Moving Mesh Methods with Applications to Blow-up ... - CiteSeerX

Moving Mesh Methods with Applications to Blow-up Problems for PDEs  Chris J. Budd

y

, Jianping Chen

z

, Weizhang Huang

and Robert D. Russell

x

,

{

Abstract In this paper, we give a short review of the moving mesh methods based

upon moving mesh PDEs, including a brief description of a new moving collocation method. Then we carry out a numerical study of a class of PDEs describing blow-up of solutions of combustion problems. The study demonstrates the usefulness of these methods as a tool for the analyst. Speci cally, we use the numerical method as an experimental approach to conjecture the solution pro le for blow-up problems and verify the conclusions obtained from a formal (and not necessarily rigorous) analysis of these PDEs.

1 The Principles Which Underly Moving Mesh PDEs (MMPDEs) 1.1 The Method of Lines (MOL)

It is now widely recognized that when solving partial dierential equations (PDEs) which have components of the solution which vary greatly over small length scales some form of adaptivity is necessary. In this paper we review some adaptive methods, based upon the method of lines for solving parabolic PDEs of the form ut = F (u; ux; uxx); 0 < x < 1 (1:1) In particular, we consider their behavior for problems that form singularities in a nite time.

This work was supported in part by the NSERC (Canada) under Grant OGP-0008781, the Nueld Fundation and the EPSRC (UK) under Grant GR/J56219, the University of Kansas General Research allocation #3297-XX-0038, and NSF (USA) under EPSCoR grant OSR-9255223. y School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. ([email protected]). z Department of Mathematics and Statistics, Simon Fraser University, Burnaby B.C. V5A 1S6, Canada. ([email protected]). x Department of Mathematics, 405 Snow Hall, University of Kansas, Lawrence, KS 66045, U.S.A. ([email protected]). { Department of Mathematics and Statistics, Simon Fraser University, Burnaby B.C. V5A 1S6, Canada ([email protected]). 

1

When solving PDEs with the method of lines, derivatives with respect to the time variable remain continuous while derivatives with respect to the other independent spatial variables are replaced by discrete approximations. Typically, such a PDE initial-boundary problem is then transformed into an ODE initial value problem. Thus if (1.1) is discretized in space we obtain a system of ordinary dierential equations (ODEs) of the general form

U_ = G(t; U ); U (0) = U where Ui(t) is an approximation of u(x; t) at the i-th mesh point xi. 0

(1:2)

It is this transformation, separating the temporal and spatial variables, that makes it possible to use dierent spatial meshes at dierent time levels. Separating the space/time discretization provides other advantages. For example, the separation can make results on stability or convergence easier to establish . Furthermore, the algorithm is straightforward to program as high quality ODE IVP software may be used, which changes the step size/method to maintain stability, and dierent spatial approximations can also be compared easily. The tradeo is that one can lose the overall optimization possible when time and space steps are chosen together. The purpose of the adaptive MOL methods is to concentrate the spatial mesh points Xi in regions where large variation of the solution or its derivatives occur, or higher accuracy is required. Two approaches are commonly used to adapt the mesh: the static approach and the dynamic approach. In the static approach, the mesh is redistributed at xed time levels. This requires an interpolation of the solution onto the new mesh points, and then the time integration should be restarted. The static approach can attain good error control but the integration restarts can be costly. In the dynamic approach (or moving mesh method), the mesh is adapted continuously in time, so that Xi is a function of time by using a separate set of mesh equations. When using such a method, interpolation is not necessary, and the moving mesh equations and physical PDE are integrated simultaneously by employing MOL with ODE (or DAE) solvers. At present weaknesses of this approach are the lack of a general stability and error analysis and diculties in extending it to higher spatial dimensions.

