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FINITE DIFFERENCE SCHEME FOR PARABOLIC PROBLEMS ON COMPOSITE GRIDS WITH REFINEMENT IN TIME AND SPACE RICHARD E. EWINGy , RAYTCHO D. LAZAROVy , AND APOSTOL T. VASSILEVy

This paper is dedicated to Seymour Parter on the occasion of his 65th birthday.

Abstract. Finite dierence schemes for transient convection-diusion problems on grids with local re nement in time and space are constructed and studied. The construction utilizes a modi ed upwind approximation and linear interpolation at the slave nodes. The proposed schemes are implicit of backward Euler type and unconditionally stable. Error analysis is presented in the maximum norm and convergence estimates are derived for smooth solutions. Optimal approximation results for ratios between the spatial and time discretization parameters away from the CFL condition are shown. Finally, numerical examples illustrating the theory are given. Key words. nite dierence scheme, error estimates, local re nement, initial value problem 65M06, 65M12, 65M15, 65M50 1. Introduction. Parabolic partial dierential equations are used to model a variety of time-dependent diusive or convective-diusive processes. When the diffusion dominates the convection, the solutions of standard discretizations tend to be fairly stable. On the other hand, when the convective properties govern the process, upstream weighting techniques combined with explicit time-stepping allow ecient resolution of the hyperbolic properties of the solution. The solution of these equations may possess highly localized properties both in space and in time. In many physical applications, these properties are due to stationary features such as wells, cracks, obstacles, domain boundaries, etc., which are xed in space. In many other cases they are moving in time { moving point loads, sharp fronts, etc. Due to the size of many applications, the local properties of the solution cannot be resolved using uniform grids even with the largest of today's supercomputers. Adaptive local grid re nement techniques are an attractive alternative for resolving the localized characteristics within a given error tolerance while saving computational resources. In this work we study theoretically the stability and the convergence of implicit nite dierence approximations with variable time steps in space. These approximations can be characterized in the following way: rst one introduces a global time discretization for the whole domain; next, in some subdomains, chosen using some adaptive mesh re nement or some a priori information on potential rapid local temporal change of the solution, one introduces time steps that are fractions of the global time step. In this way a composite time-space mesh is introduced. The rst problem that arises in such situations is the construction of a stable and accurate approximation of the time-dependent problem. Diculties arise at the interface between the subregions with dierent time steps. The available literature shows that these are the places that govern the stability and the accuracy of the whole scheme to a great extent. Based on the interfacial treatment, the advantages and disadvantages of the dierent approaches could be clearly assessed. This work was supported in part by the National Science Foundation under Grant No. INT{ 8914472 and by the Department of Energy under Contract No. DE{FG05{92ER25143. y Department of Mathematics and the Institute for Scienti c Computation, Texas A&M University, 326 Teague Research Center, College Station, Texas 77843{3404 ([email protected]). 1

