physically motivated domain decomposition for singularly perturbed equations

Physically Motivated Domain Decomposition for Singularly Perturbed EquationsAuthor(s): Jeffrey S. ScroggsSource: SIAM Journal on Numerical Analysis, Vol. 28, No. 1 (Feb., 1991), pp. 168-178Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2157939 .

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SIAM J. NUMER. ANAL. ? 1991 Society for Industrial and Applied Mathematics Vol. 28, No. 1, pp. 168-178, February 1991 009

PHYSICALLY MOTIVATED DOMAIN DECOMPOSITION FOR SINGULARLY PERTURBED EQUATIONS*

JEFFREY S. SCROGGSt

Abstract. A domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction-convection-diffusion equation with small parameter multiplying the diffusion term is presented. Convergence is established via maximum principle arguments. The equation arises in the modeling of laminar transonic flow. Decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation, effectively preconditioning the problem. This paper concentrates on the theoretical basis of the method, establishing local and global a priori error bounds.

Key words. domain decomposition, asymptotics, hyperbolic-parabolic partial differential equations

AMS(MOS) subject classifications. 65M50, 35B40, 65G99

1. Introduction. In this paper a domain decomposition algorithm for the solution of

(1) P[u]:= u,+uu,-Eu,,-ru =0,

where ? is a small positive parameter, is presented. This equation contains many of the properties that make the gasdynamic equations difficult to solve; namely, it is capable of modeling rapid variations such as shocks and boundary layers. A priori error bounds are obtained using asymptotic analysis to exploit the physics relevant to the method. These error bounds are then rigorously established via maximum principle arguments. This paper concentrates on the theoretical basis for the method; discussions of the numerical details may be found in [27] and [25]. The method is capable of obtaining solutions to (1) when the shock is not stationary, thus extending Howes' studies [18], [19] into the time-dependent regime.

This method is appropriate for problems that contain different physical mechanisms that are dominant in distinct regions, especially when resolution of the length scales associated with the mechanisms is desired. An example of this class of problems is the modeling of hypersonic fluid flow and combustion problems where chemistry is important. To accurately determine the effect of the chemistry, the viscous profiles must be determined. The asymptotic analysis discussed here identifies the dominant physical mechanisms and their associated length scales, generating a physically mean- ingful domain decomposition. In addition, the asymptotic analysis provides a means to simplify the problems in the subdomains and helps identify computational parallelism intrinsic in the physics.

* Received by the editors December 19, 1988; accepted for publication (in revised form) February 21, 1990. This research was conducted while in residence at the Center for Supercomputing Research and Development, University of Illinois. It was supported in part by National Science Foundation grant US NSF PIP-8410110, U.S. Department of Energy grant US DOE-DE-FG02-85ER25001, Air Force Office of Scientific Research grant AFOSR-85-0211, the IBM Donation to Center for Supercomputing Research and Development and the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, Lawrence Livermore National Laboratory, contract W-7405-Eng-48. This research was also partially supported by National Aeronautics and Space Administration contract NAS 1-18605 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, Virginia.

t Institute for Computer Applications in Science and Engineering, Mail Stop 132C, NASA Langley Research Center, Hampton, Virginia 23665.

168



PHYSICAL DOMAIN DECOMPOSITION 169

There is a large literature on domain decomposition methods. Among the effective approaches to domain decomposition are Berger's local adaptive mesh refinement [2], Patera's spectral element method, and the Schwartz-Schur iterative methods [5], [11]. These represent only a small sample of the literature on domain decomposition. Though there are several ways of approaching the analysis of domain decomposition methods, most of the analysis is based on the boundedness of the remainder term of a truncated Taylor series. Thus, the error bounds for these methods will be very large in problems where there are regions of extremely large gradients such as when modeling hypersonic fluid flow with shocks and boundary layers.

Methods that resolve the gradients on the appropriate local scale, on the other hand, are uniformly valid. The method presented here identifies the dominant physical mechanism and its scale for each region of the domain; hence, the method provides reliable results even when the scales become asymptotically small. Other methods developed with this approach include [3], [6], [8], [9], and [14].

