A Direct Finite-Di erence Scheme for Solving PDEs over ... - CiteSeerX

A Direct Finite-Dierence Scheme for Solving PDEs over General Bidimensional Regions A. Cortes-Medina a A. Chavez-Gonzalez J. G. Tinoco-Ruiz a 1

a

;

a Escuela

de Ciencias Fisico-Matematicas, Universidad Michoacana de San Nicolas de Hidalgo. Morelia, Michoacan, Mexico.

Abstract Classical nite-dierence methods apply only over regions that are rectangular or decomposed in several rectangular subdomains. We call a nite-dierence scheme direct if it can be applied over any region decomposed by means of a logicallyrectangular grid. We introduce a second-order and easy-to-use direct scheme to solve boundary-value problems. Deduction of the method is elementary and its results compare well with other methods.

The design of numerical methods to solve partial dierential equations (PDEs) must take in account the geometrical complexity of the physical region where the problem is de ned. The classical nite-dierence schemes are well-suited for simple regions and, in that case, their implementation is straightforward. Over dicult regions, the nite-element method is commonly used. It has a strong mathematical-background supporting it and tools for subdividing the region into simpler cells are used; however, the complexity of data structure and the amount of memory required to its implementation has suggested the search for simpler methods. The use of the Mapping Method ([8]) and nite-dierence schemes is an alternate way, it requires the knowledge of a dieomorphism {a change of coordinates{ between the unit cell (logical space) and the region. The problem is then transformed and solved, not on the original region, but on the unit cell. Changes of coordinates are known in an analytical form only for few regions; in general, they have to be calculated in a discrete way, this is done using numerical grid generation methods. A disadvantage of the mapping method is that calculations involve the use of the jacobian; if the mapping is 1

This work has been supported by U.M.S.N.H.(Grant I 1999-8.9) and CONACyT (Grant I27216E) Preprint submitted to Elsevier Preprint

24 June 2000

nearly singular in a point of the region, the jacobian is close to zero with the corresponding lack of precision. Another approach is to design schemes for discretizing partial derivatives directly on the physical region. As with the mapping method, it requires to have a grid over the region. Some such schemes have been reported by Steinberg, Shashkov, Hyman and Castillo ([10], [6]). In the present paper we introduce a simple bidimensional nite-dierence scheme applied directly on a grid over the physical region. We present some examples over irregular regions in order to show its performance. In section 1, we introduce the kind of problems to be solved. Section 2 contains a brief review of grid generations. Section 3 is devoted to explain our scheme. Finally, in section 4, some test problems are solved and we discuss the results.

1 Introduction Let be a region in the plane, and @ its boundary. We want to solve numerically the boundary-value problem:

Auxx + Buxy + Cuyy + Dux + Euy = F uj@ = u0

(1)

where A; B; C; D; E; F; u and u0 are functions from to R. In this work we will consider only polygonal regions; is a poligonal region if its boundary is a closed, non self-intersecting poligon. Let us think, for the moment, that we have p0; p1; p2; : : :; pM , M + 1 points in the interior of where the solution of (1) is to be approximated. Let pM +1; pM +2 ; : : :pM +N points on the boundary of . A nite-dierence scheme for (1) in the point p 2 is a way to discretize partial derivatives of u in a point p using a linear combination of the values on u on certain subset q0; q1; : : :; qk?1 taken from the inner points and points on the boundary (where the solution is known); that is

uxx(p)  uxy (p)  uyy (p) 

kX ?1 l=0 kX ?1 l=0 kX ?1 l=0

2

?xx l u(ql); ?xy l u(ql ); ?yyl u(ql);

ux(p)  uy (p) 

kX ?1 l=0 kX ?1 l=0

?xl u(ql); ?yl u(ql)

The set of points q0; q1; : : :; qk?1 is commonly called stencil of the scheme at point p. If we substitute these expresions and the values of A(p), B (p), C (p), D(p), E (p), F (p) in (1) we obtain a linear equation in the variables ui = u(qi); i = 0; 1; 2; : : : ; M . Moreover, if instead of p we put pi ; i = 0; 1; 2; : : : ; M , a system of M + 1 linear equations in M + 1 unknowns is obtained. Solving the system, an approximation for the uis is obtained. The quality of this approximation depends both on the location of the points and how the ?s are calculated. Grid generation methods construct the points; the election of the points for the approximation and the corresponding coecients in the linear combination depends on the particular nite-dierence scheme used. In order to simplify the notation, let us denote the qls again by pl s and let L be any of the partial derivatives. The nite-dierence expresion for L is k X (2) L(p)  ?l u(pl) l=0