2

R. E. EWING, R. D. LAZAROV, AND A. T. VASSILEV

Second, one needs to construct and study ecient solution methods for the system resulting from the composite grid. Dawson and Du [8], on the basis of Galerkin discretization, investigate a domain decomposition procedure; Ewing, Lazarov, Pasciak, and Vassilevski [15] use the discontinuous Galerkin method to construct a discretization scheme and investigate an iterative method for solving the corresponding composite-grid system. Since this issue is not in the scope of this paper, we refer the reader to [8], [15], [23], where a number of interesting results have been presented. Spatial local grid re nement that treats stationary eects of the solutions has been addressed by many authors ([5], [6], [14], [24],[25], [19]). The approach to the analysis of parabolic problems that is utilized in this paper allows the analysis of the temporal part to be separated from the analysis of the stationary part. Many of the results obtained in these works can be adopted easily in our technique, thus expanding the scope of this method to a variety of dierent problems. The problem of constructing a stable approximation on the composite grid has been studied in [8], [9], [12], [13], [15], [17], [20], [21], [23] where a variety of techniques such as nite volume methods, discontinuous Galerkin methods, and the FAC method have been employed. Ewing, Lazarov, and Vassilevski [17] derive implicit schemes on the basis of a nite volume approach by approximation of the balance equation. This approach leads to schemes that are locally conservative, and are absolutely stable. Unfortunately, this does not allow linear interpolation in time along the interface and leads to a loss of one half in the order of the convergence rate. Dawson, Du, and Dupont [9] construct schemes combining implicit approximation in the subregions and an explicit treatment of the interfaces. This leads to a fully decoupled but conditionally stable system. Herox and Thomas [20] derive schemes on the basis of the FAC method [24], [25] and present some interesting numerical experiments. Eriksson and Johnson [12], [13] construct schemes using the discontinuous Galerkin method and derive a posteriori error estimates. A similar approach is used in [15]. In this case, the schemes are absolutley stable since the stability is built into the discontinuous Galerkin method. But this method does not allow linear interpolation in time along the interface and leads to a loss of one half in the order of the convergence rate. The literature cited above indicates some time-step restrictions either for stability or accuracy when local time-stepping is performed. The scope of this paper is to provide a theoretical analysis of the stability and a priori error estimates of dierence schemes for parabolic problems when local re nement is utilized in time and in space. We investigate schemes with higher-order interpolation in time along the interface that preserves the unconditional stability and leads to more accurate approximation. The main results are summarized in Theorems 4.2 and 5.1 where the convergence rates of the schemes in the maximum norm are established. The tools of our analysis are the maximum principle arguments and the energy type a priori estimates (see [10], [29]). In addition to the stability arguments in [10], [29], Parter has also addressed important issues in convergence analysis for iterative methods for discrete systems [30] including considerations arising from boundary conditions [28]. The authors acknowledge valuable inspiration from these and other works of Parter and are pleased to dedicate this paper to him. The paper is organized as follows. In Section 2 we formulate the problem and introduce the necessary notations. In Section 3 the construction of the nite dierence scheme on grids with local re nement in time is presented. The error analysis is

FINITE DIFFERENCE SCHEME FOR PARABOLIC PROBLEMS

3

addressed in Section 4. In Section 5 we consider schemes on grids with spatial local re nement and again discuss their approximation properties. Finally, in Section 6 we present some numerical experiments that con rm our theoretical results. 2. Problem formulation and notations. Consider the general parabolic problem

(x; t) @u = Lu + f; @t (1) u(x; t) @ = 0; where

in Q (0; T ]; [0; 1]n; n = 1; 2; 3; u(x; 0) = u0 (x);

Lu r Aru + r(x; t) ru ? c(x; t)u with

A = diag(a1 (x; t) : : : an (x; t)); 0 < a?1 ai (x; t) a; r(x; t) = (r1 (x; t) : : : rn (x; t))T ; c(x; t) 0; (x; t) 1 > 0; and @ is the boundary of . All coecients are supposed to be suitably smooth functions. We de ne a discretization of Q as follows: First, is discretized using a regular grid with parameters h1 ; : : : ; hn , for each spatial direction. The spatial nodes of the grid on are then de ned by x = (x1 ; : : : ; xn ) = (x1;k1 ; : : : ; xn;kn ) = (k1 h1 ; : : : ; kn hn ), where k1 = 0; : : : ; N1; : : : ; kn = 0; : : : ; Nn ; hj = 1=Nj ; j = 1; : : : ; M . Next, we introduce closed domains f k gM k=1 , which are subsets of with boundaries aligned with the spatial discretization already de ned. Further, it is required that M [ k=1

k ;

and we set

0 = n

M [ k=1

k :

In order to avoid unnecessary complications that contribute little to the generality of our considerations, assume that dist( i ; j ) `h;

for i; j > 0;

where ` > 1 is an integer and h = maxfh1 ; : : : ; hn g. The latter actually means that there are no two neighboring nodes x and z such that x 2 i and z 2 j , for i; j > 0. The case of nested re nement can be treated in a similar manner but needs additional notation and special consideration. With each subdomain i we associate corresponding sets of nodal points (see Fig. 1): !i is de ned to be the set of all nodes of the discretization of that are in

i ; we require !i \ !j = ; for i 6= j ; i; j = 0; : : : ; M (note that no spatial re nement

4


t`0+1

r

r

r

r

r

0

b

0

r t`0 b

U `;j i r

r r

r

r

b

r

r

b

r

r

0 ;

i

!0

i ;

r

i

h1

{ slave nodes

r

!i

r

b

r t`;j i r h2 r r i r

{ grid nodes

Fig. 1. Grid with local time stepping.

