Residual Cutting Method for Elliptic Boundary Value Problems: Application to Poisson's Equation

Transcription

1 JOURNAL OF COMPUTATIONAL PHYSICS 137, (1997) ARTICLE NO. CP Residual Cutting Method for Elliptic Boundary Value Probles: Application to Poisson's Equation Atsuhiro Taura, Kazuo Kikuchi, and Tadayasu Takahashi Reprinted fro JOURNAL OF COMPUTATIONAL PHYSICS, Vol.137, No.2, Noveber, 1997 Copyright 1997 by Acadeic Press, Inc. Printed in U.S.A.

2 JOURNAL OF COMPUTATIONAL PHYSICS 137, (1997) ARTICLE NO. CP Residual Cutting Method for Elliptic Boundary Value Probles: Application to Poisson's Equation Atsuhiro Taura,* Kazuo Kikuchi,* and Tadayasu Takahashi *Therofluid Dynaics Division, Coputational Sciences Division, National Aerospace Laboratory, Chofu-shi, Tokyo 182, Japan Received April 30,1996;revised Deceber 25,1996 A new, efficient, and highly accurate nuerical ethod which achieves the residual reduction with the aid of residual equations and the ethod of least squares is proposed for boundary value probles of elliptic partial differential equations. Neuann, Dirichlet, and ixed boundary value probles of three-diensional Poisson's equation for pressure in a curved duct flow and a cascade flow have been solved. Nuerical results exhibit the effectiveness of the ethod by a high convergence rate and a high degree of robustness. The ethod is expected to be an effective nuerical solution ethod applicable to a wide range of partial differential equations with various boundary conditions Acadeic Press Key Words: coputational fluid dynaics; nuerical ethod; boundary value proble; Poisson's equation; Neuann proble. 1. INTRODUCTION In ost nuerical approaches, boundary value probles of elliptic partial differential equations are discretized and reduced to finite difference equations, after which linear systes with huge atrices are solved using appropriate nuerical procedures. However, the nuerical ethods currently available, such as SOR, CG or BCG, ADI, and the ultigrid ethod with SOR etc. see to be not accurate enough and the rate of convergence is not necessarily high for three-diensional actual engineering applications, especially for Neuann probles of Poisson's equation[1-7] /97 $25.00 Copyright 1997 by Acadeic Press All rights of reproduction in any for reserved. 247

3 TAMURA, KIKUCHI, AND TAKAHASHI 248 It is recognized that the three-diensional Neuann proble still reains a ajor difficulty in general curvilinear coordinate systes. The difficulty involved is due to the characteristics of corresponding atrices and a copatibility condition (an integral constraint) which relates the source ter of the equation and boundary conditions of the second kind, and these lower both the rate of convergence and the level of robustness. The purpose of the work reported here is to propose an efficient and accurate nuerical ethod called herein the residual cutting ethod for boundary value probles of elliptic partial differential equations. This nuerical ethod is intended to get approxiate solutions with a saller residual L2 nor. The procedure of the residual cutting ethod is described as follows. Using the residual r corresponding to the approxiate solution U at th step, we set a residual equation. We solve the residual equation within the first few iterations to get a roughly approxiated solution ψ. We then define the perturbed quantity φ which is the linear cobination of ψ and previous 1 φ 2, φ,. The coefficients are deterined by the ethod of least squares so as to reduce the residual L2 nor in the sense that r +1 < r. Then, a new approxiate solution U +1 and a new residual r +1 are obtained using the resultant perturbed quantity φ. The residual equation is updated and the above procedure is used in an iterative anner. Ultiately we get the target solution and the residual. The final residual becoes zero for well-posed Neuann, Dirichlet, and ixed boundary value probles, but for an ill-posed Neuann proble, the final residual L2 nor will only becoe a certain iniu that is not zero. Roughly speaking, in the residual cutting ethod the "residual reduction" is achieved at the expense of extra inner product operations, instead of the "error reduction" in projective type 1 ethods like CG and GMRES [8]. In fact, it is shown that A φ + 2, A φ +, and r are orthogonal to each other. Thus our ethod provides with a new aspect of iteration ethod and, hence, it is interesting to copare theoretically our ethod with popular iteration ethods as entioned above. We also reark that it ay be possible to adopt other iteration ethods as an inner iteration. This exhibits another difference of our ethod fro well-known ethods including the ultigrid ethod. The atheatical analysis on the residual cutting ethod will be published elsewhere. To show the validity of the present ethod, three-diensional Poisson's equations for pressure, in both orthogonal curvilinear and general curvilinear coordinate systes, are successfully solved for the Neuann proble, as well as for the Dirichlet and ixed boundary value probles. Nuerical results show that the residual cutting ethod has any desirable characteristics such as robustness, accuracy and efficiency in actual applications. 2. PROCEDURE OF RESIDUAL CUTTING METHOD Throughout this paper we denote by (, ) and the inner product and L2 nor in R n, respectively. Generally, in order to solve boundary value probles of elliptic partial differential equations nuerically, which are discretized into finite difference equations, it is needed to solve the linear syste of n equations given in (1): A U = f. (1)

