1549301742-The_Theory_of_Difference_Schemes__Samarskii

356 Difference Schemes with Constant Coefficients ·-----· ·------· ·---·---· (xi+1' tj) (xi-1,tj) a b c (xi, tj+1) (xi,tj+1) ·-- ...

The transfer equation 357 Remark Keeping I = ar/h = O(h), that is, T = O(h^2 ), we might have I q I< 1 +c 0 T, where c 0 > ...

358 Difference Schemes with Constant Coefficients Explicit schemes of a higher-order approximation. Of major importance is the ...

The transfer equation 359 The residual on a solution u(x, t) is Q.5 h^2 1 .. 1 h^2 II ( 2 2) 1/J = Ut + au x 0 - T U;:x. = - 2 T ...

360 Difference Schemes with Constant Coefficients The boundary-value problem. We are interested in learning more about the boun ...

The transfer equation 361 Observe that scheme (11) belongs to this family, corresponds to the case O" = 1 and has the residual 1 ...

362 Difference Schen1es with Constant Coefficients The next step is to multiply the resulting equation by 2r Yxt = 2 C!Jx - Yx) ...

The transfer equation 363 holds true due to the relation YYxx = ~(y^2 )x + ~h(y:f:)^2. An alternative form of scheme (12) rnay b ...

364 Difference Schemes with Constant Coefficients 5.6 DIFFERENCE SCHEMES FOR THE EQUATION OF VIBRATIONS OF A STRING The stateme ...

Difference schemes for the equation of vibrations of a string 365 A key role in subsequent discussions is played by a farnily of ...

366 Difference Schemes with Constant Coefficients where 1/; = A( O" u + (1 - 2 O") u + O" ii) + <p - utt is the approximation ...

Difference schemes for the equation of vibrations of a string 367 Here the approximation error for the boundary conditions is a ...

368 Difference Schemes with Constant Coefficients whose solutions are srn-.? --Irk h 2 ' X (k)(x) = V2 sin Irkx. From (9) we get ...

Difference schemes for the equation of vibrations of a string 369 By relating the initial conditions y^0 = u 0 and y~ = (y^1 - y ...

370 Difference Schemes with Constant Coefficients and (^2) Sill.^1 2 'Pk T the estirnate occurs: J2 (1 - cos 'Pd T T ( 19) SIIl! ...

Difference schemes for the equation of vibrations of a string 371 This estimate remains valid for scheme ( 4a) under the constra ...

372 Difference Schemes with Constant Coefficients Putting these together with ( 4 b) we find that (26) which allows us to derive ...

Difference schemes for the equation of vibrations of a string 373 By exactly the same reasoning as before we deduce for problem ...

374 Difference Schen1es with Constant Coefficients In Chapter 2 it has been already shown that 4 h;^2 sin^2 ( ~ Irh 1 ) > 8 i ...

Difference schemes for the equation of vibrations of a string 375 Indeed, from the first Green formula (see Chapter 2, Section 3 ...