In ? 3 the domain decomposition is presented with a novel approach to the simplification of a complicated mathematical problem. Although the analysis presented here differs from that of Dinh et al. [9], the resulting domain decomposition is similar. In ? 4 an iterative technique is discussed. Use of this technique ensures that the asymptotics has captured the behavior of the solution. In ? 5 the method is summarized by outlining the algorithm. First, the problem is described in more detail.

2. The quasilinear problem. Consider the behavior of the solution of the quasilinear parabolic equation (1) on the domain

(2) D:={(x, t)IO?x?b, O-t<T},

subject to

(3) u(x, 0) = y(x), 0<x < b,

(4) u (0, t) = ax ( t), O < t < T, and

(5) u(b, t) =,/3(t), O < t < T

The portion of the boundary along which the data is specified is denoted by

JI:= {(x, t)O ?< X < b, t = O} U {(x, t)O -< t < T, x = O, b}.

The boundary data are continuous and sufficiently smooth so that the solution to (1) is uniquely defined (for example, see [4]). Namely, the compatibility conditions

(6) a(O)=y(O) and y(b)=,8(O) must be satisfied. For simplicity, it is assumed that all boundaries are inflow boundaries, hence a(t)-' ao > 0 and 83(t) ' ,?30 < O.

The reaction term ru may, for example, arise from the effects of a variable cross-sectional area in a duct; thus, this is not a reacting flow. Howes [19] discusses the case when r(x) = -a'(x)/a(x), where a(x) is the width of the duct (see Fig. 1). The coefficient r is assumed to be bounded with bounded derivatives.

r a(x)

FIG. 1. Variable width duct.



170 JEFFREY S. SCROGGS

The problem may be assumed to be nondimensionalized such that the diffusion coefficient E is inversely proportional to the Reynolds number (see [7]). Based on free-stream conditions in transonic flow, the Reynolds number for this problem is large. Therefore, it is appropriate to exploit the smallness of the positive parameter E in the analysis.

The solution of equation (1) will be compared to the solution to the reduced equation

(7) Po[U]:= U,+ UU -rU= 0,

obtained by setting E = 0. The solution to this equation may not be uniquely defined [24]; hence, U is assumed to be the solution to (1) in the limit as 4J0. It is also assumed that the first derivatives of U are continuous except along a shock. This is satisfied by imposing the additional compatibility constraints

(8) da YdxrY 0 for(x, t)=(0,0),

(9) -+-dt dy ry =0 for (x, t) = (b, 0). dt dx

3. Asymptotic analysis. Asymptotic analysis subdivides the domain into the following two types of regions: regions where the solution is smooth, where a reduced equation may be solved; and regions of rapid variations, such as in a neighborhood of a shock, where the full equation must be solved. Asymptotics also provides a simplification of a complicated mathematical problem. For example, in some regions, the reduced equation (7) will be solved in place of (1). The regions of nearly uniform behavior of the solution are isolated into subdomains. The numerical method used on each subdomain now needs only to capture one type of behavior in the solution (i.e., smooth flow or shocks but not both). Thus, the problem as a whole is better conditioned for numerical computations. This will be discussed in greater detail in ? 5. The asymptotic analysis involves the derivation of analytic upper and lower bounds on the solution and is performed in the style of Howes [15]-[17]. The solution to the reduced problem will be presented first, followed by comparison theorems.

Let U be a weak solution of (7) with boundary data (3)-(5). Assume that U has a single shock, with the path of the shock given by the curve (x, t) = (F(t), t). The initial and boundary data are assumed to be smooth; thus, no shock is present at t = 0. Rather, F is assumed to be undefined for t < tr, where t = tr is the time U becomes discontinuous. It is natural to describe U in terms of the following functions:

UO(x, t) for O < t _tr,, U(x, t) = U, (x, t) for x < F( t) and ti-' tr,,

U U(x, t) for x > F(t) and t?_ tr

For analytic methods to determine F, see Whitham [29] or Kevorkian and Cole [20]. The assumption E > 0 implies that the solution to (7) will satisfy the entropy

condition

(10) U,(F(t), t) > s > Ur(F(t), t),

where the speed s of the shock is given by the Rankine-Hugoniot jump condition [21],

(11) s = [ U,(F(t), t) + Ur(F(t), t)]/2.




