1549301742-The_Theory_of_Difference_Schemes__Samarskii

176 Homogeneous Difference Schemes and having k 1 = 2 and k 2 = 4 to the difference problem with steps h 1 = h and h 2 =! h we c ...

Homogeneous difference schemes on non-equidistant grids 177 When u(x), f(x) and the coefficients of equation (11) are smooth eno ...

178 Homogeneous Difference Schemes on the behaviour of the solution of an original problern is available, a grid can be made so ...

Other problems 179 where a, d and <p satisfy the approximation conditions (18) of Section 2. First, we consider the simplest ...

180 Homogeneous Difference Schemes At any point x E (0, 1) condition (6) is equivalent to the pan of continuity conditions at th ...

Other problems^181 At the nodes xi = ih, i = 1, 2, ... , N - 1, we write down the three- point equation Yxcc - qo Y = -<p(x), ...

182 Homogeneous Difference Schemes which are continuous at the point x = 0 (x = 1), so that k(l - 0) = k(O + 0) = k(O), etc. Let ...

Other proble1ns 183 subject to the conditions (17) Yo= YN' For determination of Yi, i = 1, 2, ... N, we obtain the system of equ ...

184 Homogeneous Difference Schemes The natural replacement of the central difference derivative u' ( x) by the first derivative ...

Other problems 185 As can readily be observed from the foregoing, Ai > 0, B; > 0 and D; ~ 0, since bi < 0, bt > 0 an ...

186 Homogeneous Difference Schemes If q 2 c 1 > 0, then for a solution of the difference problem (21) the maxi- mum principle ...

Other problems with the coefficients (25') Xi+l/2 b; = Cl;+ 1 exp { j 1~(t) clt} , Xi x· a, = Cl; exp {- J r(t) clt}. Xi-1 /2 18 ...

188 Homogeneous Difference Schemes vVhen r = Rone of the usual boundary conditions applies here, for instance, (28) u(R) = μ 2. ...

Other problems 189 where Yi - Yi-1 Yr, i = h Yi+1 - Yi Y,., i = h the coefficients ai, di and 'Pi are chosen so that (32) 'Pi= f ...

190 Homogeneous Difference Schemes and (37) ( k u')~ = ( q u - f)o - ( k u')~ = ~ ( qu - f)o. Substituting (37) into formula (36 ...

Other problems 191 We now estimate the accuracy of scheme (38). Substituting into (38) y = z + u, where u is a solution of probl ...

192 Homogeneous Difference Schemes By the same token, so that ( 44) 1 h T' z ri-1/2 1/J, * = 12r h2 ( qu - f)i I + 0( h2) , z gi ...

Other problems^193 Substitution of wj+i = b;+i (v;+i - vj)/h into (47) leads to the relation (48) Recalling that vN = 0 and summ ...

194 Homogeneous Difference Schemes By inserting in (50) the estirnates v = O(h), 0 2 I 77i I= O(h ), we conclude that II v lie= ...

Other problems^195 The standard condition (54) is imposed for r = R. As a final result we get the difference equation (51) subje ...