This is a second-order and weakly consistent approximation ([7]) if the coef cientes ?l satisfy the equation

A? = b

0 1 1 B
x
x1 B 0 B B
y0
y1 A=B 2 B (
x0) (
x1)2 B B @
x0
y0
x1
y1 (
y0)2 (
y1)2

(3)

1


1



xk?1 CCC



yk?1 CC


(
xk?1)2 CCC



xk?1
yk?1 A


(
yk?1)2

where p = (x; y); pl = (xl; yl),
xl = xl ? x,
yl = yl ? y, and vector b on the right-hand side is 001 001 001 001 001 BB 1! CC BB 0 C BB 0 CC BB 0 CC BB 0 CC BB 0 CC BB 1! C C B C B C BB 0 CC BB 0 CC BB 0 CC BB CC ; BB C C ; B 2! C ; B 0 C or B (4) BB 0 CCC BB 0 CC BB 0 C C B C B C C B C B C B C @ 0 A @ 0 A @ 0 A @ 1! A @0A 0 0 0 0 2! 3

depending on if L(p) is ux(p); uy (p); uxx(p); uxy (p) or uyy (p), respectively. In order for the system to have a solution, k must be at least 6. In general, if k > 6, k ? 6 of the unknows have to be speci ed.

2 Grid generation Let m and n positive integer numbers, and a region. An m  n grid G over

is a set of points fPi;j ; i = 1; 2;


; m j = 1; 2;


; ng such that @ = PolygonfP1;1;


; P1;n; P2;n ;


; Pm;n ; Pm;n?1 ;


; Pm;1; Pm?1;1;


; P1;1g; that is, @ is the grid boundary fPi;j g. The i; j cell of the grid, is the quadrilateral Pi;j ; Pi+1;j ; Pi+1;j+1 ; Pi;j+1. A grid of m  n determines (m ? 1)(n ? 1) cells. The i; j cell, in turn, determines four (2) (3) (4) triangles
(1) i;j ;
i;j ;
i;j ;
i; j , so the total number of triangles determined by the grid cells is N = 4(m ? 1)(n ? 1). If explicit mention of a triangle
cab is needed, it is to be meant that the segments ca and ab are on a coordinate line and that cb is a diagonal of the corresponding cell. For a triangle in the grid, let l(
cab) = jjc ? ajj2 + jjb ?ajj2, o(
cab) = (b ? a)t(c ? a);  (
cab) = (b ? a)tJ2(c ? a) where J2 = ?01 01 .  is twice the oriented area of the triangle and o is the inner product of the sides of the triangle. A grid is called convex if each of its cells is a convex quadrilateral. The sets fP1;j ; P2;j ;


; Pm;j j = 1; 2;


; ng will be called horizontal coordinate lines; while the set fPi;1; Pi;2;


; Pi;n i = 1; 2;


; mg will be called vertical coordinate lines. Roughly speaking, a grid is smooth if their coordinate lines are smooth; that is, if the slope changes at every vertex are not too large. The main goal in grid generation is to calculate a convex grid over a given plane region, given that points on the boundary are xed. Other desirable features for a grid are smoothness, orthogonality of coordinate lines and small cell area variation. In the discrete variational approach, a function of the inner points is designed and it is expected that its minimum be attained in a grid with good geometrical properties. This minimization is performed using a large-scale iterative algorithm such as L-BFGS or Truncated Newton ([3], [9]). This kind of functions are called discrete functionals. It is out of our scope to go deeper into the details of grid generation, they can be consulted in the works by Castillo and Steinberg ([5]), Barrera, Perez and Castellanos ([1],[2],[3]). The grids in our examples were generated using the functionals of Tinoco and Barrera ([12], [13],[14]) whose optimization produces smooth and convex grids over general plane regions. They consider only functionals of the form 4

