1549301742-The_Theory_of_Difference_Schemes__Samarskii

136 Basic Concepts of the Theory of Difference Schemes Negative norms. In a priori estimates ( 41) and ( 42) we were dealing wi ...

Difference schemes as operator equations. General formulations 137 on wh and vanishing for i = 0 and i = N under the inner produ ...

138 Basic Concepts of the Theory of Difference Schemes All this enables us to derive for this norm the estimate ( 53) For its pr ...

Difference schemes as operator equations. General formulations 139 structure [y, v] = L~~l Yi vi h + ~ h (Yo v 0 + yN vN ). Then ...

140 Basic Concepts of the Theory of Difference Schemes In just the same way as before, we write down the boundary condition in ( ...

Difference sche1nes as operator equations. General formulations 141 with the inner product (y, v) and associated norm II y II = ...

142 Basic Concepts of the Theory of Difference Schemes so that v = Ty E H 1 if y E H; i=l,2, ... ,N-1, so that T*v EH ifv E H 1. ...

Difference sche1nes as operator equations. General formulations 143 and work in the space H 1 of all grid functions defined on t ...

144 Basic Concepts of the Theory of Difference Schemes (Ay, y) = (T*STy, y) = (STy, Ty]> c 1 [[Ty][^2 , meaning A > / A 0 ...

Homogeneous Difference Schemes This chapter presents the theory of homogeneous difference schemes for the solution of equations ...

146 Homogeneous Difference Schemes of the governing differential equation, a class of boundary and initial con- ditions as well ...

Homogeneous schemes for second-order equations 147 with step h. Let k( s) be a vector function defined for -m 1 < s < m 2 ...

148 Homogeneous Difference Schemes Reducing scheme (2) to the form (4) from Section 2.1 we find that b = k + k i+1 - k i-1 ! i 4 ...

Conservative schemes^149 From (3) and (4) we find that an = ~ (5k 1 - k 2 ), an+ 1 = ~ (k 1 + 3k 2 ), bn = ~ (3k 1 + k 2 ) and b ...

150 Homogeneous Difference Schemes violated, thus causing the appearance of the extra heat source (for q < 0) or sink (for q ...

Conservative schemes^151 conservation laws (of heat, mass, momentun1, energy, etc.). Usually the derivation of a differential eq ...

152 Homogeneous Difference Schemes heat being en1itted on the segn1ent [xi_ 112 , xi+ 1 ; 2 ] by heat sources with the distribut ...

Conservative schemes where (15) 1/2 clx k(x) d; = d; = j q(xi +sh) ds, -1/2 1/2 'Pi= &i = j f(x; +sh) ds. -1/2 __ cl_s_ )-l ...

154 Homogeneous Difference Schemes Homogeneous conservative schemes. In the preceding section we have designed the conservative ...

Conservative schemes 155 the inner product (y, v) = Lf:~^1 Yi vi h. Being elements of the space rh, 0 any functions y, v E Dh ar ...