876 Chapter 19.Partial Differential Equations Its symbol is therefore 12 12 (19.6.13) 12 The symbol of R is defined by considering vn to be defined everywhere on the fine grid,and then asking what is Ruh at (y)as a linear combination of these values.The simplest possible choice for R is straight injection,which means simply filling each coarse-grid point with the value from the corresponding fine-grid point. 81 Its symbol is "[1]."However,difficulties can arise in practice with this choice.It turns out that a safe choice for R is to make it the adjoint operator to P.To define the adjoint,define the scalar product of two grid functions uh and vh for mesh size h as (uhnh三h2∑h(c,y)h(,) (19.6.14) 3 工，y RECIPES Then the adjoint of P,denoted Pt,is defined by ≥ (uHIPivh)H =(PuHlUh)h (19.6.15) Now take P to be bilinear interpolation,and choose u=1 at(,y),zero elsewhere. Set Pt R in (19.6.15)and H 2h.You will find that e9兰公 (Rh)红，)=h(z,)+h(z+h,)+h(z+h,y+h)+…(19.6.16) IENTIFIC so that the symbol of R is 1 18 (19.6.17) 」 Note the simple rule:The symbol of R is the transpose of the matrix defining the symbol ofP,equation(19.6.13).This rule is general wheneverR=Pt and H=2h. Recipes Numerical 10521 The particular choice of R in(19.6.17)is called full weighting.Another popular 431 choice for R is half weighting,"halfway"between full weighting and straight Recipes injection.Its symbol is (outside 0 01 North 18 18 (19.6.18) 0 0 A similar notation can be used to describe the difference operator Ch For example,the standard differencing of the model problem,equation (19.0.6),is represented by the five-point difference star 01 0 Ch= h2 -4 (19.6.19) 0 10876 Chapter 19. Partial Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Its symbol is therefore    1 4 1 2 1 4 1 2 1 1 2 1 4 1 2 1 4    (19.6.13) The symbol of R is defined by considering vh to be defined everywhere on the fine grid, and then asking what is Rvh at (x, y) as a linear combination of these values. The simplest possible choice for R is straight injection, which means simply filling each coarse-grid point with the value from the corresponding fine-grid point. Its symbol is “[1].” However, difficulties can arise in practice with this choice. It turns out that a safe choice for R is to make it the adjoint operator to P. To define the adjoint, define the scalar product of two grid functions u h and vh for mesh size h as uh|vh h ≡ h2 x,y uh(x, y)vh(x, y) (19.6.14) Then the adjoint of P, denoted P†, is defined by uH|P† vh H = PuH|vh h (19.6.15) Now take P to be bilinear interpolation, and choose u H = 1 at(x, y), zero elsewhere. Set P† = R in (19.6.15) and H = 2h. You will find that (Rvh)(x,y) = 1 4 vh(x, y) + 1 8 vh(x + h, y) + 1 16 vh(x + h, y + h) + ··· (19.6.16) so that the symbol of R is    1 16 1 8 1 16 1 8 1 4 1 8 1 16 1 8 1 16    (19.6.17) Note the simple rule: The symbol of R is 1 4 the transpose of the matrix defining the symbol ofP, equation (19.6.13). This rule is general whenever R = P † and H = 2h. The particular choice of R in (19.6.17) is called full weighting. Another popular choice for R is half weighting, “halfway” between full weighting and straight injection. Its symbol is    0 1 8 0 1 8 1 2 1 8 0 1 8 0    (19.6.18) A similar notation can be used to describe the difference operator Lh. For example, the standard differencing of the model problem, equation (19.0.6), is represented by the five-point difference star Lh = 1 h2   0 10 1 −4 1 0 10   (19.6.19)