MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR N. N. YANENKO, B. I. KVASOV

AN ITERATIVE METHOD FOR CONSTRUCTING POLYCUBIC SPLINE FUNCTIONS

1. Let there be a grid on the interval \([a,b]\)

\[ \Delta:\quad a=x_0<x_1<\cdots<x_{n+1}=b, \tag{1} \]

at whose nodes the values of some function are prescribed

\[ f:\quad f_0,f_1,\ldots,f_{n+1}. \tag{2} \]

Suppose, in addition, that one of the conditions is satisfied

\[ \text{a) }\left. f'_x\right|_{\Gamma}=\varphi,\qquad \text{b) }\left. f''_{xx}\right|_{\Gamma}=\psi \tag{3} \]

or c) \(f\) is periodic with period \(b-a\). It is clear that one can also consider the case where condition a) is prescribed on one boundary and condition b) on the other.

We shall call a cubic spline function the solution of the boundary-value problem with boundary conditions a), b), or c) for the equation

\[ a(x)\,\partial^4 u/\partial x^4=0; \tag{4} \]

\[ a(x) \begin{cases} =0, & x\in\Delta,\\ >0, & x\notin\Delta, \end{cases} \tag{5} \]

and such that

\[ u(x)\in C_2[a,b],\qquad u(x_i)=f(x_i),\qquad i=0,\ldots,n+1. \tag{6} \]

For the numerical solution of the first boundary-value problem obtained, introduce on \([a,b]\) a grid \(\delta\) containing the grid \(\Delta\). Denote the number of interior nodes of the grid \(\delta\) by \(m\), and the finite-dimensional vector spaces corresponding to the grids \(\Delta,\delta\) by \(V_n,V_m\). Replacing (4) by a system of finite-difference equations, we have

\[ \bar a\Lambda u=0, \tag{7} \]

where \(\bar a\) is the grid sampling of the function \(a(x)\), and \(\Lambda\) is a positive difference 5-point analogue of the operator \(d^4/dx^4\).

Taking into account the boundary conditions (2), equations (7) can be rewritten in matrix form as

\[ A(Du-g)=0, \tag{8} \]

where \(D\) is a pentadiagonal matrix in \(V_m\), a finite-dimensional analogue of the operator \(d^4/dx^4\), and \(A\) is a diagonal matrix in \(V_m\), whose zero elements correspond to the nodes of the grid \(\Delta\).

To solve the system of linear algebraic equations (5), consider the iterative process

\[ u^{k+1}=u^k-\tau(Lu^k-Ag), \tag{9} \]

where \(L=AD\) is a nonnegative matrix in \(V_m\), and \(\tau\) is the iteration parameter.

In the known way, the iterative process (9) is reduced to the corresponding iterative process with a positive matrix in the \((m-n)\)-dimensional space \(V_{m-n}\), where \(V_{m-n}\) is the orthogonal complement of \(V_n\) in \(V_m\). The convergence of the process is proved in \((^3)\).

Essential for the iterative process (9) is the choice of the vector of the initial approximation \(u^0=\{u_0^0,\ldots,u_{m+1}^0\}\), carried out in such a way that

\[ u_{i_\alpha}^0=f(x_{i_\alpha}),\qquad \alpha=0,\ldots,n+1 \tag{10} \]

at the points \(i_\alpha\in\Delta\).

2. Let, in the domain

\[ G:\{a<x<b,\ c<y<d\},\qquad \overline{G}=G+\Gamma, \]

a rectangular grid be given,

\[ \Delta:\quad \begin{aligned} a&=x_0<x_1<\ldots<x_{n_1}=b,\\ c&=y_0<y_1<\ldots<y_{n_2}=d, \end{aligned} \]

on which the values \(f_{ij}=f(x_i,y_j)\) are defined.