is assumed here) . This means that the nodes of the initial discretization of which reside on @ i , i > 0, do not belong to 0 (which is open relative to ). In each !i , i = 0; : : : ; M , we de ne a subset of boundary nodes i as the nodes which have at least one neighbor not in !i . It should be noted that i contains only nodes which do not reside on the boundary of in case @ i \ @ 6= ;. Then set

!=

M [

i=0

!i :

A discrete time step i is associated with each i such that, for integers mi , (2) 0 = mi i ; i = 0; : : : ; M; m0 = 1: Consequently, discrete time levels tji for i are de ned by tji = ji , j = 1; 2; : : :. Finally, we de ne the grid points q in Q by setting

qi =

[

x2!i j =1;2;:::

(x; ji );

and

q=

for i = 0; : : : ; M ; M [ i=0

qi :

Note that (2) implies that the local re nement in time is associated with the subdomains i , i = 1; : : : ; M . Note also that at the time levels determined by 0 , all nodes of ! are nodes in q.


5

We continue by specifying the nodes in qi , between time levels t`0 and t`0+1 as

qi` =

mi [ x2!i j =0

(x; t`0 + ji ) =

mi [ x2!i j =0

(x; t`;j i );

` t`;j i = t0 + ji ; i = 0; : : : ; M:

Correspondingly, the boundary nodes of qi` are de ned by

@qi` =

mi [ x2 i j =0

(x; t`;j i );

i = 0; : : : ; M:

A grid function y(x; t) is a function de ned at the grid points of q. In order to simplify our notation, we denote the nodal values of a grid function y(x; t) between time levels t`0 and t`0+1 as `;j y(x; t) = y(x1 ; x2 ; : : : ; xn ; t`;j i ) = yk1 ;:::;kn ;

for x(x1 ; : : : ; xn ) 2 !i , i > 0, j = 0; : : : ; mi . For x 2 !0 we de ne

y(x; t) = y(x1 ; : : : ; xn ; t`0+1 ) = yk`+1 : 1 ;:::;kn De ne the forward dierence operator i by i y(x) i y(xi ; : : : ; xn ) = y(x1 ; : : : ; xi + hi ; : : : ; xn ) ? y(x1 ; : : : ; xi ; : : : ; xn ) and the related divided forward dierence operator by

yxi (x) = ihy(x) : i

Correspondingly, the backward dierence operator i is de ned by i y(x) = i y(x1 ; : : : ; xn ) = y(x1 ; : : : ; xi ; : : : xn ) ? y(x1 ; : : : ; xi ? hi ; : : : ; xn ) and the related divided backward dierence by

yxi (x) = ihy(x) : i

Also de ne the divided backward time dierence by ` `?1 @0 y` (x) = y (x) ? y (x) in !0 0

and `;j `;j ?1 @i y`;j (x) = y (x) ? y (x) ; i

j = 1; 2; : : : ; mi ; in !i ; i = 1; : : : ; M:

6


3. Construction of the nite dierence scheme. We de ne the discretiza-

tion Lh of L in (1) by (3)

Lh y(x; t) =

n X

? i (âi yxi )xi + b+i a^+1 i yxi + bi aî yxi ? c^y;

i=1

where

bi (x; t) = Bi [~ri (x; t)]; a^+1 i (x; t) = aî (x1 ; : : : ; xi + hi ; : : : ; t); (x; t)j ; Ri = 21 hiajr(ix; i t) i = (1 + Ri )?1 ;

aî = Ai [ai (x; t)]; c^ = F [c(x; t)]; r~i (x; t) = rai ((x;x;tt)) ; i 1 ? ri (x; t) = 2 ri (x; t) ? jri (x; t)j 0;

ri+ (x; t) = 21 ri (x; t) + jri (x; t)j 0; i = 1; : : : ; n: Ri is the discrete Reynolds number in the corresponding spatial direction, and Ai , Bi ,

and F are grid functionals which provide second-order approximation [31]. Examples for such functionals are

Ai [ai (x; t)] =

?1 ds hi xi ?hi ai (x1 ; : : : ; xi?1 ; s; xi+1 ; : : : ; xn ; t) ;