?1 P4 f (
(p)). F (G) = Pmi=1?1 Pnj=1 i;j p=1

IfP an order is assigned to the triangles in the grid; F can be written F (G) = N f (
). The choice of f depends on the desired properties of the optimal q q=1 grid. The following quantities are used in their formulation: q = (
q ) , lq = l(
q) , oq = o2(
q ),  = Area( ) . (m?1)(n?1)

With this notation, the expresions for some functionals are: Classical Classical Area Legth Orthogonality

P

FA2 = N1 2 Nq=1 2q P Fl = 2N1  Nq=1 lq P FO = N1 2 Nq=1 oq

Tinoco-Barrera

P

Smoothness Fs = 21N Nq=1 lqk?+2qq P Area FA =  Nq=1 k+1q i P h Bilateral Smoothness FSB = 21N Nq=1 lkq1?+2qq + lkq2??2qq i P h Bilateral Area FAB = M Nq=1 k1 +1 q + k2 ?1 q Bilateral Area-Orthogonalithy FABO = 12 FAB + 21 FO Area-Orthogonality FAO = 43 FA + 14 FO Bilateral Area-Length FABL = 12 FAB + 12 Fl Area-Lenght FAL = 43 FA + 14 FL Smoothness-Area FSA = 12 FS + 21 FA2

In these formulas k + q > 0,k1 + q > 0 k2 ? q > 0 and M is anormalization constant. The procedure to get a convex grid using Tinoco-Barrera's functionals is as follows: (1) Let G0 be a given initial grid (generated using an interpolation method, for example), and k = ??(G0 ) +  with  chosen appropiately ([11]). (2) Let Gk a grid such that

Fk (Gk ) = minfFk (G) : G 2 Dk g (3) If Gk is convex, the process is nished; otherwise, steps 1. and 2. are 5

repeated. Here Fk is used to represent some of the Tinoco-Barrera's Functionals. The former algorithm produces a convex grid in a nite number of steps (see [12]). Figures 1-5 show the graphs of some grids generated using these functionals.

Fig. 1. Region: Habana Bay (HAB). Functional: ABL. Size: 41  41

Fig. 2. Region: Habana Bay (HAB). Functional: ABO. Size: 41  41

6

Fig. 3. Region: M19. Functional: SB. Size: 41  41

Fig. 4. Region: ELE. Functional: AO. Size: 41  41

3 The scheme In this section, we'll explain how our nite-dierence scheme is constructed. As we explain in section 2, the scheme is completely de ned when we make 7

Fig. 5. Region: Soviet Union (UCH). Functional: SA. Size: 41  41

explicit the stencil and the coecients in the linear combination (2). 3.1 A Preliminary Approximation

Let us calculate a rough approximation to the coecients of every derivative in (1). A given inner point in the grid pi;j = p0 has eight neighboors: pi+1;j , pi+1;j?1 , pi;j?1 , pi?1;j?1 , pi?1;j , pi?1;j+1, pi;j+1 , and pi+1;j+1 . For simplicity, let us denote them by p1, p2, p3, p4, p5, p6 , p7, and p8 respectively, as gure 3.1 shows. It is also convenient to make a translation, taking p0 as the new origin; the eight former points are now transformed in q~l = pl ? p0; l = 0; : : : ; 8. By @! we mean the polygon formed by the line segments that join the points q~1; : : : ; q~8 and q~1. With this new origin, let us nd the intersections of the horizontal axis and @! in the positive and negative directions, and let dxp and dxn their x coordinates, respectively. Similarly, we nd the intersections of the vertical axis with @ , and we call dyp and dyn their y coordinates. Now, the x coordinates of the intersections of the line x = y with @!, are denoted by dxpyp > 0 and dxnyn < 0. Finally, the intersections of the line x = ?y with the boundary of the polygon are dxpyn > 0 and dxnyp < 0. See gure 3.1. Now, let q1 = (dxp; 0), q2 = (dxpyp; dxpyp ), q3 = (0; dyp ), q4 = (dxnyp ; ?dxnyp), q5 = (dxn ; 0), q6 = (dxnyn ; dxnyn ), q7 = (0; dyn ), and q8 = (dxpyn ; ?dxpyn). With this notation, we approximate the partial derivatives of u in pi;j as 8