4 APPLICATION TO POISSON'S EQUATION 249 Now let U be an approxiate solution of (1).The corresponding residual r is defined as (2): r = f - A U. (2) Let the solution U of (1) be given as the su of U and φ as in (3): U = U + φ. (3) Substitute (3) into (1), and using (2) we obtain the residual equation given in (4): A φ = r. (4) In the usual procedure, residual equation (4) is solved up to convergence in an iterative anner. However, in the residual cutting ethod, the ost iportant objective is to get inforation fro (4) which will reduce the residual r, not to obtain a solution of (4). That is to say, an approxiate solution ψ of (4) is obtained within the first few iterations and is used to construct the next U +1 and r -+1. We are now in a position to describe the procedure of the residual cutting ethod. Let 0 and 1 L +1. Assue that U, r, ψ, andφ j 1 (-1 j -1) are given, where φ = 0. The perturbed quantity φ is defined as (5) and the approxiate solution U +1 is defined as (6): L l 1 1 l l= 2 φ = α ψ + α φ +. (5) U +1 = U + φ. (6) Here the second ter of the right-hand side in (5) is taken to be zero for L = 1. The corresponding residual r +1 is represented by (7): r = r A φ = f A U. (7) Each of the real nubers α l (l =1,2,3,, L) in (5) is called a residual cutting coefficient, and L l 1 deterined so as to iniize the value r α A ψ α A φ +. In order to 1 l= 2 deterine the residual cutting coefficientsα l, the ethod of least squares is used, that is, the linear syste of L equations obtained fro (8) is solved: l L + l 1 α1 ψ αl φ l= 2 2 r A A = 0, l=1,2,3,,l. (8) α l 1 Making full use of the facts that the least squares atrix is syetric and A φ,

5 TAMURA, KIKUCHI, AND TAKAHASHI 250 FIG. 1. Flowchart of residual cutting ethod. 2 A φ,, and r are orthogonal to each other (see the next section), α l are siply deterined. To solve residual equation (4) approxiately, the ADI ethod is adopted because it is ore suitable for parallel vector coputers, and solutions are obtained by directly using the 3-ter recurrence relation. We here recoend that zero be set as the initial value in the ADI iteration. The procedural flow of the residual cutting ethod is (see Fig. 1) (1) Give an initial value U 0 and calculate r 0 = f - A U 0. (2) Solve A φ = r to get ψ within only a very few iterations. (3) Calculate residual cutting coefficients α l (l = 1,2,3,,L). (4) Calculate L l 1 φ = α ψ + α φ +. 1 l= 2 l 1 (5) Calculate U U φ + + = + and 1 + r r A φ ( f A U 1 ) = =. Repeating (2) to (5) until U converges, we obtain the solution U with r = 0 for well-posed Neuann, Dirichlet, and ixed boundary value probles. On the other hand, for an ill-posed Neuann proble, we obtain a certain quantity U with the residual r 0 (see Section 4). It should be noted that r +1 < r if