A useful form of the entropy condition is

(12) ,u(t) = U1(F(t), t) - Ur(F(t), t)>O for t > tI'.

The outer-region subdomains are based on the bounds on the difference U - u given in the following theorem.

THEOREM 3.1 (Howes [17]). Let u(x, t, E) be the solution to (1) and U(x, t) be the solution to (7). Both u and U satisfy the boundary data (3)-(5). Assume that the boundary data (3)-(5) satisfy the compatibility conditions (6), (8)-(9), and that a, ,8, and y, with their first and second derivatives, are all bounded. Then for E small enough

(13) Iu - Ul = O(pk exp [-f2(x, t)/E112])+ O(E)

when the derivatives of U are continuous across F, and

(14) lu - Ul = 0(,u exp [_f2/E1/2]) + O(El/2 exp [_f1E/2])+

when the derivatives of U are not continuous across F. Heref(x, t) is a distancefunction1 between (x, t) and (F, t), and 8 is an upper bound on the difference of the derivative of U normal to F.

These bounds are small except for an asymptotically small neighborhood of the shock. Thus, the outer region subdomain

(15) DOR = { (X, t)l(x, t) E D, IX -F(t)l > /\(t)}

is the portion of D where U approximates u to within 0(E). Here A(t) is the width of the internal-layer subdomain at time t. Theorem 3.1 dictates that if A(t) ? Krq (t) E1/4 n 1/2 E where K is a constant independent of E, then the a priori bound of lu- Ul = 0(E) in DOR holds. The internal-layer subdomain is the complement of DOR with respect to D,

(16) DIL = {(X, t)|(x, t) c D, Ix - F(t)l C A(t)}. This is the region where U may be a poor approximation to u and includes the internal or shock layer. Since the method is designed for small E, the internal-layer subdomain will be an asymptotically small region surrounding the shock.

The local error bounds of Theorem 3.1 are now used to establish a global a priori error bound when solving the reduced equation in place of the full equation in the outer region. The bound, as presented below, is sharp in DOR; however, the bound reflects the crude error bound of Theorem 3.1 in the region of the shock.

COROLLARY 3.2. Let u be the solution to (1) satisfying (3)-(5). Suppose v is obtained by first solving (7) in DOR subject to (3)-(5), then solving (1) on DIL with boundary data v on aDIL. Assume that the boundary data (3)-(5) satisfy the compatibility conditions (6), (8)-(9), and that a, ,/ and y, with their first and second derivatives, are all bounded. If EA = I|U - VIIA = JA 1U(X t) - v(x, t)I dx dt, is the L1 norm of the error on the region A, then for E small enough

(17) EDOR = 0(),

and (18) ED,

= O(E 1/4 ln2 1E)2

To establish this result a bound on the size of the solution in the internal layer is established via a maximum principle argument. From there, the proof follows directly from Theorem 3.1.

'Without loss of generality, f = Cdx - F(t)(.




Proof. Relation (17) follows directly from applying the L1 norm to the bound (13) of Theorem 3.1.

Let

w(t) =K1+K2t.

The constant K2 may be chosen independent of E such that

P[w]?_ P[v] = 0

for (x, t) c DIL, and the constant K1 may be chosen independent of E such that

(o?- v

for (x, t) c aDIL. Thus, the conditions of the form of the maximum principle theorem presented in the Nagumo-Westphal lemma [28] are satisfied, and wc -? v for all of DIL. A symmetry argument can be used to establish a lower bounding function which has the same form as a). In addition, a similar argument can be used to establish upper and lower bounds for u in DIL. Thus,

iu - vI = 0(1)

for (x, t) c DIL. Since the area of DIL is O(E1/4 In112 E), applying the LI norm, the relation (18) follows. 0

Remark. In the process of establishing the rigorous error bounds above, some of the physics has not been captured in an optimal way. For example, multiple-scale asymptotic analysis [10], [20], [23] identifies 1/e as the appropriate scaling in a neighborhood of the shock (4 = (x - F(t))/E is the appropriate scaled variable). In addition, computations indicated errors much smaller than those reflected in the bounds; thus, sharper error bounds may be expected.