P4 11 00 00 11 00 11

P3

1 0 0 1

P2 11 00 00 11 00 11

P

5

11 00 00 11 00 11

1 0 0 1

P

0

1 0 0 1

P1

11 00 00 11

P

11 00 00 11 00 11

7

11 00 00 11 00 11

P6

P

8

Fig. 6. dxnyp

dyp

00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 dxpyp 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 1111111111111111111111111111 0000000000000000000000000000 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 1111111111111111 0000000000000000 00000000000000000000000 11111111111111111111111 dxn 111111111111111111111 dxp 000000000000000000000 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 1111111111111111 0000000000000000 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000 1111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 dxpyn 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111

dxnyn

dyn

Fig. 7.

  uxx = 2 dxp (duxp(q1?)dxn ) + duxp(qd0xn) ? dxn (udxp(q5?)dxn ) ; u(q2 ) u(q4 ) uxy = 12 ( dxpyp (dxpyp ?dxnyn ) + dxnyp (?dxnyp +dxpyn ) (q6 ) (q8 ) ? dxnyn(duxpyp + dxpyn (duxnyp ? d ?dxpyn ) ;  xnyn )  u(q3) uyy = 2 dyp(dyp?dyn) + duyp(qd0yn) ? dyn(udyp(q7?)dyn) In general the qk s are not points in the grid, and the values of u at those points have to be interpolated as a linear combination of the values at the pk s. This can be done using, for example, a bilinear map ([8]). The corresponding ux =ud(xpq1?)?dxnu(q; 5) uy = u(dqyp3)??duyn(q7) ;

9

coecients depend only on the pk s and not on the particular values of u there. Once we substitute these linear combinations, the corresponding coecients of the u(pk )s in the dierent derivatives are obtained. This is a preliminar choice, only useful to be used as a nite-dierence scheme on rectangular grids; however, these values are useful to obtain the de nitive coecients in the scheme we propose next.

4 Calculating coecients Let us calculate a second-order approximation for the value of a derivative of a function u in a given grid point pi;j . As we have mentioned before, this requires at least six node points. It is important that these points are close enough to the point where the approximation is to be done. In our case, for every inner point on the grid, we have  eight natural neighboors; if we also consider the point pi;j itself, we have 96 = 84 possible choices of such six points. For the moment, let us suppose the choice has been made and to simplify the following discusion, let us agree in naming the points as q1; q2; : : :; q6. We need, therefore to provide the values of ?7, ?8 and ?9. Equation (3) becomes

0 1 BB
x1 BB
y1 BB (
x1)2 @


x1
y1 (
y1)2

1
x2
y2 (
x2)2
x2
y2 (
y2)2

1 0 ?1 1 0 P9 ?l 1


1 P9 l=7?
x C B B



x6 C ?2 C C B C B Pl9=7 ?l
yl C B C B C



y6 C ? 3 l=7 l l C B C B C (5) P = b ? 9 2 2 C B C B


(
x6) C B ?4 C ?l (
xl ) C l =7 B P 9 ? (
x )(
y ) C @ P



x6
y6 A @ ?5 A l l A l=79 l 2 2


(
y6) ?6 l=7 ?l (
yl )

where b is vector (4) and has to be chosen according to the derivative to be represented. Our preliminary calculation of coecients is useful at this point to calculate the right-hand side of (5); the corresponding values of ? will be assigned. Now, among all the possible choices for the three points on the right-hand side, we agree to take q9 as the point pi;j always. The following diagrams ilustrate choices for q1; : : :; q6 that we have found adequate:

p2p 3p1 456

32p 4p1 56p

432 5p1 p6p 10

p32 4p1 p56

p32 4p1 56p

32p 4p1 p56

3p2 4p1 5p6

321 ppp 456

2p1 3pp 456

3p2 pp1 456

432 pp1 5p6

321 4pp 5p6

The selected points are those numerated from 1 to 6, those marked p are xed and the corresponding coecients are taken from the preliminary approximation; as we have said before, the center point is never taken for the left-hand side of (5). We solve the twelve systems of equations and make an average of the corresponding solutions. This average is our election of the coecients. When this work is made for all the derivatives ux; uy ; uxx; uxy and uyy involved in our dierential problem, the i; j linear equation is 8 X yy xy uk (A0?xk + B0?yk + C0?xx k + D0 ?k + E0 ?k ) = F0 k=0

where A0; B0; C0; D0; E0 and F0 are, respectively, the values of A; B; C; D; E and F evaluated at the point p0.