1.2 Development of Mesh Equations

Several features of the computation and analysis of the adaptive method of lines motivate the development of eective equations for moving the spatial mesh points Xi (t):  Computation of the mesh points deserves virtually as much consideration as computation of the physical solution itself. When a solution varies drastically in a small region, it is usually essential for the success of the computation that more mesh points should be distributed in that region.  The discrete mesh equations are dicult to analyze and compare. However, the continuous form of a moving mesh equation can provide a useful framework for 2

analysis and comparison of methods.  A well chosen mesh can inherit natural spatial structures present in the original PDE (such as scaling properties).  A useful general tool for determining the mesh equation is equidistribution. The Equidistribution Principle. The basic idea of equidistribution, dating back to de Boor [2] and Dodson [5], takes some measure of the error and presumes that a good choice for a mesh is one for which the contributions to the error over the mesh intervals are equalized, or distributed equally. Based on this principle, de Boor [2] gave the following algorithm: Given a smooth monitor function M (x; t; u(x; t)) > 0, x 2 [0; 1], the mesh points Xi (t), i = 0; :::; N are taken to satisfy the integral identity

or equivalently

XiZ+1 (t)

Z1

Xi (t)

0

Mdx  N(t) ; (t) := Mdx: XZi (t) 0

M (x; t)dx = Ni (t)

(1:3)

Continuous equidistribution equation. By considering a mesh function x(; t) of a computational coordinate  2 [0; 1] so that Xi (t) = x(i=N; t), the above algorithm can be interpreted in a continuous form such that x(; t) satis es xZ(;t)

Mdx = (t)

(1:4)

0

with

x(0; t) = x(1; t) ? 1 = 0 ; (1:5) i.e., we nd a coordinate transformation x = x(; t) such that the transformed solution U (; t) to the PDE is smooth in , and then use a uniform mesh in the computational coordinate . From (1.4), we get, @ M @x (; t) = 0: (1:6) @ @ !

Discretization of (1.6) and the PDE (1.1) leads to a system of DAEs for the solution Ui and the mesh points Xi.

3

2 Moving Mesh Partial Dierential Equations (MMPDEs) It is well known [12] that moving mesh methods based on (1.6) can be unstable and that some sort of smoothing of the mesh is often necessary in order to obtain nonoscillatory, reasonably accurate solutions. We use smoothing in both the temporal and spatial variables. Temporal smoothing: By introducing a suitably small relaxation time  after which the mesh is to reach equidistribution, linearizing (1.6) at t +  and dropping higher order terms in  , we get [14] that

@ @ M @x @t @ @

@ M @x : = ? 1 @ @ Two simpli ed versions of (2.1) often used are @ M @ x_ = ? 1 @ M @x "

!#

!

!

and

@

!

@

 @

@

@ x_ = ? 1 @ M @x @  @ @ 2

!

x 1 x_ = ? [ Mdx ? (t)] Z



(2:2) (2:3)

2

for which

(2:1)

(2:4)

0

where we use x_ for (@x)=(@t). In the latter case the relaxation time becomes =M . Solving these equations has the additional advantage that we may start with an initial mesh Xi (0) which is not well equidistributed. Spatial mesh smoothing: Spatial mesh smoothing can be done by using arti cial diusion for smoothing M (see [12]), i.e. instead of using the monitor function M , we use M~ which satis es the equation ~ M~ ? 1 @@M = M: (2:5) 2

2

2

For example, if we replace M by M~ in (2.2) and use the identity (2.5) we obtain (after some manipulation) the mesh equation

@ @

8 >
:

1

2



9 @2 1 @ x _ = 1 @x 2 @ @ +  @ >

M

> ;

= 0:

(2:6)

This equation is then discretized and solved as before. Such smoothing leads to longtime regularity of the mesh. Indeed it is proven in [12] that 4

Theorem 1 For the MMPDE (2.6), there exist certain constants  ;  , depend1

ing upon M such that, (i)

2

@x (; 0)e? t  @x (; t)  @x (; 0)e? t +  ; @ @ @  1

0

(ii)



@ x (; t)= @x (; t)   : @ @

2

2

This result has the discrete analogue

Theorem 2 For the central nite dierence approximation of (2.6), (iii) e? t [ xi (0)h? xi(0) ]  xi (t)h? xi(t)  e? t [ xi (0)h? xi(0) ] +  : +1

+1

(iv)

+1

1

0

q

1+  h ?1   xxi(t)(t?) ?x xi((tt))   ? ;  := ; h := N1 : i i? 1+  h +1 +1

1

q

1

4

2 2

4

2 2

The result (ii) is a continuous version of the result that an initially locally quasiuniform mesh remains so for all future times. The numerical method of Dor and Drury [6] can be described by using these results. Particular discretizations of (2.1) { (2.6) lead to most of the previous moving mesh methods (in one spatial dimension), e.g. those of Anderson [1], Flaherty et al.[7], Greenberg [9], Hindman-Spencer [10], Hyman-Larrouturou [11], Madsen [16], and Petzold [18]. To avoid a breakdown of these methods, it is essential that the mesh so obtained has no node-crossings. Theorem 1 shows also that for moving mesh method (2.3), (2.6) no node-crossing occurs, i.e. @@ x(; 0) > 0 implies @@ x(; t) > 0.

