G. I. Gershengorn

On the Construction of Difference Analogues of Certain Boundary-Value Problems

(Presented by Academician A. A. Dorodnitsyn, 24 XI 1966)

A differential equation is given

\[ \sum_{\alpha=1}^{p} f_\alpha \frac{\partial^{k_\alpha}u}{\partial x_\alpha^{k_\alpha}} = f_0,\qquad a_\alpha \leqslant x_\alpha \leqslant b_\alpha, \tag{1} \]

and boundary conditions

\[ \partial^\lambda u/\partial x_\alpha^\lambda\big|_{x_\alpha=a_\alpha} = \varphi_{a_\alpha}^{\lambda}, \qquad \lambda=0,\ldots,\nu_\alpha-1; \tag{2} \]

\[ \partial^\gamma u/\partial x_\alpha^\gamma\big|_{x_\alpha=b_\alpha} = \varphi_{b_\alpha}^{\lambda}, \qquad \gamma=0,\ldots,\mu_\alpha-1; \tag{3} \]

\[ f_{\alpha,0}=f_{\alpha,0}(u;\,x_1,\ldots,x_p), \]

\[ \varphi_{a_\alpha,b_\alpha}^{\lambda,\gamma} = \varphi_{a_\alpha,b_\alpha}^{\lambda,\gamma} \left( u;\ x_1,\ldots,x_p;\ \partial u/\partial x_\alpha,\ldots, \partial^{\lambda-1,\gamma-1}u/\partial x_\alpha^{\lambda-1,\gamma-1} \right)\big|_{x_\alpha=a_\alpha,b_\alpha}, \]

\[ \nu_\alpha+\mu_\alpha=k_\alpha>1. \]

The solution \(u(x_1,\ldots,x_p)\) is sought by the finite-difference method at \(N\) interior nodes \(x_{\alpha j}\) of a rectangular grid; \(\alpha=1,\ldots,p;\ j=1,\ldots,r_\alpha;\)

\[ N=\prod_{\alpha=1}^{p} r_\alpha. \]

Replacing
\(\partial^{k_\alpha,\lambda,\gamma}/\partial x_\alpha^{k_\alpha,\lambda,\gamma} \big|_{x_\alpha=x_{\alpha j},a_\alpha,b_\alpha}\)
(for \(j=1,\ldots,r_\alpha-k_\alpha+2\)) by difference operators
\(\delta_{\alpha j,a_\alpha,b_\alpha}^{(k_\alpha,\lambda,\gamma)}\), we obtain a system of \(N\) difference equations.

The purpose of the note is to construct operators
\(\delta_{\alpha j,a_\alpha,b_\alpha}^{(k_\alpha,\lambda,\gamma)}\)
whose prescribed accuracy is attained with the smallest possible \(N\). By the accuracy of \((^1)\) is meant an integer \(m>0\) such that
\[ \varepsilon= \delta_{\alpha j,a_\alpha,b_\alpha}^{(k_\alpha,\lambda,\gamma)}x_\alpha^i - \partial^{k_\alpha,\lambda,\gamma}x_\alpha^i/ \partial x_\alpha^{k_\alpha,\lambda,\gamma} \big|_{x_\alpha=x_{\alpha j},a_\alpha,b_\alpha} =0 \]
for \(i=0,\ldots,m\), and \(\varepsilon\ne0\) for \(i=m+1\).

Let us consider the approximation error of any one of the derivatives
\(\partial^k u/\partial x_\alpha^k\) entering (1) by an \((r_\alpha+2)\)-point difference expression. For simplicity of notation we omit the indices \(\alpha\) and take \(a=-1,\ b=1,\ r=n+k-2\). If \(u(x)\in C^{(n+2k-1)}[-1,1]\), then

\[ \varepsilon^{(k)}(x_j) = \delta_j^{(k)}u-\frac{\partial^k u(x_j)}{\partial x^k} = \]

\[ = \sum_{i=0}^{k-1} \frac{k!\,u^{(n+k+i)}(\xi_i)} {(k-i)!\,(n+k+i)!}\, \omega_{n+k}^{(k-i)}(x_j), \qquad -1\leqslant \xi_i\leqslant 1, \tag{4} \]

where

\[ \omega_{n+k}(x) = (x^2-1)\prod_{j=1}^{\,n+k-2}(x-x_j). \tag{5} \]

It is not difficult to show that (4), obtained from (2) for Lagrange polynomials, remains valid also in the case of Hermite polynomials. Therefore it is permissible to take

\[ x_j=-1 \quad \text{for } j=n+1,\ldots,n+\nu-1, \tag{6} \]

\[ x_j=1 \quad \text{for } j=n+\nu,\ldots,n+k-2. \tag{7} \]