1

Bi [~ri (x; t)] = h1

Z xi

Z x i + hi 2

r~i (x1 ; : : : ; xi?1 ; s; xi+1 ; : : : ; xn ; t)ds;

i xi ? h2i

1

F [c(x; t)] = h : : : h 1 n

Z

x1 + h21

x1 ? h21

:::

Z x n + hn 2

xn ? h2n

c(s; t)ds;

or their approximations obtained by some quadrature rule. Such a discretization has been used by many authors [2], [31], [32]. It is known that, in addition to second-order truncation error on smooth functions, these schemes comply with the maximum principle. Although the maximum principle theorem is a standard result found in many sources, e.g., [31, pp. 244{247], we now state it because it is referenced often in what follows. The nite dierence stencil at any grid point p in q includes the node p itself and some of its neighbors, denoted by N (p). The nite dierence equation could be viewed as an expression of the form

W (p)y(p) ?

X

s2N (p)

V (p; s)y(s) = f (p);

where y(s) is a grid function, W (p) is the weight coecient of p, and V (p; s) is the weight of the link between nodes s and p. Then, it is said that a given nite dierence


7

scheme complies with the requirements of the maximum principle if

W (p) > 0; V (p; s) > 0; 8s 2 N (p); X V (p; s) 0; D(p) = W (p) ? s2N (p)

for every p 2 q. This notion corresponds directly to the widely used notion of M matrices, which is another form of the same principle; e.g. see [33]. In order to conclude the introduction of our nite dierence scheme, we have to de ne Lh for the boundary nodes @qi` , i > 0, at time levels t`;j i = `0 + ji , `;j j = 1; : : : ; mi . It is obvious that at time ti , when Lh is applied to the points of i , not all space-time positions required in (3) correspond to actual nodes in q. For such cases, we de ne

mi ? j j `+1 ` y(x; t`;j i ) = m y (x; t0 ) + m y (x; t0 );

(4)

i

i

where x 2 0 and is a neighbor of a point in i . At time level t`0+1 , the corresponding i t`0+1 are included in the stencil of the nodes in points from @qi` at time level t`;m i @q0` . In Fig. 1 the slave nodes represent the missing space-time positions in the stencil of nodes in @qi` , i > 0. The values there are computed by the interpolation formula (4). We denote by u(x; t) the exact solution of (1) and by U (x; t) the numerical approximation to u(x; t) obtained by the following nite dierence scheme: the discrete problem for advancing from t`0 to t`0+1 in qi` , i = 0; 1; : : : ; M , is given by (5)

(x; t`0+1 )@0 U `+1 `;j (x; t`;j i )@i U U (x; t) f^

= = = =

LhU `+1 + f^`+1; LhU `;j + f^`;j ; 0; F [f (x; t)]:

in q0` ; in qi` for i = 1; : : : ; M; for grid points x 2 @ ;

Note that because of the properties of the linear interpolation in time (4) and the earlier observation for the properties of the spatial discretization of Lh , it is easy to conclude that the nite dierence scheme (5) complies with the requirements of the maximum principle. Because of (4) and the approximation at the nodes in @q0 , however, this scheme is not conservative. 4. Error analysis. In order to proceed with the analysis, the discrete inner product and L2 -norm of grid functions are de ned, respectively, by (y; v) =

X

x2!

h1 h2 : : : hn y(x)v(x)

and

kyk0;! = (y; y) 21 :

We also use the standard notation for the discrete H 1 -norm of a grid function in the Sobolev space H01 (!) (the subscript means that the grid function vanishes at the grid points on @ ):

kyk21;! = kyk20;! +

n X i=1

kyxi k20;! :

8


The following assumption can thus be made: Assumption 4.1. The discrete bilinear form associated with Lh de ned in (3) is coercive in H01 (!); i.e., 9 > 0 such that

?(Lh'; ') k'k21;! ;

8 ' 2 H01 (!):

Remark 4.1. There are important cases for which one can show coercivity of the discrete operator Lh associated with problem (1). For a pure diusion problem, this property follows immediately from the requirements for A and c(x; t) in (1). Another class of problems that ts in this framework are convection{diusion problems with small convection. Using standard perturbation arguments, one can show coercivity for such problems as well. Our third example is the class of convection{diusion problems with considerable advection terms of the form