Asymptotics has identified two subdomains and provided a simplification for the computational problem on one subdomain. The domain decomposition algorithm described thus far requires a priori knowledge of the location of the shock. To remove this restriction, the domain decomposition is combined with a functional iteration. The iteration guarantees that the domain decomposition places the shock in the correct position, thereby allowing the development of a computational method for which rigorous error bounds are available.

4. Iteration. The domain decomposition algorithm described in the previous sec- tion will be treated as a single step in an iterative process. Convergence of the iteration (19) in the outer-region subdomain to a solution of (7) will be established. In addition, a global a priori error bound for the method will be presented.

Denote the iterate by Uk+l. Each step of this iteration is a two-stage process requiring the solution of the following linearized form of equation (7)

( 19 ) U,~~~~k+ + Uk Uk+I1r k+I1 (19) Ut? IUkUk? ~rCTk =

in DOR in the first stage, then the form of equation (1)

(20) U, + Ik+ Ik A- k+IUAlrUk+l0

in DIL for the second stage. Boundary data for equation (20) is provided by the solution of (19) in the outer-region subdomain. The equation may be linearized over one or more timesteps in the computational method, providing the conditions in the theorem below are satisfied.




THEOREM 4.1. Let U1, U2, U3, be the set of iterates of (19) in the subdomain DOR satisfying the boundary data (3)-(5) with initial guess U?. Assume U? satisfies (3)-(5) and is Lipschitz continuous on D. Let

8 =SUp I Uk_ Uk-11 D

Then

(21) u Uk+ I k < C e A (e -1)

for (x, t) c DOR, where C, A, and R are known positive constants. Continuity and boundedness of the iterates are established first. LEMMA 4.2. Let Uk+l be the solution to equation (19) on the subdomain DOR,

where Uk is Lipschitz continuous. Then,

|u I < K e

for (x, t) E DOR, where K and R are constants independent of x, t, and E. Proof Consider the transformation (x, t) -(, T) defined by

(22) t =T,

and

axk A

(23) - = UCk (Xk( T), T),

with initial conditions

(24) xk(o)=, b > > O,

(25) xk(y0l(4))=?, = < O

(26) Xk (Y- (()) =?, b.

Here (4, T) = (yo(T), T) is the image of the curve (x, t) = (0, t), and (4, T) = (Yb(T), T) iS the image of the curve (x, t) = (b, t). Under this transformation, (19) becomes

(27) U rU

Since Uk is Lipschitz continuous, the transformation (22)-(26) is uniquely defined, and (27) may be used in place of (19).

The coefficient r is bounded; hence, there is a constant K independently of x, t, and E such that c = K eRr is an upper bound for Uk?l. In addition, wp = -c is a lower bound. Since t = T the desired result is established. O

Based on the assumption of the boundedness of U'k, the boundedness of Ujk+1 is established in Lemma 4.3.

LEMMA 4.3. Suppose that the conditions of Lemma 4.2 are obtained. Then Ux and Ut 1 are bounded independent of x, t, and E in DOR

In the proof of this lemma, an equation governing Uk+1 will be derived. Then, a form of the maximum principle will be shown to apply after a change of the dependent variable.

Proof Let w = Uk+1. The equation

(28) w,(r U) w + U dx




is derived by taking the partial derivative with respect to x of (19) and then applying the transformation (22)-(26). Boundary data for w may be obtained by differentiating (3)-(5).

Let w = eA"v. The equation governing v is

v(r_ U_-A)v+- Uk. dx

Choose A = -max (0, infD (r(x) - Ux)), so that the coefficient of v in this equation will be nonnegative.

