1549301742-The_Theory_of_Difference_Schemes__Samarskii

76 Basic Concepts of the Theory of Difference Schemes Example 4. The first boundary-value problem for the heat con- duction equa ...

Difference approximation of elementary differential operators 77 or y.j+1 - (2 + h 2 z-1 T ) y.i+1 i + y.j+1 i+1 = -h2 pi z ' O& ...

78 Basic Concepts of the Theory of Difference Schemes corresponding to different values of the parameter his of great interest a ...

Difference approximation of elementary differential operators 79 holds with constant M > 0 independent of I h I and n > 0 ...

80 Basic Concepts of the Theory of Difference Sche1nes At the next stage we look for the residual 1/; = u,c -<p on a solution ...

Difference approximation of elementary differential operators 81 we focus our attention on the three-point difference scheme Yxx ...

82 Basic Concepts of the Theory of Difference Schemes Example 1. The third boundary-value problem. for an ordinary second-order ...

Difference approximation of elementary differential operators 83 since μ 1 + u'(O) - u u(O) = 0. This provides support for the v ...

84 Basic Concepts of the Theory of Difference Schemes where 'P = 'P/ = f ( X;, tj). In giving the difference approxirnation of t ...

Difference approximation of elementary differential operators 85 Evidently, a special attention in approximating problem (51) is ...

86 Basic Concepts of the Theory of Difference Schemes Example 4. The three-layer difference scherne for the heat con- duction eq ...

Difference approximation of ele1nentary differential operators 87 into the formula u(x, r) - u ou I T2 a2u I 0 (x) = r - 0 + - 2 ...

88 Basic Concepts of the Theory of Difference Schemes which can be rewritten as 1 Y;=SY;-1, s=--- 1 +ah ' i = l, 2, ... , Yo = U ...

Stability of a difference scheme^89 It is well-known that the general solution to equation (3) is of the form (5) Yi=As~+Bs~. Pu ...

90 Basic Concepts of the Theory of Difference Schemes Table 1 contains the values Yi at several nodes of the grid wh for the fou ...

Stability of a difference sche1ne 91 so that With these entries, we look for a solution of problem (6) in the form n u(t) = 2= o ...

92 Basic Concepts of the Theory of Difference Sche111es which implies that (8) From such 1·easoning it seems clear that the solu ...

Stability of a difference scheme^93 expresses the stability of the problem (10) solution. The condition maxk I qk I < 1 is fu ...

94 Basic Concepts of the Theory of Difference Schemes Assuming this to be the case, we deduce in light of the obvious relations ...

Stability of a difference scheme 95 Table 2 Variant (^1) Variant 2 J Yj zj J Yj Z· J 8 -5.52 .10-^3 5.52. 10-^3 8 0.227 0.227 9 ...