functions C(). But it is impossible to dene such a restriction for all u 2 L 2 (). Let us use henceforth the notation [z] h R h (z) 2fX Y g : Convergence of the approximating solution u h to u is now meant inthe sense that lim ku h ; [u ] h k =0: h!0 y We suppress indices of norms: kzk kzk z2 2fX Y Xh Y h g. 2

5 Also the denition of consistency makes no diculties. We introduce the local discretization error at a point u 2 D R h X : h (u) F h ([u] h ) ; [F (u)] h Denition 1 The family ff h ()g is said to be consistent of order p with F () at u i for some constant M, independent of h, k h (u)k Mh p : (3) A further denition will be useful because our original problem (1) as well as the discretized problems (2) might have more than one isolated solution. Hence, the original solutions have to be related to the right discretized ones. Denition 2 We call the family ff h ()g a proper discretization of F () for the solution u i there existsaradius >0, independent of h,such that the following two conditions are satised for all relevant values of h : (i) There are solutions u h of (2) such that B (u h) D Fh and F h () restricted on B (u h ) is one-to-one. (ii) [u ] h 2 B (u h) 2. Classical denitions of stability. There are several attempts to dene nonlinear stability in the literature, see [Ste73], [Kel75], [LS88a] or [LS88b]. For a summary of these dierent notions see [LS88a]. They all require the existence of a stability constant S, independent of h, such that a stability inequality ku h ; v h kskf h (u h ) ; F h (v h )k (4) holds for u h v h in some subset D h D Fh. Let us call D h the domain of stability. It is exactly this domain of stability which distinguishes all the notions of nonlinear stability mentioned above. Clearly, ifwechoose D h too large it may be dicult to nd a suitable stability constant S satisfying (4). On the other hand, if we choose D h too small, e.g. diam(d h )=O(h q ), it may occur that we must impose a condition k[u ] h ; u hk = O(h q ), which itself guarantees convergence and our stability analysis becomes meaningless. 3

6 Let us select H. B. Keller's denition of stability. He chooses D h = B (u h) y, the open ball of radius about u h. Here the stability threshold must be positive and independent ofh. The problem now ishowto determine and S. Let us assume that the linearized problem at u h is stable with stability constant S 0 independent of h, which is equivalent to the condition k[d u h F h ] ;1 ks 0 : If, in addition, the Frechet derivative d u h F h has a Lipschitz constant K 0 independent of h in some open ball B 0 (u h), i.e. kd uh F h ; d vh F h kk 0 ku h ; v h k u h v h 2 B 0 (u ) h (5) then it can be shown (see [LS88b]) that we maychoose an S>S 0 to get the condition arbitrarily = minf 0 (S ;1 0 ; S ;1 )=K 0 g : We have transformed the problem of nding and S into another problem, namely nding S 0 and K 0.Itmaybeeasy to handle the linearized problem, but to determine K 0 means in practice that we must estimate the norm of the second derivative d 2 u h F h, which for a wide series of applications does not even exist. Nevertheless, this denition of stability might be useful. An important lemma of Stetter gives us the connection to convergence. Lemma 1 ([Ste73]) Let X h Y h be nite dimensional Banach spaces and let F h : B (u h) X h! Y h beacontinuous mapping with F h (u h)=0. If the bound (4) holds on D h = B (u h) for some constant S, independent of h, then the inverse F ;1 h : F h (B (u ))! h B (u h ) is Lipschitz continuous with Lipschitz constant S and B =S (0) F h (B (u h)) : Now we may state a convergence result. Theorem 1 Let the family ff h ()g be consistent of order p>0 with F () at u and stable on the stability domain B (u h) with stability constant S. If, y In fact he chooses D h = B ([u ] h ), but this dierence is only of technical nature. 4