6 APPLICATION TO POISSON'S EQUATION 251 and only if (r, A ψ ) 0. In view of this, the residual cutting ethod is interpreted as the residual reduction ethod and it will be clear that the property of the ethod is quite different fro those of CG etc. It is also entioned that the nuber L for α l sees to be enough with 3 L 6 to solve ost probles (see Section 6). It is noticed that the residual cutting ethod needs L + 1 inner products per step. (When L = 1, the nuber of new inner products is equal to that of the CG ethod.) Although this ay be a tie-consuing process, it is ephasized that high convergence rates are achieved at the expense of those operations. 3. ORTHOGONALITY AND CONVERGENCE In this section U, r,ψ, andφ denote the vectors in the residual cutting ethod. PROPOSITION 1 (Orthogonality). Let 0 and suppose that (r, A ψ ) 0. Then we have + j ( φ ) j k ( φ φ ) r 1, A = 0, + 1 L j, 1 L + 1; (9) A, A = 0, + 1 L j, k, j k, 2 L + 1. (10) + Proof. It follows fro the ethod of least squares that ( 1 + r, A ψ ) = 0 and( 1 j r, A φ ) = 0 1 for in { - 1, + 1 -L} j - 1. (1 L + 1), where φ = 0.Since φ is the linear cobination of ψ j + and φ, it is obvious that( 1 r, A φ ) = 0. Hence (9) is proved. To prove (10), let 1 and 2 L + 1. By solving the proble of least squares, we have that j j ( A φ A φ ) ( r A φ ), =, = 0, L j - 1. Here we used relation (9) with + 1 replaced by. This copletes the proof. Q.E.D. At this point we ention that in the GMRES algorith φ is deterined so as to achieve orthogonality in the following sense: ( j k, ) φ φ = 0,0 j, k, j k. PROPOSITION 2 (Convergence). Let 1 L +1 be fixed. Suppose that A is nonsingular, r 0 for 0, and there is a constant 0 < θ < 1, independent of, such that r - A ψ θ r, 0. (11) Then U converges to the target solution U as.

7 TAMURA, KIKUCHI, AND TAKAHASHI 252 Proof. As is easily checked, the ethod of least squares assures that + 1 r r A ψ. Cobining this with (11), we obtain that r + 1 θ r, which eans that r 0 as. Since r = f- A U and A is nonsingular, it is proved that U converges to the target solution U as. Q.E.D. We notice that condition (11) is satisfied for boundary value probles of the three-diensional Poisson's equation in a cubic doain. In fact, if we use the standard finite difference equation (the7-point difference schee), then the corresponding coefficient atrix A becoes the su of three atrices which are utually coutative. By the coutativity, it will be an eleentary exercise to check condition (11) [9]. The reark entioned here suggests that the convergence of U ay be also affirative in general case. 4. COMPATIBILITY CONDITION IN THE NEUMANN PROBLEM It is well known that in the Neuann proble of Poisson's equation, the existence of a solution requires the satisfaction of a copatibility condition (an integral constraint) [5,6] which relates the source of the equation and boundary conditions of the second kind. The copatibility condition is a consequence of the divergence theore (Green's theore). Now we assue that A is syetric, positive seidefinite, and rank (A) = n -1. We further assue that 1 = (1,1,,1) t is the eigenvector corresponding to the inial eigenvalue λ = 0 of A. We then have (A U, 1) = 0 for all U, (12) which is regarded as a atrix version of the divergence theore. In the for of Eq. (1), the copatibility condition is expressed as (13): (f, 1) = 0. (13) It is evident that condition (13) is a necessary and sufficient condition for Eq. (1) to possess a solution [10]. For convenience, if (f, 1) = 0, Eq. (l) is said to be well-posed and if (f, 1) 0, Eq. (1) is said to be ill-posed (not well-posed). In usual nuerical ethods, it is necessary to fulfill condition (13), for instance, by odifying the source. On the other hand, in the residual cutting ethod, condition (13) is not necessarily required, because the ethod is also applicable to ill-posed probles. In fact, r certainly decreases and it is seen that U and r converge to certain U and r, respectively. In the case of ill-posed proble, itturns out that r 0. Furtherore, if the coputation is executed by odifying the right-hand side as shown in (14),