Lu(x) = r Aru(x) + r(x)u(x) : Under the assumption that the advection coecient satis es

r r(x) 0;

ri (x) 2 W11+ ( ) for i = 1; : : : ; n;

coercive discretizations of L are discussed in [2], [22]. De ne the error of the above scheme by

e(x; t`0 ) = u(x; t`0 ) ? U `(x); x 2 !0 ; `;j `;j e(x; t`;j i ) = u(x; ti ) ? U (x); x 2 !i ; i = 1; : : : ; M: Using a Taylor series expansion, we get `+1

(6)

e = 0;

on @ ;

(x; t`0+1 ) @e(x;@t0 ) ? Lh e(x; t`0+1 ) = K0(x; t`0+1 )(0 + h2 ); in q0` ; 0 @e ( x; t`;j i ) ? Lh e(x; t`;j ) = Ki (x; t`;j )(i + h2 ); (x; t`;j ) i i i @i in qi` n@qi`; i = 1; : : : ; M; 2 @e(x; t`;j i ) ? Lh e(x; t`;j ) = Ki (x; t`;j ) i + h + 0 ; (x; t`;j ) i i i @i h2 in @qi` i = 1; : : : ; M:

In order to get an estimate for the error e(x; t) that takes into account the special presentation of the local truncation error in (6), we need two types of auxiliary functions i and i . The grid functions f i (x)gM i=0 are solutions to the problems (7)

?Lh i (x) = i (x); i (x) @ = 0;


9

where i (x) is the characteristic function of !i n i and 0 (x) is characteristic function of !0, and fi (x)gM i=1 are solutions to the problems (8)

?Lhi (x) = i (x); i (x) @ = 0;

where i (x) is the characteristic function of i . M Lemma 4.1. The solutions f i (x)gM i=0 and fi (x)gi=1 of (7) and (8) respectively exist and are non-negative, and the following estimates hold: (9)

(10)

8 > > >
! i > > :

8 > > >
! i > > :

C; in IR1 ; C j log hj 21 ; in IR2 ; Ch? 21 ; in IR3 ; Ch; in IR1 ; Chj log hj 21 ; in IR2 ; Ch 21 ; in IR3 :

Proof. Note that if we consider (7) and (8) with homogeneous right-hand sides, because of Assumption 4.1, these algebraic problems determine unique solutions, equal to zero. Therefore, independently of the right-hand side, (7) and (8) always have unique solutions. The solutions determined by (7) and (8) are non-negative because the righthand sides are non-negative, including on the boundary, and Lh complies with the requirements of the maximum principle, as noted before. Multiplying both sides of (7) by i and integrating over yields

?(Lh i ; i ) = i (x);

i

ki (x)k0;! k i k0;! :

Because of coercivity (Assumption 4.1), we conclude that

k i (x)k21;! ki (x)k0;! k i (x)k1;! : Therefore,

k i (x)k1;! C1 ki (x)k0;! C: Now, using known imbedding theorems [7], [26], [27] that imply max j (x)j D(h)k i k1;! ; x2! i where

D(h) =

8 > > > < > > > :

C; in IR1 ; C j log hj 21 ; in IR2 ; in IR3 ; Ch? 21 ;

10


we get (9). To obtain an upper bound for i (x), we consider rst a subdomain with a simple boundary in a two-dimensional domain (see Fig. 2). As shown there, the boundary of i-th subdomain consists of the pieces i(j) , j = 1; : : : ; 4, i.e.

i =

4 [

j =1

i(j) :

With each piece i(j) we associate a function i(j) such that (see Fig. 2) 8
X > > > > > > > > i=0 > >
j log hj 21 q > > > > > > > > > > :

2

Ci (i + h2 ) + Ii hi + h0 Ci (i + h2 ) + Ii

with constants Ci and Ii independent of the discretization parameters i and h. Proof. De ne M 2 X 0 2 i (x)Ii i + h + h2 ; (x) = i (x)Ci (i + h ) + i=1 i=0

M X

where

Ci = max Ki (x; t) and Ii = max Ki (x; t) : ` ` `

qi n@qi

@qi

Using induction over `, it easy to observe that

(x; t`0+1 )@0 (x) ? e(x; t`0+1 ) ? Lh (x) ? e(x; t`0+1 ) 0; and

in q0` ;