5 Experiments and Conclusions We used our scheme to solve three test boundary problems over some irregular regions. Of course, in order to evaluate performance, exact solutions to these problems were known a priori. For a grid of m  n knots, the system of (m ? 1)  (n ? 1) linear equations is solved using Gauss-Siedel's iterative algorithm with an overrelaxation parameter of 1:6. We stop iterating when the maximum of the absolute values of the dierences on values for ui;j from one step to the next was smaller than 10?8 .

Example 1 Our rst problem is uxx + uyy = 10e2x+y uj@ = 2e2x+y whose analytic solution is u = 2e2x+y .

11

(6)

Example 2 The equation to solve is: (d1 cos2() + d2 sin2())uxx +(2(d1 ? d2) cos() sin())uxy +(d1 sin2() + d2 cos2())uyy +(cos2() @d@x1 + sin2() @d@x2 + cos() sin() @d1@x?d2 )ux +(sin2() @d@y1 + cos2() @d@y2 + cos() sin() @d1@y?d2 )uy

(7)

=f uj@ = sin(x) sin(y) where

d1 = 1 + 2x2 + y2 d2 = 1 + x2 + 2y2  = 8 and f is chosen such that the analytic solution is sin(x) sin(y).

Example 3 Third equation is again (7), but in this case d1 = 1 + 2x2 + y2 + y5 d2 = 1 + x2 + 2y2 + x3  = 4 its analytic solution is again sin(x) sin(y).

These three PDEs were solved over several grids of 4 dierent regions that where optimized with Barrera-Tinoco's functionals. Results are summarized in the tables, where the following notation is used: 12

Functional It identi es the optimizing functional used to get the grid L2 error The norm of the approximation error: E2 = qP euclidian 2 i;j Ei;j Ai;j , where Ei;j = juexact(xi;j ; yi;j ) ? ui;j j and Ai;j is the area of the i; j cell order This is the order of the error, it is obtained by calculating the base 2 logarithm of the quotient between the L2 error for a given grid and the L2 error for a grid with double points on each boundary. (Q2 = log2(E2M N =E22M 2N ) Size We used grids with the same number of vertical and horizontal lines. Size is this number. Functional Size 21 ABL 41 81 21 ABO 41 81 21 AL 41 81 21 AO 41 81 21 S 41 81 21 SA 41 81 21 SB 41 81

Table 1. Example calculations for ELE grid First PDE Second PDE L2 error order L2 error order 6.038E-04 5.833E-04 1.587E-04 1.928 1.244E-04 2.229 3.129E-05 2.342 3.514E-05 1.824 9.171E-04 5.766E-04 2.581E-04 1.829 1.159E-04 2.315 6.425E-05 2.006 4.704E-05 1.301 7.600E-04 9.166E-04 2.150E-04 1.822 1.074E-04 3.093 4.942E-05 2.121 3.371E-05 1.672 7.630E-04 1.192E-02 2.260E-04 1.755 1.325E-04 6.491 5.523E-05 2.033 5.643E-05 1.232 4.530E-04 6.321E-04 1.333E-04 1.765 1.913E-04 1.724 3.055E-05 2.126 4.426E-05 2.112 6.859E-04 4.511E-04 1.739E-04 1.980 9.595E-05 2.233 3.139E-05 2.469 2.422E-05 1.986 3.980E-04 5.253E-04 9.508E-05 2.066 1.359E-04 1.951 1.431E-05 2.732 3.282E-05 2.050

Third PDE L2 error order 5.985E-04 1.291E-04 2.213 3.552E-05 1.862 5.481E-04 1.027E-04 2.416 4.360E-05 1.235 7.910E-04 1.092E-04 2.857 3.430E-05 1.670 6.238E-03 1.239E-04 5.654 5.024E-05 1.302 6.619E-04 1.899E-04 1.801 4.383E-05 2.115 4.643E-04 1.001E-04 2.213 2.242E-05 2.159 5.670E-04 1.431E-04 1.987 3.356E-05 2.092