8 APPLICATION TO POISSON'S EQUATION 253 A U = f - r, (14) then the sae solution U as the original ill-posed proble is obtained. This also justifies the nuerical procedure proposed in [6]. Thus the residual cutting ethod is advantageous enough for practical use, and the Neuann proble can be actually dealt with in the sae anner as the Dirichlet proble. 5. APPLICATION TO POISSON'S EQUATION In order to exaine the validity of the residual cutting ethod, several nuerical coputations are carried out for Neuann, Dirichlet, and ixed boundary value probles of three-diensional Poisson's equation for pressure. Poisson's equation for pressure p can be written in the coordinate syste (ξ1,ξ2,ξ3) as in (15): p 3 ij Jg = S. (15) i, j= 1 ξ i ξ j Here J is the Jacobian of the Cartesian coordinate syste (χ1,χ2,χ3) with respect to (ξ1,ξ2,ξ3) and g i j is defined as in (16) and (17), respectively: J ( x1, x2, x3) ( ξ, ξ, ξ ) = (16) g ij 3 ξ ξ i j =, i, j = 1,2,3. x x (17) k = 1 k k The probles to be solved are shown in Table I. In case a the pressure distribution in a curved duct is coputed as a fundaental test and in case b the pressure distribution in a turboachinery cascade is coputed to verify the applicability TABLE I Test Probles of Three-Diensional Pressure Field Neuann proble Dirichlet Mixed boundary Well-posed Ill-posed proble value proble Curved duct flow a-1 a-2 a-3 orthogonal curvilinear coordinate syste Cascade flow general curvilinear coordinate syste b-1 b-2 b-3 b-4

9 TAMURA, KIKUCHI, AND TAKAHASHI 254 FIG. 2. Coputational grid for a cascade. and robustness of our ethod. In the ixed boundary value proble b-4, boundary conditions of the first kind are set on two surfaces and boundary conditions of the second kind are set on the other four surfaces. The coordinate systes used for case a and case b are the orthogonal curvilinear and general curvilinear coordinate systes, respectively. Both of coputational grids are nonstaggered and the coputational grid for a cascade is shown in Fig. 2. Equation (15) is discretized using the standard central differencing. Naely, the 7-point and 19-point difference equations are respectively obtained for the orthogonal curvilinear and general curvilinear coordinate systes. We apply the residual cutting ethod with U replaced by p. It is noted that the right-hand side f in Eq. (1) is coposed of the discretization of the source S in Eq. (15) and boundary conditions. In all probles in Table I we set p 0 = 0 at interior grid points as a very rough initial value. In this case it is seen that r 0 = f. To solve residual equations, the ADI ethod of Wachspress-Habetler type [11] is used. In each sweep of the ADI procedure, difference ters corresponding to cross-derivatives are siply handled as ters of the right-hand side, because our ain interest lies in robustness. For convenience, relative residua1 ε and residual cutting ratio δ at th step are defined as r ε = (18) f

10 APPLICATION TO POISSON'S EQUATION 255 δ + 1 r r =. (19) r The residual cutting ratio δ in in inner ADI is also defined as in (20) (see Fig. 6): r r A ψ δin =. (20) r The severe convergence criterion as given in (21) is applied: e p p = < 10. (21) p In order to copare CPU ties in Fig.4 and Fig.5, results of the ADI ethod without the residual cutting step are used for a noralization of CPU tie. 6. NUMERICAL RESULTS 6.1. Effects of the Nuber L of α l on Convergence Characteristics To explain the typical convergence characteristics in the residual cutting ethod, the three-diensional well-posed Neuann proble a-1 in Table I was solved. Figure 3 shows the FIG. 3. Effects of the nuber L of residual cutting coefficients α l on the convergence characteristics (well-posed Neuann proble a-1, N = 2).