3 The Moving Collocation Method (MOVCOL) We have used two basic discretizations to produce the moving mesh systems based on the method of lines. In the moving nite dierence (MFD) approach, we discretize the MMPDEs and the physical PDEs in  with equal spacing, and integrate the resulting system. In the moving collocation approach, we discretize the MMPDEs in  with three-point nite dierences on a uniform mesh, discretize the physical PDEs in x with Hermite cubic collocation on the corresponding nonuniform mesh, and integrate the resulting system. This moving collocation approach from [13] (and implemented in the code MOVCOL) is sketched below. The motivation for it is that the moving mesh points do not need to be resolved with the same accuracy as the solution to the 5

PDE itself, so it is reasonable to use a lower order discretization for the MMPDE than the physical PDE. We now consider a second-order parabolic PDE in divergence form

@ G(t; x; u; u ; u ; u ); x t xt @x xL(t) < x < xR(t); ta < t  tb

F (t; x; u; u;ut; uxt) =

(3.1)

which satis es the conservation law xR

Z

xL

Fdx = Gjx

xR ? Gjx=xL

:

=

(3:2)

Given a mesh (determined from MMPDE discretization)

X (t) := xL(t) <


< XN (t) := xR(t);

(3:3)

1

the piecewise cubic Hermite collocation solution is de ned as

v(x; t) =

vi(t) (s i ) + vx;i(t)Hi(t) (s i ) +vi (t) (s i ) + vx;i (t)Hi(t) (s i ); (3.4) on x 2 [Xi(t); Xi (t)]; i = 1;


; N ? 1, where s i := (x ? Xi (t))=Hi(t); Hi (t) := Xi (t) ? Xi(t) (3:5) and the i are the standard shape functions. The collocation scheme we use satis es 1

( )

+1

3

( )

2

( )

+1

4

( )

+1

( )

+1

the conservation law and is similar to standard cubic Hermite collocation except that cell averaging is enforced. Speci cally, a cell average is taken on each half of [Xi; Xi ], e.g. +1

Xi+1Z+Xi )=2

(

Fdx = Gi

? Gi ;

=

+1 2

Xi

(3:6)

Taking the piecewise linear approximation

F

F (Xi ; t)(x ? Xi )=(Xi ? Xi ) (3.7) +F (Xi ; t)(x ? Xi )=(Xi ? Xi ) (3.8) and using two-Gauss point Xij (j = 1; 2) quadrature in (3.6), after some simpli cation 1

2

2

1

1

2

2

1

we get

F (X i ;i

( 1 2)

; t) = H1 ?(1  p2 )Gi  p4 G 3 3 i

=

1+1 2

6

+ (1  p2 Gi 3

+1

!

:

(3:9)

This discretization satis es the discrete analogue of (3.2) NX ?1 H

i

i=1

2 (F (Xi ; t) + F (Xi ; t)) = GN ? G : 1

2

p

(3:10)

1

The truncation error for (3.9) is ( 3=864)(@ =@x )Hi . Note that this approach is a standard collocation when the PDE is in non-divergence form. A three-point discretization of MMPDE (2.6) is: 4

4

3





(1=Mi = ) Yi = ? ( + 1)(Yi = ? 2Yi = + Yi? = ) ?(1=Mi? = ) Yi? = ? ( + 1)(Yi = ? 2Yi? = + Yi? = ) = 0;

(3.11)

Yi = := ? (X_ i ? X_ i)=(Xi ? Xi ) + (1=(Xi ? Xi); Y? = := Y = ; YN = := YN ? =

(3.12)

+1 2



1 2

+1 2

+3 2

1 2

+1 2

+1 2

1 2

1 2



3 2

where +1 2

+1

1 2

1 2

2

+1

+1 2

+1

1 2

and the monitor function is de ned by

Mi

=

+1 2

:= M (t; x; v; vx; vxx; vt; vxt)jx

Xi +Xi+1 )=2;

=(

(3:13)

We now consider an application of this method to two problems which develop singularities.