Define an upper bounding function as ,= K,(e 2 -D)

where the constants K1 and K2 will be chosen. Then z = c- - v satisfies

(29) z7 = A(x, t)z + B(x, t),

where A = r- UX'-A and B = [K2-(r- Ux-A)]cIi+(KiK2/2)-(dr/dx) Uk. The constants K1 and K2 may be chosen so that A, B, and z(x, 0) are nonnegative. For example, choose

I' ~~jk ) ( drAk K2 = max 2, sup (r- Ux-A) and K, = max 0, sup- Uk ,sup 2v) DOR / DOR~ dx ni

Under these conditions z is positive and cOi is an upper bound for v. A lower bounding function may be obtained in a similar way. Set q = K3(eK4T-

If K2 and A are chosen as before and

K3 =Min 0, inf- Uk,inf 2 (DoRdX n

then a) - v is nonpositive. Thus, wp is a lower bounding function for v. Since T ? T, the boundedness of Uk+1 independent of x, t, and E follows. The

boundedness of U,k+ follows from the boundedness of the terms in (19). Therefore, the desired result is established. O

Given that U? is Lipschitz continuous, Lemma 4.3 states that all of the remaining interates will be Lipschitz continuous. The significance of this is that the characteristic transformation (22)-(26) is uniquely defined. This is used in the proof of the theorem.

Proof of Theorem 4.1. The equation governing z = Uk - k+ in the characteristic coordinate system (22)-(26) is

(Uk-1 _ (bk) ^ k+1 + rZ.

An exponential change of variable will transform the problem into a form to which the maximum principle applies. Let

-A r

where A = -max (0, inf r(x)). Thus, it is seen that the equation for w is

w=At t(k-I Ck)CJk?1? A wT = e ( U - U )U + rw,

where r=r+A, and w=Oon 171. An upper bound for w may now be defined. Let

w (T) = 8~ N (eRT1),




where R = sup r - inf r = sup r^. The constant r will be determined shortly. Taking the partial derivative with respect to T,

(c) 5 q + Rw.

An equation for h = t) - w is

(30) h [ = [qeA( - k 1 - Uk) Uk+l + (R )o] + Ph.

Since w _ O and w =O on HI, the function h is nonnegative on rl. Choose r = max [0, infD eA" U k+]. Thus, the source term in (30) is nonnegative, and h ?-0 in DOR.

Thus, w(T)? w(4, T) for (x(4, T), t(4, T)) in DOR. But t = 7; hence,

AK ek(eRt-1)I>Z= uk?1 _uk

for K = rq/R. Using symmetry arguments, -&) can be shown to be a lower bound on z, leading

to the desired result. 0 This theorem provides an upper bound on the latest time for which the iteration

converges. Apply the infinity norm to (21) to obtain -k+1 _k j? C Cjk - (k-1jj

Then the following corollary provides the conditions for convergence. COROLLARY 4.4. Suppose that the conditions of Theorem 4.1 are obtained. Restrict

the domain to DOR. Let Tmax be the largest positive number such that

C= sup C eAt(eRt-1)-1 0_t T Tax

When O< t < T < Tmax the sequence of iterates defined by (19) converges to a solution of (7) satisfying the data (3)-(5).

Proof With the restriction of t < Tmax the iteration is a contraction mapping, and the result follows. 0

A statement of a global a priori error bound for the computational method is presented in Corollary 4.5 below. As with Corollary 3.2, the bound is sharp in DOR;

however, the bound is crude in the region of the shock. COROLLARY 4.5. Let u be the solution to (1) satisfying (3)-(5). Suppose each iterate

Uk is obtained by first solving (19) in DOR subject to (3)-(5), then solving (20) on DIL with boundary data Uk on dDIL. Suppose Tmax> T>0, and let v = limkO uk = U?. If the width of DIL is \== 0( /4 ln1/2 E), then for E small enough, the error satisfies

(31) EDOR = 0(E),

and

(32) EDIL = 0(E1/4 ln1/2 )

To prove Corollary 4.5, it will first be established that U? is the desired solution of (7). The bound in (31) will follow. Then, Lemma 4.6 will be applied to establish the bound (32).