11 TAMURA, KIKUCHI, AND TAKAHASHI 256 FIG. 4. Optial nuber L of residual cutting coefficient α l (well-posed Neuann proble a-1, N = 2). behavior of relative residual ε with the nuber L of α l as a paraeter, taking up the nuber of residual cutting steps as the abscissa. Through this trial, the nuber N of inner ADI cycles are fixed as N = 2. The ADI ethod itself is slow in convergence, while the residual cutting ethod drastically converges when L 3. However, it sees that the effects of L becoe saturated near L = 3 or 4 in taking into account the CPU tie. Making clear the effects of the nuber L of a l the nuber M of residual cutting steps and CPU tie T up to convergence are plotted in Fig. 4, taking up L as the abscissa. It is seen that L = 3 or 4 is sufficient as far as the CPU tie is concerned. Alost no difference is noted even if a larger L is taken, and it is rather eaningless to eploy too large L because the load of inner product operations becoes greater Influence of the Nuber N of Inner ADI Cycles on Convergence Characteristics The nuber N of inner ADI cycles is next discussed in the well-posed Neuann proble a-1. The nuber N is varied fro 1 to 10, fixing the nuber of α l at L = 3. The nuber M of residual cutting steps and noralized CPU tie T up to convergence are plotted in Fig. 5, taking up N as the abscissa. Fro these results, it is clear that the CPU tie does not vary so uch according to N, but as a atter of course, M decreases with an increase in N. The optiu nuber of N concerning CPU tie efficiency depends upon the load of inner ADI and the outer residual cutting procedure. So it is possibly affected by the perforance of coputer and the tuning level of progra. However, within the entire test case treated in this paper, N = 2 is recoendable as the nuber of ADI cycles.

12 APPLICATION TO POISSON'S EQUATION 257 FIG. 5. Optial nuber N of inner ADI cycles (well-posed Neuann proble a-1, L = 3) Fluctuation of Residual Cutting Ratio δ The fluctuation of residual cutting ratios δ given in (19) and δin given in (20) is shown in Fig. 6 for N = 2 and L = 2,3,4. The solid line and the dotted line signify δ andδ in, respectively. In general, the residual cutting ratio δ becoes larger with the larger L and a higher convergence rate is attained. For exaple, when L = 4 is used as shown in Fig. 6c, the residual cutting ratioδ in inner ADI is alost doubled by the outer residual cutting procedure. This eans that the outer residual cutting procedure is very effective in reducing residual. Additionally speaking, the residual cutting ratio δ takes as a value 0 < δ 1, and if δ is closer to unity, a higher convergence rate is realized. Due to the load of calculation of residual cutting coefficients, it sees likely that the best result can be expected with N = 2 and L = 3, considering the CPU tie. Results obtained fro the test on the two-diensional Neuann proble also bring about good efficiency for N = 2 and L = 3, being the sae as in the three-diensional case. Glancing at Fig. 6, it can be seen that the residual cutting ratio δ oscillates. However, it can be iagined that there exists an averaged cutting ratio, which suggests that the large residual rapidly decreases. The residual cutting ratio hereby shown is one of the useful indexes for the convergence rate of a solving ethod used, when several nuerical ethods are copared Convergence Characteristics of an Ill-posed Neuann Proble In the next stage, the ill-posed Neuann proble a-2 will be solved using the residual cutting ethod. The results shown in Fig. 7 are the convergence characteristics in

13

14 APPLICATION TO POISSON'S EQUATION 259 of the proble a-2 with N = 2 and L = 3. The relative residual converges to its iniu value of 10-6 ; that is, it does not converge to zero. The result verifies nuerically that the source of the proble and boundary conditions of the second kind were set so as to not satisfy the copatibility condition (13). The final nonzero residual represents the degree of ill-posedness of the proble. On the other hand, the relative error of pressure p converges up to the convergence criterion of Coparing the well-posed proble a-1 and ill-posed proble a-2, it is seen that the latter converges faster. Incidentally, according to Eq. (14), if the final nonzero residual in the proble a-2 is subtracted fro the right-hand side in a-2, and it is solved again. one can obtain the sae solution with a zero final residual. In general engineering applications, it is not necessarily easy to set Neuann probles as well-posed probles. In any case, it will be understood that the residual cutting ethod has favorable characteristics suitable for actual engineering applications Boundary Condition Dependency The convergence characteristics of the Neuann proble a-1 and Dirichlet proble a-3 for the pressure distribution in a curved duct are shown in Fig. 8. The residual cutting ethod is also valid for the Dirichlet proble for which it attains a convergence rate three ties higher than that of the Neuann proble. Generally, Dirichlet probles are easier to solve than Neuann probles, and the convergence rate of ixed boundary value probles shows the interediate characteristics between the forer and the latter, according to the nuber of boundary conditions of the second kind Initial Value Dependency In order to exaine robustness, the initial value p 0 at interior grid points in all the test probles is set to be zero as the ost rough initial value. However, the residual cutting ethod is stable even if such a rough initial value is provided. Usually in Navier-Stokes coputation, the convergence of Poisson's equation is drastically enhanced because of the fact that its initial value is iproved according to the progress of coputation Coputational Grid Dependency To show the general validity of the residual cutting ethod, let us exaine the convergence in the Neuann probles b-1, b-2, Dirichlet proble b-3, and ixed boundary value proble b-4 of Poisson's equation described in the general curvilinear coordinate syste. In each case, N = 2 is used as the nuber of inner ADI cycles and L = 5 is used as the nuber of residual cutting coefficients, the reason for which will be referred to later. Figure 9 shows the convergence characteristics of relative residual and relative FIG. 6. Fluctuation of residual cutting ratio.