4 Problems With Solutions That Blow-up in a Finite Time Many partial dierential equations modelling physical phenomena have solutions which blow up (become in nite) in a nite time. Typically this is associated with the formation of a singularity or with the breakdown of an asymptotic description of the problem. Some examples of problems which exhibit blow-up are  the temperature of a reacting or combusting medium:

ut = uxx + up; (p > 1) or ut = uxx + eu ;

(4:1)

 the temperature of a reacting nonlinear medium: ut = (juxj ux)x + eu; or ut = uuxx + up

(4:2)

 the self-focusing phenomena described by the nonlinear Schrodinger equation: i t +
+ j j  = 0 : (4:3) 2

7

In a problem exhibiting blow-up typically (but not always) the solution becomes in nite at a single (blow-up) point x in a nite (blow-up) time T  1, that is as

t!T

u(x; t) ! 1 and u(x; t) ! u(x; T ) < 1; if x 6= x (4:4) Close to the point x the solution develops a singular spike of increasing height and decreasing width as t approaches T . If a xed mesh method is applied to solve such problems then, as t approaches T , the width of the spike will ultimately become smaller

than the mesh size and the method will cease to resolve the blow-up behavior, leading sometimes to numerical solutions which blow-up over the whole domain rather than at a single point. Thus it is essential to use an adaptive method, which can move points into the spike, to compute the solutions of a blow-up problem accurately. Blow-up at a xed point To show the eectiveness of this approach we initially consider the model problem: ut = uxx + up; 0 < x < 1 (4:5) u(0; t) = u(1; t) = 0; u(x; 0) = u (x) > 0 (4:6) where u (x) is both suciently large to ensure that blow up occurs and has a single maximum so that the blow-up is at the single point x, with 0 < x < 1. Behavior of the solution In the absence of boundary conditions, the partial dierential equation (4.5) is invariant under the rescaling: (T ? t) = (T ? t); (4.7) (4.8) u = ? p? u; (x ? x) =  (x ? x); (4.9) It was originally thought that close to blow-up the solutions of (4.5) were self-similar, that is they were invariant under this rescaling and took the form 0

0

1

(

1)

1 2

u(x; t) = (T ? t)? p? v(y); y = (T ? t)? (x ? x) for an appropriate function v(y). 1

(4.10) (4.11)

1

1 2

It is now known that the form given in (4.10,4.11) is not quite correct and instead the blow-up peak is approximately self-similar under (4.7-4.9) with (4:12) u(x; t) = ( p ?1 1 ) p? (T ? t)? p? (1 + ( p 4?p 1 ) )? p? : Here (x; t) = (x ? x)(T ? t)? j log(T ? t)j? = and (4.12) is valid provided that the variable  (the ignition kernel) is not large. We note that these pro les are independent of the initial data. The variable  (or indeed 1

(

1

8

1

1)

2

1 2

1 2

1

1

the variable y) are natural computational coordinates for solving (4.5), and in Berger & Kohn [3] the problem was rescaled using (4.7-4.9) to make explicit use of y and solved using a static remeshing approach. We now describe a somewhat simpler algorithm based upon MMPDEs. Numerical results using MMPDEs [4] The scaling transformation (4.7-4.9) plays an important role in the solution of (4.5), and it is natural to seek a numerical method which is also invariant under such a rescaling. This is not possible with a xed mesh method but can be achieved with an adaptive method. In particular, if we use the MMPDE6 of [4] with

@ x_ = ? 1 @ M @x : @  @ @ !

2

(4:13)

2

then this equation is invariant under (4.7-4.9) if

M = up?

(4:14)

1

Remarkably, we nd that if problem (4.5) is solved together with (4.13) then the computed mesh points Xi (t) close to the singularity lie exactly along the trajectories for which (Xi (t); t) is constant. That is, there are functions W (), y() independent of t, such that close to the peak the computed solution U (; t) and coordinate transformation x(; t) take the form

U (; t) = (T ? t)? p? W (); x(; t) = x + (T ? t) j log(T ? t)j y() : (

1

1 2

1 2

1)

(4:15)