Theoretically, the scheme is second-order, tables show the that empirical order is quite close to 2 in most cases. However, in some isolated cases, particularly with orthogonal grids, the order is small or even negative, the later means that we had a better approximation with the grid with less points. In regions with not so complicated shape, the method is robust, no matter which functional is used to generate the grid. In this cases, the errors are comparable. However, on dicult regions, such as HAB and UCH, near-orthogonal 13

Functional Size 21 ABL 41 81 21 ABO 41 81 21 AL 41 81 21 AO 41 81 21 S 41 81 21 SA 41 81

Table 2. Example calculations for HAB grid First PDE Second PDE Third PDE L2 error order L2 error order L2 error order 1.575E-03 5.075E-04 5.214E-04 3.482E-04 2.177 1.069E-04 2.247 1.157E-04 2.172 7.496E-05 2.216 3.963E-05 1.432 3.965E-05 1.545 1.281E-03 1.225E-03 9.242E-04 3.657E-04 1.808 2.188E-04 2.485 2.067E-04 2.160 1.079E-04 1.761 8.367E-04 -1.935 7.358E-04 -1.832 1.907E-03 6.500E-04 6.352E-04 3.610E-04 2.401 1.541E-04 2.077 1.626E-04 1.966 7.138E-05 2.339 3.329E-05 2.210 3.458E-05 2.234 1.785E-03 9.008E-04 9.245E-04 5.696E-04 1.648 2.728E-03 -1.599 1.593E-03 -0.785 1.208E-04 2.237 4.891E-04 2.480 3.343E-04 2.252 2.146E-03 1.309E-03 1.332E-03 4.905E-04 2.129 3.283E-04 1.995 3.358E-04 1.988 1.070E-04 2.197 6.937E-05 2.243 6.933E-05 2.276 1.509E-03 5.994E-04 6.049E-04 3.634E-04 2.054 1.554E-04 1.948 1.625E-04 1.896 8.503E-05 2.096 3.524E-05 2.141 3.703E-05 2.133

Functional Size 21 ABL 41 81 21 ABO 41 81 21 AL 41 81 21 AO 41 81 21 S 41 81 21 SA 41 81 21 SB 41 81

Table 3. Example calculations for M19 grid First PDE Second PDE L2 error order L2 error order 2.038E-03 2.813E-03 4.725E-04 2.109 2.341E-04 3.587 1.276E-04 1.889 7.189E-05 1.704 1.585E-03 2.332E-03 3.113E-04 2.348 1.043E-03 1.160 9.086E-05 1.777 1.355E-04 2.945 2.136E-03 1.005E-03 4.912E-04 2.121 2.572E-04 1.966 1.107E-04 2.150 2.615E-04 -0.024 1.763E-03 1.056E-03 4.029E-04 2.129 3.783E-04 1.481 9.320E-05 2.112 1.102E-04 1.779 2.712E-03 1.044E-03 7.135E-04 1.927 3.124E-04 1.741 1.746E-04 2.031 8.939E-05 1.805 2.126E-03 9.285E-04 5.168E-04 2.041 2.805E-04 1.727 1.132E-04 2.191 7.109E-05 1.980 2.175E-03 9.229E-04 5.264E-04 2.047 2.366E-04 1.964 1.159E-04 2.183 6.119E-05 1.951

14

Third PDE L2 error order 4.441E-03 2.308E-04 4.266 7.161E-05 1.688 2.096E-03 9.556E-04 1.133 1.261E-04 2.922 9.699E-04 2.570E-04 1.916 2.536E-04 0.020 1.017E-03 3.710E-04 1.455 1.125E-04 1.721 9.406E-04 2.813E-04 1.741 8.176E-05 1.783 9.084E-04 2.763E-04 1.717 7.043E-05 1.972 8.701E-04 2.254E-04 1.949 5.886E-05 1.937

