1549301742-The_Theory_of_Difference_Schemes__Samarskii

196 Homogeneous Difference Sche1nes It follows from the foregoing that ,/ * y). -~ -, μi 1 r. z An estimate similar to that esta ...

Other problems^197 We introduce on the segrnent 0 ::; r < R the equidistant grid w h = {r; = ih, i = 0, 1, ... , N, hN = R} a ...

198 Homogeneous Difference Schemes After replacing the variable by r = r; + sh we find that r;+1/2 1/2 h ~ 2 j Jr^2 dr = : 2 j J ...

Difference Green's function 199 3.6 DIFFERENCE GREEN'S FUNCTION Difference Green's function. Further estimation of a solution o ...

200 Homogeneous Difference Schemes where ex(x) and f3(x) are solutions of the relevant Cauchy problems: Lex= 0, (5) L {3 = 0, 0 ...

Difference Green's function 201 under the agreement that this expression solves the equation Ayi = -tp;. From the equality Ayi = ...

202 Hmnogeneous Difference Schemes If cxk > 0 for k = 1, 2 ... , i - 1, then a; cxx. ' i > 0 and ex; > cx;_ 1 > 0. I ...

Difference Green's function 203 as required. Formula (11) means that Gik > 0 for i, k f 0, N and Gik = Gki· Moreover, Gik as ...

204 Hon1ogeneous Difference Schen1es Lemma For the Green function G(x,~) related to problem (6)-(7) the uniform estimates (19) ( ...

Difference Green's function we arrive at (22) By the same token, 1 Go:v(x,~) < -. C1 205 Let v(x,~) = G 0 (x,~) - G(x,~). The ...

206 Homogeneous Difference Schemes Proof Putting μ = T/x and applying then formula (18), we deduce by the preceding lemma that ( ...

Higher-accuracy sche1nes 207 3.7 HIGHER-ACCURACY SCHEMES An exact scheme. For equation ( 1) in Section 1 it is possible to make ...

208 Homogeneous Difference Schemes where A; and B; are some numbers which will be specified in the sequel, v;(x) and v;(x) are l ...

Higher-accuracy schemes We will prove the indicated properties. 1) Property 1) follows directly from problem staternent (5 )-( 6 ...

210 Homogeneous Difference Sche1nes The function v~(x) admits an alternative form x;+1 (11) v;(x) = j G(x,~)f(~) d~, where G(x, ...

Higher-accuracy schemes 211 We now introduce a local coordinate system at a point x = X; by merely setting x = x;+sh, s = (x-x;) ...

212 Homogeneous Difference Schemes The coefficients d( x ), IP( x) can be calculated by one and the same formula 1P(x) = F [p(x ...

Higher-accuracy schemes^213 One trick we have encountered is to take only a finite number of the mem- bers in series ( 19): m m ...

214 Homogeneous Difference Schemes at some isolated points on each segment [xi-I, xi+il· Such schemes permit one to retain the o ...

Methods for designing difference schemes 215 2) the variational difference methods (the Ritz rnethod and Bubnov- Galerkin rnetho ...