This result demonstrates the power of the MMPDE methods to inherit natural spatial structures in the original PDE. A careful asymptotic analysis presented in [4] gives W () = ( p ?1 1 ) p? [cos(( ? 12 ))] = p? (4:16) and (4:17) y() = 2 p ?p 1 tan(( ? 12 )) + O( ) where we note that jy()j ! 1 as  ! 1 or  ! 0. To demonstrate this behavior, we consider an example with p = 2, u = 20 sin x for which x = 0:5, and T = 0:082372. Using  = 10? (relaxation time de ned in (2.1)), ip = 1 (three-point spatial smoothing), atol; rtol = 10? (absolute and relative error tolerance, respectively, for DASSL) and 41 mesh points, we compute solutions up to ku(x; t)k1 = 6:0  10 . From (4.15{4.16), we expect that U (; t) ! cos (( ? 1 )); as t ! T (4:18) max U (; t) 2 1

2 (

1

1)

s

0

5

6

5

2

9

1

0.8

U/Umax

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Xi

Figure 1: Convergence of scaled solutions in computational coordinate Ut = Uxx + U^2 700000 ’p2s4e3i1’ ’p2s4e3i2’ ’p2s4e3i3’ ’p2s4e3i4’ ’p2s4e3i5’ ’p2s4e3i6’ ’p2s4e3i7’

600000

500000

U(x,t)

400000

300000

200000

100000

0 0

0.2

0.4

0.6

0.8

1

x

Figure 2: Physical solutions In gure 1 we present computations (using the moving collocation method) of the function U (; t) made in the computational plane for kuk1 = 1:8  10 ; 3:75  10 ; 7:5  10 ; 1:5  10 ; 3:0  10 ; 6:0  10 where the solid line is the curve cos ( ? 1=2). These results are in close agreement with (4.18). In contrast, in Figure 2 we see the formation of the peak in the physical plane. It is clear that the adaptive method is clustering points eectively within the peak and resolving it well. 4

4

5

5

5

4

2

Blow-up in a degenerate problem A model for a uid in a channel with a temperature dependent viscosity, derived by Ockendon [17], takes the form

xqut = uxx + up;

u(0; t) = u(1; t) = 0; u(x; 0) = u (x) > 0 : 0

(4:19)

This system has been studied by Floater [8] and Lacey [15], and some numerical calculations (using a dierent method from ours) are reported in Stuart & Floater [20]. Rather less is known about this problem than the model problem (4.5), and numerical calculations are important in resolving its structure. We now show that 10

MMPDE methods are eective in solving (4.19) and present some formal arguments for the form of the resulting solution, which agree closely with the computed pro le. In Floater [8], it is shown that if u (x) is suciently large then blow-up occurs and that if p  q +1 the blow-up point is at the origin so that there is a sequence x(t) ! 0 such that u(x(t); t) ! 1 as t ! T . It is conjectured that if p > q + 1 then blow-up occurs at an interior point x 6= 0. Accordingly we examine the two cases q = 1; p = 3 and q = 2; p = 3 taking various initial data. 0

Case 1.Blow-up at an interior point when q = 1; p = 3: To study this case, four dierent initial conditions were used u (x) = 20 sin(x) (4.20) u (x) = 100 sin(x) (4.21) ? x u (x) = 30x(1 ? x)e (4.22) ? x u (x) = 60x(1 ? x)e : (4.23) We take 41 mesh points, ip = 1;  = 10? and atol = rtol = 10? (as de ned before). In each case the solution forms a singularity as predicted at an interior point x > 0. The resulting spatial pro les when kuk1 = 1:6  10 are given in Figure 3. 1

2

3

3

4

4

6

4

1 ’p3c1e3i7’ ’p3c2e3i7’ ’p3c3e3i7’ ’p3c4e3i7’

0.9 0.8 0.7

U/Umax

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

X

Figure 3: Physical solutions of dierent initial conditions of xut = uxx + u The computations also show that both the blow-up time and blow-up point will be aected by the initial conditions. There is evidence showing that the blow-up time is more closely related to the L norm than in nity norm k
k1 of the initial conditions. In principle the numerical method could be used reliably to determine x and T as functions of the initial data, but we do not do this here. As t approaches T , the equation (4.19) close to the blow-up point eectively becomes (x)q ut = uxx + up (4:24) which is a rescaling of (4.5) and is invariant under the transformation (4.7, 4.8, 4.9). In particular, we expect the solution to have a symmetric peak centred on x and that 3