Functional Size 21 ABL 41 81 21 AL 41 81 21 AO 41 81 21 SA 41 81

Table 4. Example calculations for UCH grid First PDE Second PDE L2 error order L2 error order 2.907E-03 8.457E-04 8.396E-04 1.792 7.589E-04 0.156 2.751E-04 1.610 8.080E-05 3.232 4.590E-03 3.088E-02 1.346E-03 1.769 1.424E-03 4.439 5.148E-04 1.387 9.441E-05 3.914 3.627E-02 4.853E-03 4.982E-03 2.864 5.187E-03 -0.096 4.586E-04 3.441 6.755E-03 -0.381 7.099E-03 2.557E-03 1.529E-03 2.215 2.887E-04 3.147 4.080E-04 1.906 7.371E-05 1.969

Third PDE L2 error order 8.960E-04 2.701E-04 1.730 8.318E-05 1.699 8.995E-03 5.507E-04 4.030 1.177E-04 2.226 3.897E-03 3.510E-03 0.151 2.038E-03 0.785 1.166E-03 3.208E-04 1.862 8.806E-05 1.865

grids produce bad results unlike smooth grids which behave well. This could be due to the highly-distorted cells near the boundary of those grids.

References [1] Barrera-Sanchez,P., Castellanos-Noda, L., Perez-Domnguez, A. Metodos Variacionales Discretos para la Generacion de Mallas. DGAPA-UNAM, Mexico, D.F., 1994. [2] Barrera-Sanchez,P., Perez-Domnguez, A., Castellanos-Noda, L. Curvilinear Coordinate System Generation Over Plane Irregular Regions. Vnculos Matematicos No. 133, Facultad de Ciencias, UNAM, Mex. 1992. [3] Castellanos, J. L. Generacion Numerica de Redes usando Newton Truncado. Tesis Doctoral, I.C.I.M.A., Ministerio de la Ciencia, Tecnologa y el Medio Ambiente. La Habana, Cuba. 1994. [4] Castillo, J. E. On Variational Grid Generation. Ph. D. Thesis, The University of New Mexico, Albuquerque, New Mexico. 1987. [5] Castillo, J. E., Steinberg, S. and Roache, P.J. Mathematical Aspects of Variational Grid Generation II. J. Comp. and Appl. Math. 20, 127-135. 1987. [6] J. E. Castillo, J. M. Hyman, M. J. Shashkov, S. Steinberg. High Order Mimetic Finite Dierence Methods on Nonuniform Grids. Proceedings of the Third International Conference on Spectral and High Order methods, Houston Journal of Mathematics, Houston, A. V. Ilin and L. R. Scottt (eps), 347-361. 1996. [7] Celia Michael A. and G. Gray William. Numerical Methods For Dierential Equations. Prentice Hall. Englewood Clis , New Jersey. 1992.

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[8] Knupp, Patrick M., and Steinberg, Stanley. Fundamentals of Grid Generation. CRC Press. Boca Raton, Ann Arbor, London, Tokyo. 1993. [9] More J.J., Thuente D.J. On line search algorithms with guaranteed sucient decrease. Math. and Computer Science Division Preprint MCS-P153-0590, Argonne National Lab., Argonne, Ill. 1990. [10] M. Shashkov. Conservative Finite - Dierence Methods on General Grids. Boca Raton Fla. CRC Press. 1996. [11] Tinoco,J.G. Funcionales Discretos para la Generacion de Mallas Suaves y Convexas sobre Regiones Planas Irregulares. Tesis Doctoral. CIMAT., Guanajuato, Gto., Mexico. 1997 [12] J. G. Tinoco-Ruiz, P. Barrera-Sanchez. Smooth and Convex Grid Generation over General Plane Regions. Mathematics and Computers in Simulation (46), pp 87-102. Elsevier Science - North Holland. 1998 [13] J. G. Tinoco-Ruiz, P. Barrera-Sanchez. Area Control in Generating Smooth and Convex Grids over General Plane Regions. J. Computational and Applied Mathematics (103)1, pp 19-32. Elsevier Science - North Holland. 1999. [14] J. G. Tinoco-Ruiz, P. Barrera-Sanchez. Area Functionals in Plane Grid Generation. Proceedings of the 6th. International Conferences on numerical grid Generation and Computational Field Simulation. London, England, 1998, pp 293-302. International Society of Grid Generation. 1998.

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