`;j `;j ` (x; t`;j i )@i (x) ? e(x; ti ) ?Lh ( (x) ? e(x; ti ) 0; in qi ; for i = 1; : : : ; M:

Moreover,

(x) ? e(x; t) @ 0; 8 t 0:

Therefore, using the maximum principle, it follows that i in qi` for i = 0; : : : ; M: e(x; t`;m i ) (x); Repeating the same argument for (x) + e(x; t) yields

i e(x; t`;m i ) (x);

in qi` for i = 0; : : : ; M:

In view of (9) and (10), we conclude (11). Remark 4.2. According to (11), in the case of 1D or 2D problems for 0 = h this scheme is of optimal or almost optimal order. Remark 4.3. In the case of 3D problems, one can improve the error estimate (11) by using a discrete H 2 -coercivity assumption for the dierence operator Lh (see, e.g.,

12


Bahvalov [3] or Hackbusch [18]) combined with stability arguments in maximum norm (see, e.g., Andreev [1] and Dryja [11]). Remark 4.4. The structure of the error estimate in (11) shows that the scheme properly takes into account the local characteristics of the solution, thus providing a uniform error over the domain . Remark 4.5. Our approach can be easily extended to grids with non-regular spatial discretization, provided that no local re nement in space is involved. 5. Schemes on grids with local re nement in time and space. The subject we discuss here is how to construct schemes with locally re ned grids in time and space. In view of the notations used in the previous sections, the re nement in space is to be introduced in the subdomains i , i > 0. This constitutes space{time subregions with locally re ned grids in time and space. In terms of a variety of possible industrial applications, e.g., [4], [14], such schemes are much more interesting because the space and time local re nement techniques result in very ecient numerical approximations within a given error tolerance. Generally speaking, the utilization of local re nement in space along with local re nement in time is a much more dicult problem. The diculties here greatly depend on the dimension n of IRn and the treatment of the interface nodes. For example, in IR1 , the local re nement in space does not involve diculties other than the computation of the discrete Reynolds number Ri at the interfaces accordingly. Moreover, the arguments used above can be applied directly here, resulting in an estimate similar to (11). In fact, for most of the numerical experiments presented in the next section, we used re nement in both time and space. We de ne the nite dierence scheme for higher spatial dimensions in a similar way. The two-dimensional example in Fig. 3 shows that in order to de ne our scheme at the interface points, we use linear interpolation in space and time. As noted above, this interpolation produces schemes that comply with the maximum principle. Thus, the values at the T-slave nodes are obtained by linear interpolation in time between the corresponding nodes in q. The values at the S-slave nodes are obtained by linear interpolation in space between the corresponding spatial positions in q using the spatial analog of (4). Further, the values at ST-slave nodes are obtained by linear interpolation in time between the corresponding S-slave nodes. The need for interpolation in space in addition to interpolation in time when Lh is applied to the nodes of @qi` in IRn , n = 2; 3, gives rise to certain complications in the error and stability analyses of the nite dierence scheme thus de ned. As pointed out before, we require interpolation formulas that produce schemes complying with the maximum principle. This limits the application of our theory to piecewise constant and piecewise linear interpolations. On the other hand, the attempt to apply the more general technique of a priori estimates in L1 (H 1 )-norm leads to diculties even for linear interpolation in time (see [17]). However, the technique used in the previous section (known as separation of steady-state singularities) permits us to split the truncation error due to interpolation along the interfaces between the coarse and ne grid regions and estimate its contribution using only the coercivity in H01 (!) of the elliptic part of Lh . Note that now ! is a composite mesh which contains locally re ned spatial regions. There is substantial research devoted to investigation of the properties of discrete elliptic operators de ned on locally re ned grids. Such results are presented in [5], [6], [19], [24], [25] for a number of illuminating applications. In view of these results, we make the assumption that the composite grid operator Lh is coercive in H01 (!). To present the basic idea of our analysis, consider a simple parabolic operator