7 in addition, it is a proper discretization for F () at u (i.e. [u ] h 2 B (u ) h ), then 1=p ku h ; [u ] h kmsh p h : MS The order of convergence is not smaller than the order of consistency. Proof. Since by (3) k h (u )k = kf h ([u ] h ) ; [F (u )] h k = kf h ([u ] h )kmh p we see that h ( MS )1=p implies F h ([u ] h ) 2 B =S (0). Thus, Lemma 1 applies and the result follows by the stability inequality (4). From the proof of Theorem 1 we learn the following: the inverse F ;1 h is not required to be Lipschitz continuous on the whole range F h (B (u h )) to imply convergence for the numerical method. It would suce for F ;1 h to be Lipschitz continuous on B =S (0) only. This is exactly the point wherewe want to modify the denition of nonlinear stability. 3. Stability by range. Since we only consider isolated solutions u of (1), the Frechet derivative d u F : X! Y is boundedly invertible. If F () is continuously dierentiable y, then, by the inverse mapping theorem (see e.g. [Rud73]), there exists a ball B (u ) such that F : B (u )! F (B (u )) is an isomorphism. This allows us to dene an inverse mapping F ;1 on F (B (u )). It is the inverse mapping we want to compute, therefore we think stability in some sense has to be dened for this inverse mapping. Denition 3 Let F h (u h)=0 and let R h Y h beanopen set with 0 2 R h. The family ff h ()g is said to be stable on the stability range R h with stability constant S, independent of h, i for all relevant h > 0 (i) F ;1 h : R h! X h can be uniquely dened with u h 2 F ;1 h (R h ), (ii) F ;1 h () is Lipschitz continuous on R h with Lipschitz constant S. Again, we get the convergence result. yrecall that in general F () need not be continuously dierentiable since it is only dened on a subset D F, but we will see that this requirement (at least the continuity) is essential for the family ff h ()g. 5

8 Theorem 2 Let the family ff h ()g be consistent of order p>0 with F () at u and stable on the stability range B r (0) with stability constant S. If, in addition, it is a proper discretization for F () at u with Sr, then r 1=p ku ; h [u ] h kmsh p h : M Proof. As in the proof of Theorem 1 inequality (3) yields k h (u )k = kf h ([u ] h )kmh p : Thus, h (u ) is in B r (0) if only h (r=m) 1=p. F ;1 h () is Lipschitz continuous on B r (0) with Lipschitz constant S. This implies F ;1 h (B r (0)) B Sr (u h) B (u h). Since the discretization is proper we conclude [u ] h 2 F ;1(B h r(0)) and [u ] h = F ;1( h h(u )). The result follows now by the Lipschitz continuity of F ;1 (). h 4. Nonlinear stability by linearized stability. We want tohave a practicable tool that allows us to verify the assumptions of Theorem 2. It would be helpful if we could work with the linearized problems L h (u n ) F h (v h )+d vh F h (u h ; v h )=0 (6) for suitable v h 2 X h. Clearly, the linear family fl h ()g is stable in all above senses i there is a constant S, independent of h,suchthat We will now state our main theorem. k[d vh F h ] ;1 ks: (7) Theorem 3 Let the family ff h ()g satisfy the following conditions for all relevant values of h : (a) ff h ()g is a proper discretization for F () at u on B (u h). (b) F h () is continuously dierentiable on B (u h). (c) The linearized families fl h ()g of (6) are stable with stability constant S uniformly for v h 2 B (u ). (d) The constants and S are independent of h. 6

