Variable-grid method

A method for the numerical solution of problems in mathematical physics, where the difference grid on which the equations for the basic problem are approximated does not remain fixed; it tracks the changes in the boundaries of the working regions during the calculation. A very simple difference grid is obtained as the points of intersection of straight lines parallel to the axes of a Cartesian coordinate system (a rectangular grid), and this maximally simplifies the difference equations in which the basic problem is formulated. However, there are considerable difficulties in giving an accurate representation of boundaries of complicated shape together with the boundary conditions there, and these are frequently unsurmountable if computer facilities are restricted. These difficulties become particularly acute in non-stationary problems in mathematical physics where the boundaries of the working regions are moving and undergo considerable deformations. A substantial part of an algorithm for handling such a problem lies in constructing a coordinate system in which the coordinate lines coincide with the boundaries.

In applications of the variable-grid method, at each instant the working region is divided into a finite number of cells which do not overlap and which fill the entire region without gaps. When there are two spatial variables, it is most convenient to divide the working region using two families of lines to give rectangular cells. Then it is simple to enumerate the cells (the enumeration is analogous to that of matrix elements by rows and columns).

The calculation of the coordinates of the nodes in such a grid can be treated as the difference analogue of seeking functions $x(\xi,\eta)$ and $y(\xi,\eta)$ that provide a single-sheeted mapping into the region of the physical ($x,y$)-plane in which the calculation is to be performed, of a certain parametric region in the ($\xi,\eta$)-plane, for example, the unit square $0\leq\xi\leq1$, $0\leq\eta\leq1$. The boundary values of these functions define a certain one-to-one correspondence between the points on the sides of the parametric square and on the boundaries of the physical region. This correspondence should be made available to the computer, which calculates the nodes of the grid at the boundary of the working region on the basis of the detailed content of the problem.

If the region is of simple shape, the coordinates of the nodes can be calculated from explicit formulas based on interpolation or on a concrete form of the mapping. In the case of a complicated region, one has to use an iterative process in which the position of a node is recalculated at each iteration in relation to the positions of the neighbours. Such iterative processes are usually modelled on solving a system of partial differential equations. In constructing these processes, one often has to use conformal or quasi-conformal mappings in some form.

In problems in mathematical physics (such as in gas dynamics) the structure of the solution can be characterized by the presence of zones in which there are sharp changes in the flow parameters. The dimensions of such zones may be substantially less than the characteristic linear dimensions, while their exact positions may not be known in advance, or they may vary during the calculation. For this reason, one designs algorithms that enable one to use grids that are more closely spaced in such zones.

In order to set up reliable numerical algorithms, the problem of constructing a grid is often formulated as a task of minimizing a certain variational functional. The functional may contain control parameters, the adjustment of which provides a certain freedom and a possibility for adjusting the grid to the features of a particular problem.

References

[1]	, Numerical solution of multi-dimensional problems in gas dynamics , Moscow (1976) (In Russian)
[2]	A.F. Sidorov, , Numerical methods in the mechanics of continuous media , 8 : 4 (1977) pp. 149–156 (In Russian)
[3]	Yu.P. Meshcheryakov, V.P. Shapeev, , Numerical methods in the mechanics of continuous media , 9 : 2 (1978) pp. 91–103 (In Russian)
[4]	N.T. Danaev, , Numerical methods in the mechanics of continuous media , 10 : 4 (1979) pp. 60–74 (In Russian)
[5]	P.D. Tomas, D.F. Middlkoff, Raketn. Tekhn. i Kosmonavtika , 18 : 7 (1980) pp. 55–61

References

[a1]	W. Hackbusch, "Multigrid methods and applications" , Springer (1985)