2

11

the pro le of this peak should be described by (4.24). Correspondingly, the monitor function which helps to resolve the peak and which keeps (4.13) invariant is the same as given in (4.14), namely M (u) = u , and this was used for the computations. An extension of the previous analysis combined with the monitor function u gives a pro le for u(x; t) which is a rescaling of (4.12) with U (; t) and x(; t) taking similar forms to (4.15). In the case q = 1; p = 3 we then have 2

2

W () = p (x) cos(( ? 1=2)); Y () = 2 (x)? tan(( ? 1=2)) 1 2 q 2 3

(4.25) (4.26)

1 2

1 2

x Ut = Uxx + U^3 1 ’p3q1e3i2’ ’p3q1e3i3’ ’p3q1e3i4’ ’p3q1e3i5’ ’p3q1e3i6’ ’p3q1e3i7’ cos(pi*(x-.5))

0.8

U/Umax

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

xi

Figure 4: Scaled solutions with smoothed monitor The results of our computations obtained using a smoothed version M~ of the monitor u as a function of the computational function are presented in gure 4, which gives umax coordinate . These results appear to deviate from the pro le given in (4.26), but this is an artifact caused by the departure of the smoothed equations from the scaling invariance. If instead we project the results onto the computational coordinate that would be obtained without smoothing, we obtain the results presented in gure 5 which show clear convergence (away from the boundary) to the pro le given in (4.25).

Case 2 Blow-up at the boundary when q = 2; p = 3 For this case Floater's analysis predicts that blow-up will occur at the boundary such that the maximum value of u(x; t) (umax) will be attained at a point x(t) with x(t) ! 0 as t ! T . In Floater [8] it is stated that as t ! T , then for all  > 0 (T ? t) q < x(t) < (T ? t) q 1 +2

1 +2+



:

(4:27)

Using our numerical method we can now make this result more precise. To do this we make some formal conjectures as to the form of the solution and then obtain evidence 12

x Ut = Uxx + U^3 1 ’p3q1e3u2’ ’p3q1e3u3’ ’p3q1e3u4’ cos(pi*(x-.5)) ’p3q1o2’ ’p3q1o3’ ’p3q1o4’

0.8

U/Umax

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

xi

Figure 5: Scaled solutions with unsmoothed monitor for these conjectures numerically. The PDE (4.19) is invariant under the rescaling (T ? t) = (T ? t); (4.28) (4.29) x =  q x; ? u =  q p? u ; (4.30) and a natural set of variables to study (4.19) are coordinates related to this rescaling with q = 2; p = 3. Accordingly we set 1 ( +2)

2 ( +2)(

1)

x = (T ? t)? y; u = (T ? t)? w(y; s); s = ? log(T ? t) ; 1 4

1 4

(4:31)

Under this change of variables we have

ws = F (w) = wyy ? y4 (ywy + w) + w 2

w(0) = 0;

3

(4:32)

w(y; s) ! Ky ; y ! 1 ;

(4:33)

where K is a suitable constant. The conditions in (4.33) are derived from the boundary conditions in (4.19). A similarity solution of (4.19) invariant under (4.28 - 4.30) would satisfy the ordinary dierential equation F (w) = 0 (4:34) However, numerical experiments indicate that apart from the two special solutions w(y) = 0 and w(y) = py , all of the solutions of (4.34) grow exponentially as y ! 1. Accordingly, we seek (following the method adopted from that described in Berger & Kohn [3]) an approximately self-similar solution of (4.32) in the form 2

w = g(s)f (z); where z = g(ys) ; 13

(4:35)

To allow a consistent asymptotic expression we require that the function g(s) should grow with s so that g(s) ! 1; s ! 1 (4:36) Substituting (4.35) into (4.32) gives _ ? (f ? zfz ) ; ? z4 (zfz + f ) + f = g? fzz + gg 2

3

4

3

(4:37)

For large s ( t close to T ) the condition (4.36) implies that (4.37) reduces to the rst order equation ? z4 (zfz + f ) + f = 0 : (4:38) This reduction in order of the PDE is typical of blow-up problems. The equation (4.38) has the exact solution (4:39) f (z) = p1 p z ; 2 1+z and consideration of the higher order term in the equation (4.37) gives (to leading order) (4:40) g(s) = As where A is a constant. Combining these results we have the following conjectured solution pro le for u: x u(x; t) = p1 (T ? t)? (4:41) 2 1 + A T ?t xjlog T ?t j 2

3

4

1 4

1 2 q

4

4(

)