15 TAMURA, KIKUCHI, AND TAKAHASHI 260 FIG. 7. Convergence characteristics of the ill-posed Neuann proble a-2 (N = 2, L = 3) : curved duct. FIG. 8. Convergence characteristics of the well-posed Neuann proble a-1 and the Dirichlet proble a-3 (N = 2, L = 3) : curved duct.

16 APPLICATION TO POISSON'S EQUATION 261 FIG. 9. Convergence characteristics of the well-posed Neuann proble b-1 and the ill-posed Neuann proble b-2 (N = 2, L = 5) : cascade. error in the well-posed Neuann proble b-1 and ill-posed Neuann proble b-2. The relative residual ε and relative error e, in case of the well-posed proble b-1, reach to the target well well points. On the other hand, in the case of the ill-posed proble b-2, the relative residual ε ill does not converge to zero, while the relative error, e ill rapidly converges. It is the sae as in the case of the orthogonal curvilinear coordinate syste. Coparing ε of the well-posed proble a-1 in the orthogonal curvilinear coordinate syste (see Fig. 8) and ε well of the well-posed proble b-1 in the general curvilinear coordinate syste, it is seen that the nuber of steps for the latter is approxiately twice that of the forer. However, in ill-posed probles, it is seen that the difference between the orthogonal curvilinear coordinate syste and the general curvilinear coordinate syste is very sall fro the viewpoint of general engineering applications. The convergence characteristics of relative residual ε and relative error e in the Dirichlet proble b-3 are shown in Fig. 10. The sae rapid convergence is noted with both of the, and the convergence is alost never inferior to the proble a-3 in the orthogonal curvilinear coordinate syste (see Fig. 8). Coparing the Neuann proble with the Dirichlet proble, a considerable degree of difference is seen in the convergence characteristics. In fact, it will be understood that the convergence in the Dirichlet proble is generally several ties ore rapid than that in the Neuann proble. It is the sae in the case of the orthogonal curvilinear coordinate syste. The convergence characteristics in the ixed boundary value proble b-4 with two surfaces having boundary conditions of the first kind and the other four surfaces

17 TAMURA, KIKUCHI, AND TAKAHASHI 262 FIG. 10. Convergence characteristics of the Dirichlet proble b-3 and the ixed boundary value proble b-4 (N = 2, L = 5) : cascade. having boundary conditions of the second kind is also shown in Fig. 10. How the convergence is ade indicates the interediate characteristics between the Dirichlet proble and the well-posed Neuann proble. In ters of condition nuber, this is expressed as AD A - 1 D < AM A- 1 M < AN A- 1 N where AD denotes the coefficient atrix corresponding to the Dirichlet probleand so on, and AN is regarded as an operator on R n-1. As entioned above, L = 5 was adopted as the nuber of residual cutting coefficients for the probles b-1, b-2, b-3, and b-4 in the general curvilinear coordinate syste. It has been found fro identical tests with those in the orthogonal curvilinear coordinate syste that L = 5 or 6 is enough both for the nuber of residual cutting steps and ore iportantly for CPU tie. Such requireent for L sees to be owing to the treatent of difference ters corresponding to cross-derivatives as entioned in Section Nuber of Grid Points Dependency In order to show how the residual cutting ethod behaves as the nuber of grid points is increased, the Dirichlet and well-posed Neuann probles for a cascade are exained varying the nuber of the grid points. The fine and coarse grids ade fro original one retaining the physical doain. It sees that the nuber M of steps up to convergence (e ) becoes

