1549301742-The_Theory_of_Difference_Schemes__Samarskii

56 Basic Concepts of the Theory of Difference Schemes For a difference function uh, we turn to the difference Yh - 1l h, which g ...

Difference approximation of elementary differential operators 57 form, that is, to compose the set of nodes adjacent to x at whi ...

58 Basic Concepts of the Theory of Difference Schemes fixed. Substituting the preceding series into (1), (2), and (4) yields v= ...

Difference approximation of elementary differential operators 59 With the aid of the expansion of the function v(x) in Taylor's ...

60 Basic Concepts of the Theory of Difference Schemes number of the nodes will increase. However, the way of achieving a higher- ...

Difference approximation of elementary differential operators 61 where~ is the mean value of x on the segment [a, x]: ~=a+e(x-a) ...

62 Basic Concepts of the Theory of Difference Schemes Since g 2 ( s) > 0, Lagrange's formula provides support for the represe ...

Difference approxirnation of elementary differential operators 63 where 8(1+0.5s) 3 for - 2 < - s -< -1 , 8 (1 +0.5s) 3 -4 ...

64 Basic Concepts of the Theory of Difference Schemes (x,t+r) (x-h,t+r) (x,t+r) (x+h,t+r (x-h,t) ( x, t) (x+h,t) ( x, t) a b (x- ...

Difference approximation of elementary differential operators 65 formulae for L~~ v, L),^1 ) v and L~~) v the following expressi ...

66 Basic Concepts of the Theory of Difference Schemes 82 v 82 v Example 5 L v = Bt 2 - Bx 2 • In that case the values of a grid ...

Difference approximation of elementary differential operators 67 In a similar manner one can wt·ite down on the pattern 5.a the ...

68 Basic Concepts of the Theory of Difference Schemes Having stipulated the condition h± < 21i, the well-established expansio ...

Difference approximation of elementary differential operators 69 The next step is to introduce an operator L in the space H 0 an ...

70 Basic Concepts of the Theory of Difference Schemes Remark 3 If the grid wh is non-equidistant, that is, h = (h 1 , ••• , hN ) ...

Difference approximation of elementary differential operators 71 The same is still valid in the grid norm of the space L2: ( ) N ...

72 Basic Concepts of the Theory of Difference Schemes implies that 111/J I le -l) < M h^2. This 111eans that the approxin1ati ...

Difference approximation of elementary differential operators 73 vh(x, t), thereby specifying vhT on the grid wh 7 and reducing ...

74 Basic Concepts of the Theory of Difference Schemes The meaning of Lh v = vxx as the difference approxin1ation of the operator ...

Difference approximation of elementary differential operators 75 in correspondence with problem (32). Here the right-hand side ' ...