(

)

which in turn implies that

umax(t) = 21 (T ? t)?
Aj log(T ? t)j ;

(4:42)

x(t) = (T ? t)
Aj log(T ? t)j ;

(4:43)

1 4

1 4

1 4

and

1 4

ux(0; t) = p1 (T ? t)? :

(4:44) 2 The result (4.43) is entirely consistent with (4.27). The latter result (4.44) is particularly useful as we deduce from it that 1 2

ux(0; t)? = 2(T ? t) (4:45) so that a plot of ux(0; t)? as a function of t should be linear with slope ?2 with an intercept at t = T . These results have assumed a general functional form for u(x; t) without proof. To corroborate these conjectures we integrate the solution of 2

2

14

x^2 Ut = Uxx + U^3 0.0003 ’p3q2e3i’

0.00025

Ux(0,t)^(-2)

0.0002

0.00015

0.0001

5e-05

0 0

2e-05

4e-05

6e-05

8e-05 t

0.0001

0.00012

0.00014

0.00016

Figure 6: Estimation of T ? t as a function of ux(0; t) (4.19) with initial data u (x) = 20 sin x and the same parameters as before. The monitor function we use is based upon the invariance of the PDE (4.19) to the scaling transformations described in (4.28 { 4.30). An appropriate monitor function which leaves (4.13) invariant under this rescaling is 0

q

p?1)

(4:46) and we use this to study the solutions which blow-up at the origin. This is in contrast to our previous use of the monitor function M = up? for those problems with blow-up in the interior. In gure 6 we plot ux(0; t)? as a function of t, which clearly shows that (4.45) is an accurate description of the solution behavior. By using ux(0; t)? as an estimate for (T ? t) we may then give plots of log(umax(T ? t) ) and log(x(T ? t)? as functions of log(j log(T ? t)j). These are presented in Figure 7 and Figure 8. Both plots show the slowly growing behavior (linear with gradient ) expected of approximately self-similar rather than self-similar solution behavior. Thus our method

M =u

( +2)( 2

;

1

2

1 2

1 4

2

1 4

1 4

x^2 Ut = Uxx + U^3 1.58 ’p3q2e3t’ 1.56 1.54

Log( Umax (T-t)^(1/4) )

1.52 1.5 1.48 1.46 1.44 1.42 1.4 1.38 2.2

2.4

2.6

2.8 3 Log( | Log(T-t) | )

3.2

3.4

3.6

Figure 7: Estimation of umax as a function of T ? t 15

x^2 Ut = Uxx + U^3 1.58 ’p3q2e3t’ 1.56 1.54

Log( Umax (T-t)^(1/4) )

1.52 1.5 1.48 1.46 1.44 1.42 1.4 1.38 2.2

2.4

2.6

2.8 3 Log( | Log(T-t) | )

3.2

3.4

3.6

Figure 8: Estimation of x as a function of T ? t (with 41 points) is eective in dierentiating between these two forms of behavior. Other methods (such as the one presented in [3] ) require many more mesh points to make the same dierentiation. Finally we compute the function   h(; t) = u (ux ((tt)); t) : (4:47) max From (4.41, 4.43) we deduce that this function should be approximately constant in time and equal p (4:48) h(; t) = 2 p1 +  : In Figure 9 we present h(; t) for times when umax = 3:75  10 ; 7:5  10 ; 1:5  10 ; 3:0  10 ; 6:0  10 ; 1:2  10 . The convergence to the pro le in (4.48) is clear, and our numerical method gives con rmation of the asymptotic calculation. 4

3

4

4

4

3

5

5 Conclusions The numerical solution of dierential equations with an adaptive method can be interpreted as involving (explicit or implicit) computation of a coordinate transformation. Frequently the success of the computation depends critically on the properties of this coordinate transformation and the corresponding mesh. This is true for the computation of the blow-up problems. The basic numerical method we use involves collocation with cell averaging for the PDE and a low order discretization for the MMPDE. It can eciently handle the general divergence form of PDEs and general boundary conditions and is ideal for analyzing blow-up problems since a continuous approximation to the physical solution is obtained. MMPDEs are ideal candidates for mesh evolution, since they adjust the mesh in a way which utilizes the asymptotic features of the solutions, also preserving smoothness and 16