18 APPLICATION TO POISSON'S EQUATION 263 FIG.11. Multigrid-like property of residual cutting ethod: cascade. independent of the nuber of the grid points both for Dirichlet and Neuann probles (see Fig. 11). The ultigrid-like property of the Dirichlet proble is established at a relatively sall nuber of grid points, copared with that of Neuann probles. The ultigrid-like property confirs the rapid decrease of the large residual as entioned in Subsection 6.3. Thus the residual cutting ethod has a ultigrid-like property and this is quite advantageous for actual engineering applications. 7. CONCLUSIONS The residual cutting ethod is proposed as a nuerical ethod for boundary value probles of elliptic partial differential equations. The residual equation is set and its approxiate solution is obtained within the first few iterations. The desired perturbed quantity is detained by the linear cobination of the approxiate solution and previous perturbed quantities. The coefficients, called residual cutting coefficients, are deterined by the ethod of least squares so as to reduce the residual L2 nor. The residual equation is updated and the above procedure is repeated up to convergence. The residual cutting ethod is quite siple, but provides with a high convergence rate and a high level of robustness for various types of boundary value probles including Neuann probles. It is recognized that Neuann probles reain a difficulty, especially in general curvilinear coordinate syste. However, in our solving ethod, Neuann probles in curvilinear coordinate systes are successfully dealt with in the sae anner as Dirichlet probles. The echanis of residual

19 TAMURA, KIKUCHI, AND TAKAHASHI 264 reduction is nothing but the fact that the inner solver iplicitly decreases the residual and the outer residual cutting step explicitly decreases the residual. The ADI ethod is used as an inner solver of the residual equation and no special acceleration is done. Accordingly, if adequate acceleration paraeters are eployed nuerical results ay be iproved to a higher convergence rate. By nuerical coputations of three-diensional Poisson's equation for pressure, the following results were obtained: (1) The residual cutting ethod can, despite its siple algorith, solve elliptic partial differential equations steadily, with high accuracy and at a high rate of convergence. (2) The residual cutting ethod does not necessarily require the so-called copatibility condition which is a ajor difficulty in solving Neuann proble. In fact, it is shown that the target solution of an ill-posed Neuann proble is sae as that of the corresponding well-posed Neuann proble which is reasonably preset. (3) The residual cutting ethod copensates for the ADI ethod, because the ADI ethod itself is not necessarily available for three-diensional probles in general curvilinear coordinate systes. (4) The nuber N of inner ADI cycles is enough with N = 2. Also, the nuber L of residual cutting coefficients is enough with L = 3 in the orthogonal curvilinear grid syste and with L = 5 in the general curvilinear grid syste. Fro a general point of view, N =2 and 3 L 6 are recoended and it is not required that N and L are taken any larger. (5) In ters of CPU tie, the Dirichlet proble ost rapidly converges, followed by the ixed boundary value proble, with the Neuann proble being the slowest to reach convergence. (6) There exists no dependence on initial values, and steady coputation is carried out fro a roughly approxiated initial value. (7) The residual cutting ethod has a ultigrid-like property and this is quite advantageous for actual engineering applications. REFERENCES 1. P. J. Roache, Coputational Fluid Dynaics, Herosa, Albuquerque, NM, G. Meurant, Parallel Coput. 5,267 (1987). 3. A. Brandt, Math. Coput. 31,333 (1977). 4. S. Silvaloganatha, Nuer. Method Lainar Turbulent Flow 6,1591 (1989). 5. S. Abdallah, J. Coput. Phys. 70, 182 (1987). 6. W.R. Briley, J.Coput. Phys. 14,8 (1974). 7. B.J. Alfrink, Proceedings of the Second International Conference on Nuerical Method in Lainar and Turbulent Flow, Pineridge Press, Swansea, UK, Y. Saad and M.H. Schultz, SIAM J. Sci. Stat. Coput. 7, 856 (1986). 9. R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, London, J.Jr. Douglas and C.M. Pearcy, Nuer. Math. 5,175 (1963). 11. E.L. Wachspress and G.J. Habetler, J. Soc. Indust. Appl. Math. 8, 403 (1960)