9 Then the family ff h ()g is stable on the stability range B =S (0) with stability constant S. Corollary 1 Let the assumptions of Theorem 3 hold. If, in addition, ff h ()g is consistent of order p with F () at u then ku h ; [u ] h k = O(h p ) : The order of convergence is not smaller than the order of consistency. To proof our main theorem, it suces to proof the following, more general lemma. Lemma 2 Let X Y be arbitrary Banach spaces and let the mapping f : B (u ) X! Y satisfy the conditions (a) f(u )=0 (b) f() is continuously dierentiable and one-to-one on B (u ). (c) d u f : X! Y is a linear isomorphism and k[d u f] ;1 ks, uniformly for u 2 B (u ). Then the following holds. (i) f : B (u )! V f(b (u )) is a dieomorphism. In particular f ;1 () can be uniquely dened on V. (ii) B =S (0) V (iii) f ;1 is Lipschitz continuous with Lipschitz constant S on B =S (0). Proof. Because f() onb (u ) is one-to-one, we get (i) from the inverse mapping theorem. For the rest we may assume that Y nv is nonempty, for otherwise (ii) isalways satised and (iii) follows from Taylor's formula f ;1 (y) ; f ;1 (y 0 )= Z 1 0 d ty+(1;t)y 0f ;1 (y ; y 0 )dt (8) (see e.g. [Die69, Theorem ]) together with hypothesis (c). Thus, let y 2 Y nv and r 1 = kyk + 1, i.e. B r1 (0)nV 6=. By the inverse mapping theorem we know that there is a 0 <r 0 <r 1 with B r0 (0) V.Wedene r =supfs 2 [r 0 r 1 ]:B s (0) V g : (9) 7

10 Thus, B r (0) is the biggest ball around zero, which liesentirely in V. We maynowchoose >0 arbitrarily. B r+ (0) is no more a subset of V. We maythereforechoose y 2 (Y nv ) \ B r+ (0). Let T y = fty :0 t<1g and t = supf : T y V g : If we could walk on T y beginning at zero, we would leave V for the rst time at the point y = t y. Clearly, y =2 V by the denition of t, but y = V nv and by (9) we have r ky kr + : (10) If we choose a sequence fy k gt y V with limit y, then the sequence fv k = f ;1 (y k )g is well dened in B (u ). Since T y is convex, we mayapply Taylor's formula to get v k ; v m = Z 1 and together with hypothesis (c) 0 d tyk +(1;t)y m f ;1 (y k ; y m )dt k m 1 : kv k ; v m ksky k ; y m k k m 1 : (11) By this, both fy k g and fv k g are Cauchy sequences. Let v = lim k v k.itis impossible for v to be an element of B (u ), for else, by thecontinuity of f(), y = f(x ) =2 V,which is a contradiction. Hence, v (u ), i.e. kv ; u k =.Wemaychoose y m = 0 in (11) and pass to the limit then we have, recalling (10), = kv ; u ksky ks(r + ) : Since can be chosen arbitrarily, (ii) is proven. B r (0) is convex thus again, we can apply Taylor's formula and verify (iii) byhypothesis (c). This completes our proof. 5. Numerical examples. Let us look at the nonlinear problem ;u xx + g(u) ; f = 0 x 2 [0 ] (12) u(0) = u() = 0 8

11 where g :IR! IR is a contraction (i.e. g() is Lipschitz continuous with Lipschitz constant L<1 ) and f 2 L 2 (0 ). Here F (u) =Au + g(u) ; f, where A is an unbounded self-adjoint linear operator with dense domain D F in L 2 (0 ) and spectrum (A) =fk 2 : k 1g. Problem (12) has a unique solution u 2 L 2 (0 ) (see [Bra91]). If f is continuous, i.e. f 2 C 0 (0 ), then u 2 C 2 (0 ). In the following we will always assume that f is continuous. Let us discretize (12) with central dierences on the grid fih : i = 0 ::: n+1 h= =(n +1)g.We get a system with u h f h 2 IR n F h (u h )=A h u h + g(u h ) ; f h =0 (13) and A h = 1 h ;1 ;1 2 ; ;1 2 ;1 ; IR nn : Let us use the notation U u h. Then by g(u) we denote the vector [g(u 1 ) g(u n )] T 2 IR n. The boundary conditions U 0 = U n+1 = 0 imply that we canwork with the reduced grid fih : i =1 ::: ng. The restriction from L 2 (0 ) on this reduced grid is surely dened on the dense set C 0 (0 ). We write f h =[f] h [f(h) f(2h) f(nh)] T. Let us use the discrete L 2 - norm on grid functions kuk =( nx i=1 U 2 i h) 1=2 : We rst show that the family ff h ()g is a proper discretization for the solution u of (12) with = 1. As one can easily check, this can be done by showing that the system (13) has a unique solution U 2 IR n.itiswellknown that A h has eigenvalues k = 4 sin 2 (kh=2)=h 2 k=1 ::: n. This implies ka ;1 k = h 2 h 4sin 2 (h=2) : From now onwe only admit values of h so small that a condition h 2 L 4 sin 2 (h=2) L0 < 1 9