LEMMA 4.6. Let v be the solution to (1) on DIL. Suppose that the data specified on aDIL are bounded with bounded derivatives. Assume that aDIL is at least C2. Then

lvl < K,

for (x, t) E DIL-




Proof. Make the change of dependent variable w = e-7Av, where A satisfies r= r-A _ ro < 0. Then

PA [ w] := w, + eA WWX - rw-wW- = 0,

where w= e-'v on aDIL. Define

c = K = max (0, sup e VI). aDIL

Then

PA[]] roK - PA[w]

In addition, c ' e-A'v - w for (x, t) c aDIL. These conditions allow the application of the Nagumo-Westphal lemma to conclude cOi ' w for (x, t) c DIL. By symmetry arguments, cp = -cOi may be shown to be a lower bounding function on w. Setting K = eATK, the result is established. 0

Proof of Corollary 4.5. In the outer region it is seen that v= U?, where bounds for u -v = u - U are given in (13)-(14). Thus, there is a constant KOR such that u- U'l E KOR for (x, t) E DOR. Applying the L1 norm to u- U?, relation (31) is established.

From Lemma 4.6, there is a constant KIL, such that Iul + Ivl- KIL in DIL, where KIL is independent of E. Thus,

EDIL KIL ds. .DIL

Since the area covered by DIL is of size O(E1/4 ln1/4 E), relation (32) holds. 0 The iteration has been presented for a scalar equation; however, it may be used

to solve systems of equations [26]. In this case, the iteration can provide a means to decouple the equations, increasing the potential parallelism of the method.

5. Algorithm. The iteration, domain decomposition, and simplifications can be combined to form an algorithm. Recall from ? 4 that the iteration is formed by solving (19) in DOR followed by solving (20) in DIL. The solution to the latter equation is subject to Dirichlet boundary conditions obtained from the solution in DOR. Relation (21) places an upper bound on the temporal size of the domain for which this iteration may be used. If the restriction on t is less than the time at which the solution is desired, then the algorithm would include an outer iteration on each of the domains f,= { (x, t) I 0x _ b, T -1 - t < T,} in succession, using the converged solution at t = T,_ for the initial data on the domain fl. The number of temporal subdivisions N (for T = Tn of magnitude unity) need not be large to satisfy (21) because the constants in (21) do not have large magnitude.

In the most general case the internal layer subdomain will change from iteration to iteration within each Qn. (The theory presented in this paper restricts consideration to a stationary internal-layer subdomain; however, experimental results demonstrate that this is not a constraint in the numerical method [27].) Thus, there must be a dynamic means to locate the internal layer boundary. For example, a method that uses the magnitude of the Jacobian of the transformation defined by (22)-(26) is outlined in [27]. An alternative detection scheme is to place the boundary where a given tolerance for the magnitude of Uk is reached. This is based on the value of Uk being large and negative in a neighborhood of the shock, while it is bounded independently of E in the outer-region (see Lemma 4.3).




The problems in the subdomains are solved by first simplifying them and then applying standard numerical schemes to the resulting well-conditioned problems. As described in ? 3, the simplification for the problem in DOR is the dropping of the viscous terms. Combining the scale 1/e (see the remark following Corollary 3.2) with the shock-following translation x-F(t) creates a simplified problem for (20). Thus, the new spatial coordinate in DIL is

(33) x~= (x- F)/ E.

Computational experiments have demonstrated an internal layer size no larger than O(E); thus this refinement is feasible. Since resolution of the shock with a uniform grid would require refinement on the order of 1/ E throughout D this refinement strategy increases the efficiency of the method by adaptively refining only in the neighborhood of the shock. The algorithm is independent of the choice of the particular numerical schemes used; thus, the algorithm may incorporate various discretizations depending on the problem and/or computational environment used. Some candidates will be mentioned here. The discretizations discussed in [1], [13], [22], and [30] would be effective on these simplified problems. Additional parallelism may be exploited, for example, by solving the matrix equations arising in these discretizations [12].

Acknowledgments. The author expresses his appreciation for the financial and moral support of Ahmed Sameh during the research of this project. The assistance of Dan Sorensen, Gerald Hedstrom, Raymond C. Y. Chin, and Fred Howes was invalu- able. The author would also like to acknowledge the useful comments of the referees.

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physically motivated domain decomposition for singularly perturbed equations

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