We define a two-dimensional spline function as the solution of a boundary-value problem in the domain \(G\), under analogous assumptions of specifying on \(\Gamma\) the first or second normal derivatives, or periodicity of the solution with respect to one or both variables, for the equation

\[ a(x,y)\,[\partial^4 u/\partial x^4+\partial^4 u/\partial y^4]=0, \tag{11} \]

where

\[ a(x,y) \begin{cases} =0, & (x,y)\in\Delta,\\ >0, & (x,y)\notin\Delta, \end{cases} \tag{12} \]

under the condition

\[ u(x)\in C_2(\overline{G}),\qquad u(x_i,y_i)=f_{ij}. \tag{13} \]

Introducing, analogously to item 1, a two-dimensional rectangular grid \(\delta\), the spaces \(V_{n_1 n_2}, V_{m_1 m_2}\), and replacing (11) by a system of finite-difference equations, we arrive at a system of difference equations in the space \(V_{m_1 m_2}\)

\[ \Lambda u=\bar a[\Lambda_1u+\Lambda_2u]=0, \tag{14} \]

where \(\bar a\) is the grid sample of the function \(a(x,y)\); \(\Lambda_1,\Lambda_2\) are positive difference approximations of \(\partial^4u/\partial x^4,\partial^4u/\partial y^4\).

For the solution, consider an iterative process in fractional steps of splitting type

\[ u^{n+1/2}=u^n-\tau\bar a\Lambda_1u^{n+1/2}, \]

\[ u^{n+1}=u^{n+1/2}-\tau\bar a\Lambda_2u^{n+1} \tag{15} \]

or else of the stabilizing operator

\[ B\,\frac{u^{n+1}-u^n}{\tau} = B_1B_2\,\frac{u^{n+1}-u^n}{\tau} = \bar a\Lambda u^n = \bar a(\Lambda_1+\Lambda_2)u^n \tag{16} \]

(see \((^2)\)). The proof of convergence of the iterative process (15) is analogous to item 1. The proposed method is suitable for any \(p\)-dimensional domain

\[ G:\{a_i<x_i<b_i;\quad i=1,\ldots,p\},\qquad \overline{G}=G+\Gamma \]

with an irregular grid \(\Delta\). In this case the problem consists only in choosing the function \(a(x_1,\ldots,x_p)\)—nonnegative, sufficiently smooth, vanishing at the nodes of the grid \(\Delta\)—and the vector of the initial approximation such that \(u^0=f\) at the points \(i_\alpha=\{i_{\alpha_1},\ldots,i_{\alpha_p}\}\in\Delta\).

We note that the number \(q\) of subintervals of the grid \(\delta\) falling within one interval of the grid \(\Delta\) does not depend on the fineness of the grid. In the case of a one-dimensional cubic spline one may set \(q=4\), and then the solution of the iterative scheme (9) is the exact solution of problem (4)—(6).

A series of numerical experiments was carried out for the iterative process (15). As the domain \(\overline{G}\), the square \(\{0 \leq x, y \leq 1\}\) was considered. The grid \(\Delta\) was taken to be uniform. As the multiplier \(a(x,y)\) in (12), functions \(a(x,y)=b_1(x)b_2(y)\) were taken, where \(b(s)\) are functions of varying smoothness (from \(C_0\) to \(C_\infty\)) that assume zero values at the points of the one-dimensional grid \(\Delta\).

For comparison of the results with the same data, a bicubic spline function was constructed according to the procedure described in \((^3,{}^4)\).

The computations gave good agreement, although the proposed definition of a spline function is not equivalent to the definition from \((^4)\) already in the two-dimensional case.

Novosibirsk State University

CITED LITERATURE

\(^{1}\) Yu. A. Kuznetsov, DAN, 184, No. 2, 274 (1969).
\(^{2}\) N. N. Yanenko, The Method of Fractional Steps for Solving Multidimensional Problems of Mathematical Physics, “Nauka,” 1967.
\(^{3}\) J. L. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and their Applications, N.Y.—London, 1967.
\(^{4}\) C. De Boor, J. Math. and Phys., 41, 3, 212 (1962).