12 can be satised. System (13) has a unique solution i the function G h : IR n! IR n dened by G h (U) =;A ;1 h (g(u) ; f h ) has a single xed-point. Since g() is a contraction, we have kg h (U) ; G h (V )k ka ;1 h kkg(u) ; g(v )k LkA ;1 h kku ; V kl 0 ku ; V k hence G h () isacontraction, too. Now the result follows by the contraction theorem. Next we verify that the linearizations of (13) are uniformly stable with stability constant L 0 S = L(1 ; L 0 ) provided that g( ) is continuously dierentiable. From (13) we derive d U F h = A h + diag(g 0 (U)) = A h [Id+ A ;1 h diag(g0 (U))] : Thus, by the Banach lemma [Nas76, p. 333] kd U F ;1 h k ka ;1 h k 1 ;kg 0 (U)k 1 ka ;1 k L0 h L(1 ; L 0 ) : Now all assumptions of Theorem 3 hold for = 1. We conclude that the family ff h ()g is stable on the stability range IR n with stability constant S = L 0 =(L(1 ; L 0 )). It is a well-known property of the central dierence discretization that it is consistent of order p =1 provided the solutions u lie in C 3 (0 ), which is true if both f() and g() are continuously dierentiable y. The solutions U of (13) therefore converge to u with order not smaller than p =1. We mention that we have never imposed a condition on the Lipschitz continuity of the derivative d U F h as in (5). This is the main advantage of our analysis. y If f() and g() are twice continuously dierentiable then u 2 C 4 (0 ) which implies consistency of order p = 2. 10

13 We conclude this paper with a remark on partial dierential equations. Exactly the same kind of analysis as we have made for equation (12) can be done on related two-dimensional problems such as u tt ; u xx + g(u) ; f = 0 u(0 t)=u( t) = 0 u 2-periodic : For more details on these problems see [Bra91]. References [Bra91] R.A. Brawer. Numerik und Existenzsatze schwacher Losungen nichtlinearer partieller Dierentialgleichungen. PhD thesis, ETH Zurich, [Die69] [Kel75] J. Dieudonne. Foundations of Modern Analysis. Academic Press, New York, H.B. Keller. Approximation methods for nonlinear problems with application to two-point boundary value problems. Math. Comp., 29:464{474, [LS88a] J.C. Lopez-Marcos and J.M. Sanz-Serna. Denition of stability for nonlinear problems. In K. Strehmel, editor, Numerical Treatment of Dierential Equations, Teubner, Leibzig, [LS88b] J.C. Lopez-Marcos and J.M. Sanz-Serna. Stability and convergence in numerical analysis III: Linear investigation of nonlinear stability. IMA J. Numer. Anal., 8:71{84, [Nas76] M.Z. Nashed. Perturbations and approximations for generalized inverses and linear operator equations. In M.Z. Nashed, editor, Generalized Inverses and Applications, Academic Press, New York, [Rud73] W. Rudin. Functional Analysis. Tata McGraw-Hill Company Ltd., New Dehli,

14 [Ste73] H.J. Stetter. Analysis of Discretization Methods for Ordinary Differential Equations. Springer, Berlin,

Extension and simple proof of Lekner's summation formula for Coulomb forces

Research Collection Woring Paper Extension and simple proof of Lener's summation formula for Coulomb forces Author(s): Sperb, René Peter Publication Date: 00 Permanent Lin: https://doi.org/0.399/ethz-a-0048407