2

`+1

t0

r

r

r r

0

r

b

r

r

r

r

!0 h0;1

{ T-slave nodes { grid nodes

i

r

r

r

i

r

r

2 0 ; b

r r

r

t`0 r

U `;j i

r

b

r

b

r r

4

2

r

r

r

i r; !i

r

hi;1

4

r r

13

r

r

b

r `;j t i r 2 r h r 0;2 r i r hi;2 r

2 { S-slave nodes 4 { ST-slave nodes

Fig. 3. Grid re ned in space and time.

with elliptic part equivalent to the Laplacian. In this case, using again a Taylor series expansion yields `+1

e = 0; on @ ;

(x; t`0+1 ) @e(x;@t0 ) ? Lh e(x; t`0+1 ) = K0 (x; t`0+1 )(0 + h20 ); in q0` ; 0 @e(x; t`;j i ) ? Lh e(x; t`;j ) = Ki (x; t`;j )(i + h2 ); (x; t`;j ) i i i i @ i (12) in qi` n@qi`; i = 1; : : : ; M; `;u ) 2 + h2 @e ( x; t `;j `;j `;j 0 i (x; ti ) @ ? Lhe(x; ti ) = Ki (x; ti ) i + h0 + h2 0 ; i 0 in @qi` i = 1; : : : ; M; where

hi = jmax h =1;n i;j is the largest space discretization parameter associated with i . Following the approach in the previous section, the grid functions (7) and (8) are de ned in the same way. It easy to see that because of the coercivity of Lh in H01 (!),

14


the estimates of Lemma 4.1 hold. Again, constructing the function

(x) =

M X i=0

i (x)Ci (i + h2i ) +

where

M X i=1

2

2

i (x)Ii i + h0 + 0 h+2 h0 ; 0

Ci = max Ki (x; t) ; Ki (x; t) and Ii = max ` ` `

qi n@qi

@qi

yields the following theorem: Theorem 5.1. Let the solution u(x; t) to (1) be a suitably smooth function. Then, the discretization scheme proposed in (5) is stable and the following estimates for the error hold: 8 M > X > > > > > > > i=0 > > >
j log hi j 12 (13) max q > > > > > > > > > > :

2 Ci (i + h2i ) + Ii h0 i + h20 + h0 ; 0

Ci (i + h2i ) + Ii

with constants Ci and Ii independent of the discretization parameters i and hi . Remark 5.1. The analysis of parabolic operators with more complex elliptic parts could be approached in the same way provided that Assumption 5.1 holds. Note also that instead of linear interpolation formulas, one can use the computationally cheaper constant interpolation. The nite dierence scheme thus obtained would be stable but the corresponding error estimates at the interfaces would be worse than (13). It is also worth mentioning that we may consider space{time regions with local re nement where the boundaries of the re ned in space subregion do not necessarily coincide with the boundaries of the subregion re ned in time. The analysis presented above covers such cases as well. This could be useful for many applications. 6. Numerical results. The main goal in testing the properties of the proposed schemes is to understand their behavior in terms of stability and accuracy. We focus our attention primarily on the eects of local time stepping in order to investigate the in uence of the interfaces and the intermediate time steps in the re ned regions on the stability and the convergence properties. As the structure of the error in (11) and (13) suggests, the most delicate places which determine to a great extent the accuracy of the scheme are the interfaces of the re ned regions. From this point of view, one can easily conclude that if the solution does not change much near the interfaces, the contribution of the interfacial terms to the total error of the scheme should be negligible. On the other hand, if the solution changes substantially near the interfaces, the interfacial error will govern the total approximation error. This is the basis for setting up the experiments described below. First, we experimentally assess the properties of the constant and linear interpolations in time for 1D problems. In this case we use local re nement in space as well, which does not introduce any additional interfacial error { there is no interpolation