monotonicity of the continuous coordinate transformation. For the computation of the blow-up solutions, our MMPDE methods are shown to be very successful. Computations using a few mesh points can distinguish self-similar or approximately self-similar solutions. Major characteristics can be captured by our computations. In numerical computation, it is generally desirable for the numerical schemes to preserve key properties of the PDEs themselves. In the case of the blow-up computation, choosing M (monitor function) to preserve scaling invariance can make the mesh evolution inherit the properties of the solution and therefore mimic the asymptotic solution behavior. In this case the numerical method is suciently powerful to give strong support to some formal arguments for the form of the function u(x; t) satisfying (4.19), in particular, that the solution is approximately self-similar. It is interesting to note that such a veri cation allows a most precise analytic computation of the form of the solution, and that in this problem formal asymptotic and numerical methods are working closely together.

References [1] D. A. Anderson, AIAA Paper 83-1931, 1983, p.311 (unpublished). [2] C. de Boor \Good approximation by splines with variables knots. II" in Springer Lecture Notes Series 363, Springer-Verlag, Berlin, 1973. [3] M. Berger and R. V. Kohn, \A rescaling algorithm for the numerical calculation of blowing-up solutions." Comm. Pure Appl. Math. pp.841-863, 1988. [4] C. J. Budd, W. Huang, and R. D. Russell, \Moving Mesh Methods for Problems with Blow-up." SIAM J. Sci. Comput., to appear. [5] D. S. Dodson, \Optimal order approximation by polynomial spline functions." Ph.D. thesis, Purdue University, 1972. [6] E. A. Dor and L. O'c Drury, \Simple adaptive grids for 1-D initial value problems." J. Comput. Phys. 69, pp.175 { 195, 1987. [7] J. E. Flaherty, J. M. Coyle, R. Ludwig, and S. F. Davis, in \Adaptive Computational Methods for Partial Dierential Equations" edited by I. Babuska, J.Chandra, and J. E. Flaherty (SIAM, Philadelphia, 1983), p.144. [8] M. S. Floater, \Blow-up at the Boundary for Degenerate Semilinear Parabolic Equations." Arch. Rational Mech. Anal. 114, pp. 57{77 , 1991. [9] J. B. Greenberg, \A new self-adaptive grid method." AIAA J. 23, 317 (1985). [10] R. G. Hindman and J. Spencer, AIAA Paper 83-0450, 1983, p.1 (unpublished). 17

[11] J. M. Hyman and B. Larrouturou, Los Alamos, Report LA-UR-86-1678, (1986) (unpublished). [12] W. Huang and R. D. Russell, \Analysis of Moving Mesh Partial Dierential Equations with Spatial Smoothing", manuscript (1993). [13] W. Huang and R. D. Russell, \A Moving Collocation Method for Time Dependent PDEs", manuscript (1995). [14] W. Huang, Y. Ren, and R. D. Russell, \ Moving Mesh Methods Based on Moving Mesh Partial Dierential Equations." J. Comp. Phys. 112, pp. 279{290, 1994. [15] A. A. Lacey, \The form of blow-up for nonlinear parabolic equations." Proc. Roy. Soc. Edinburgh 98, pp.183-202, 1984. [16] N. K. Madsen, in \PDE Software: Modules, Interfaces and Systems." edited by B. Engquist and T. Smedsaas (North-Holland, Amsterdam, 1984). [17] H. Ockendon, \Channel ow with temperature-dependent viscosity and internal viscous dissipation." J. Fluid Mech. 93, pp. 737{746, 1979. [18] L. R. Petzold, Appl. Numer. Math. 3, p. 347, 1987. [19] Y. Ren and R. D. Russell, \Moving Mesh Techniques Based upon Equidistribution and Their Stability." SIAM J. Sci. Stat. Comput. 13, pp. 1265{1286, 1992. [20] A. M. Stuart and M. S. Floater, \On the computation of blow-up." Euro. J. Appl. Math. 1, pp. 47{71, 1990.

18

x^2 Ut = Uxx + U^3 1 ’p3q2e3i2’ ’p3q2e3i3’ ’p3q2e3i4’ ’p3q2e3i5’ ’p3q2e3i6’ ’p3q2e3i7’ sqrt(2)*x/sqrt(1+x**4)

0.9 0.8 0.7

U/Umax

0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

12

14

x/x*

Figure 9: Estimation of the asymptotic solution pro le as a function of x=x

19