By Mikhail Shashkov
This new ebook bargains with the development of finite-difference (FD) algorithms for 3 major varieties of equations: elliptic equations, warmth equations, and gasoline dynamic equations in Lagrangian shape. those equipment should be utilized to domain names of arbitrary shapes. the development of FD algorithms for all sorts of equations is completed at the foundation of the support-operators strategy (SOM). this system constructs the FD analogs of major invariant differential operators of first order comparable to the divergence, the gradient, and the curl. This publication is exclusive since it is the 1st e-book no longer in Russian to give the support-operators ideas.
Conservative Finite-Difference tools on common Grids is totally self-contained, offering all of the history fabric invaluable for knowing. The booklet presents the instruments wanted through scientists and engineers to resolve quite a lot of useful engineering difficulties. An abundance of tables and graphs help and clarify tools. The booklet info all algorithms wanted for implementation. A 3.5" IBM appropriate laptop diskette with the most algorithms in FORTRAN accompanies textual content for simple use.
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Extra info for Conservative Finite-Difference Methods on General Grids
2. THE MAIN IDEAS OF FINITE-DIFFERENCE ALGORITHMS x i ::: + 0 ,... H I I'"'::: 0 . ::: I H ::: I H ::: H .... 1; 0 I ~ + i ::: H + .. ,... i ::: H ::: ~ ::: H I ::: H .... Ii ~ t H + 0 . I~ . , H I ::: ~ H ...... r:-i IU :a I! ~ ,... H I I .. ""\ H "i:' ..... i .... ::: I H (Ii ~+ 0 0 0 0 I H ,... I~I . H el~+ . : ,... I ~ 0 I .. "' H II ""\ 25 26 CHAPTER 1. INTRODUCTION ( I . V) ( M. NJ n. I,) ( x .. y . IL j I ) I ( I. I ) ( M. 8: Tensor product grid. 1 2-D case Grid in 2-D The simplest example of a grid in 2-D is called the tensor product grid in a rectangle.
6 helps in understanding how operator D,, acts. Next, the important question is: What is the relationship between differential operators and its difference analog? In the simplest case the statement of this question is: let grid function uf E H N be the projection of the continuous function u( x) and let, for simplicity, uf = u(xi). Then, what we can say about truncation error or error of approximation for a differential operator by a difference operator IS I dx ,,. 1/J = du uf+ 1 - uf Xi+I - Xi du' u(xi+1) - u(x;) = dx :z:• It is important to note that the value of 1/J depends on the choice of point x*, and in the general case, instead of~; J:z:• we must consider projection of the continuous function du/dx to the space of grid functions HG, because D,.
Ym(l) which is closer to the exact solution than the previous approximation. There are many different iterative methods (see (111]). The simplest example of the iterative methods is the Gauss-Seidel method. Assume that we know the k-th iterative approximation. _,a1 . Yj(k) 1 + f 1· j=2 Assuming that all i = 2 we obtain =f. 0, we can then use this equation to find y~k+l). For Therefore, in obtaining the second component of the new approximation we can use the second equation of the original system and the already computed first component of the new approximation.