15

in space. The model problem we started with is the heat equation with constant coecients. The following function is used as an exact solution: (14)

u(x; t) = exp(20t?2) exp(?37x2 + 66x ? 30);

which represents a bump with maximum around x = 0:75. In the interval [0, 21 ], this function is close to zero, changing negligibly in time. On the contrary, in the interval [ 12 ,1], it changes substantially in time. It is therefore useful for simulating real problems with local behavior. The rst series of experiments of the re ned region is (0:3; 1). The re nement in time uses the factors of 4, 6, etc. In this case, the scheme behaves as a nite dierence scheme on a regular grid with error O(f + h2f ), where f and hf are the discretization parameters in the re ned region. The next, more interesting set of experiments is when the re ned region is (0:75; 1), where the solution changes substantially in time. In practice, one does not use local re nement with interfaces crossing the very place where the local phenomena is observed. However, this is a way to test the properties of the scheme in the so-called \worst case". In other words, this will resemble the case when the local process approaches the boundaries of the re ned region, which could happen in many applications. Moreover, as (11) and (13) suggest, the behavior of the scheme in this case is governed by the interfacial terms of the error, which is a good test to distinguish the properties of the dierent interpolation in time used at the interface. In the following, hc and c denote the discretization parameters of the coarse region. In Tables 1, 2, and 3, the results from the experiments with dierent relations between c and hc are presented. In all experiments, the ratio c =f is held equal to 4. The results of the experiments are shown on the basis of the comparison of two interpolations in time|piecewise constant and piecewise linear. These results show the fact that the treatment of the interface is of crucial importance for the convergence of the scheme. They also indicate the advantages of linear interpolation. We have to point out that in the case of hc = c shown in Table 3, the scheme with constant interpolation in time performs better than expected, because the theory predicts O(1) error in such case. In fact, we have made experiments with other exact solutions, for instance sin(2x) sin(2t), where the asymptotic behavior is consistent with the theory. Experimental results with ratio c =f equal to 8, 10, 16 show the same asymptotic behavior with smaller maximum norms of the error. In order to check the stability of the scheme, we made runs for up to 5120 coarse time steps with dierent ratios between c and f . Again, the results are consistent with the theory. Table 1 c

1 h? c

20 40 80 160

= h2c

Constant Max-norm Reduction 1.716e-3 0.885e-3 1.94 0.450e-3 1.96 0.226e-3 1.99

Linear Max-norm Reduction 6.912e-7 1.938e-7 3.57 0.499e-7 3.88 0.125e-7 3.97

16

R. E. EWING, R. D. LAZAROV, AND A. T. VASSILEV Table 32 c

1 h? c

16 64 256 1024

= hc2

Constant Max-norm Reduction 1.004e-2 0.465e-2 2.15 0.190e-2 2.44 0.073e-2 2.58

Linear Max-norm Reduction 3.390e-4 0.486e-4 6.97 0.063e-4 7.62 0.008e-4 7.82

Table 3 c

1 h? c

64 128 256 512 1024

= hc

Constant Max-norm Reduction 1.351e-2 1.081e-2 1.25 0.831e-2 1.30 0.622e-2 1.34 0.457e-2 1.36

Linear Max-norm Reduction 4.244e-4 2.148e-4 1.97 1.077e-4 1.99 0.539e-4 2.00 0.269e-4 2.00

In the higher dimensional case, the accuracy of the proposed schemes, according to (11) and (13), is in uenced by two more factors: the imbedding constant and the spatial interpolation error at the interfaces. The latter suggests that the interface contribution to the total error of the scheme may reduce the accuracy of the scheme in space to rst order when the corresponding derivatives there are large enough. Such an experiment is shown in Fig. 4. The higher error along the interface is clearly indicated. The exact solution used is the two-dimensional analog of (14):

u(x1 ; x2 ; t) = 0:2 sin(0:5x1 ) sin(0:5x2 ) +2 exp(t ? t2 ) exp(?37x21 + 66x1 ? 30) exp(?37x22 + 66x2 ? 30): Its graph is shown in Fig. 4. We might point out that all error estimates are in maximum norms, which are known to be much more demanding than many other norms, e.g. L2 . However, for truly local solutions, our schemes prove to be very ecient. Of course, there are other more accurate schemes, such as the well known Crank{ Nicolson scheme, that could be used as a basis for developing local re nement techniques. In fact, we have performed many experiments with such schemes, using more accurate interfacial interpolations, e.g. piecewise quadratic, see [16]. The results obtained are rather encouraging, however the analysis of these schemes is beyond the scope of the techniques used in this paper.

Acknowledgments. The authors would like to thank Professors S. McCormick and M. Dryja for their valuable discussions and the referees for their remarks which helped improve the presentation substantially.


17

Fig. 4. 2D test function.

Fig. 5. Interface error in 2D.

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