With this choice of nodes, the need to approximate the boundary conditions (2), (3) disappears. Moreover, the number \((n+k-2)\) of internal nodes is reduced by \((k-2)\) without loss in the accuracy of the operator \(\delta_j^{(k)}\). If any one of the \(n\) remaining internal nodes \(x_j\) is taken equal to \(1\), \(-1\), or \(x_{j+1}\), then an additional unknown appears in \(\delta_j^{(k)}\) (respectively \(u^{(\mu)}(1)\), \(u^{(\nu)}(-1)\), or \(u'(x_j)\)), so that the system of difference equations approximating (1)—(3) becomes underdetermined. In connection with this it is necessary to impose the condition

\[ -1<x_1<\ldots<x_j<x_{j+1}<\ldots<x_n<1. \tag{8} \]

It follows from (4) that, in order to increase the accuracy of the operator \(\delta_j^{(k)}\) by 1 for fixed \(n\), it is necessary to “get rid” of the term containing \(u^{(n+k)}(\xi_i)\), i.e., to choose such a distribution of nodes for which

\[ \omega_{n+k}^{(k)}(x_j)=0 \quad \text{for } j=1,\ldots,n. \tag{9} \]

Taking into account (5)—(8) and applying Rolle’s theorem successively to the polynomials \(\omega_{n+k}(x)\), \(\omega'_{n+k}(x),\ldots,\omega_{n+k}^{(k)}(x)\), one can prove that if

\[ \omega_{n+k}^{(k)}(x_j)=0, \quad \text{then } \omega_{n+k}^{(k-1)}(x_j)\ne 0. \tag{10} \]

According to (4), (9), and (10), an increase in the accuracy of the operator \(\delta_j^{(k)}\) for fixed \(n\) is possible by no more than 1, and only for \(k\ge 2\).

The choice of nodes in accordance with (6)—(9) is optimal in the sense that the operator \(\delta_j^{(k)}\) exactly differentiates polynomials

\[ \sum_{i=0}^{m} c_i x^i \]

with the smallest possible number \(n=(m-k)\) of internal nodes on the \(x\)-axis. Suppose that the optimal distribution of nodes is adopted for each of the directions \(x_\alpha\), with \(n_\alpha+k_\alpha=m=\mathrm{const}\). Then accuracy \(m\) of the operators \(\delta_{\alpha j}^{(k_\alpha)}\) will be attained with the smallest possible number \(N_m\) of unknowns in the difference analogue of problem (1)—(3),

\[ N_m=\prod_{\alpha=1}^{p} (m-k_\alpha). \]

Direct use of (9) for computing the optimal nodes is difficult. From (5)—(7) and (9) it follows that \(\omega_{n+k}(x)\) satisfies the differential equation

\[ (x+1)^\nu (x-1)^\mu \omega_{n+k}^{(k)}(x) = \frac{(n+k)!}{n!}\,\omega_{n+k}(x). \tag{11} \]

At the Computing Center of Irkutsk University, for some \(k,\nu,\mu,n\), solutions of (11) were obtained by the method of undetermined coefficients. Investigation of the roots of the solution was carried out only for the special case \(k=2\nu=2\mu\). According to the relations 8.934.1, 8.935.1, and 8.960 given in [3], for \(k=2\nu=2\mu\) the solution of (11) is the polynomial

\[ \omega_{n+2\nu}(x) = \frac{(n+2\nu)!}{(2n+2\nu)!} \frac{d^n (x^2-1)^{n+\nu}}{dx^n} = q_1(n,\nu)(x^2-1)^\nu C_n^{\nu+1/2}(x) = \]

\[ = q_2(n,\nu)(x^2-1)^\nu P_n^{(\nu,\nu)}(x), \]

where \(C_n^{\nu+1/2}(x)\) and \(P_n^{(\nu,\nu)}(x)\) are the Gegenbauer and Jacobi polynomials, respectively; \(q_{1,2}(n,\nu)\) are the reciprocals of the coefficients of \(x^n\) in the polynomials \(C_n^{\nu+1/2}(x)\), \(P_n^{(\nu,\nu)}(x)\).

Applying Rolle’s theorem successively to \(d^i(x^2-1)^{n+\nu}/dx^i\), \(i=0,\ldots,n\), it is not difficult to prove that \(\omega_{n+2\nu}(x)\) has \(n\) distinct real roots inside \([-1,1]\).

After solving equation (11) and computing the \(n\) optimal interior nodes \(x_i\), one can, according to \((^2)\), construct difference expressions approximating the derivatives \(\partial^s u(x_j)/\partial x^s\):

\[ \delta_j^{(s)}u=\sum_{i=1}^{n} A_{ji}u(x_i)+\sum_{\lambda=0}^{\nu-1} B_{j\lambda}u^{(\lambda)}(-1)+\sum_{\gamma=0}^{\mu-1} C_{j\gamma}u^{(\gamma)}(1), \]

where

\[ A_{ji}=\left\{\frac{\omega_{n+k}(x)}{(x-x_i)\omega_{n+k}'(x_i)}\right\}^{(s)}_{x=x_j}, \]

\[ B_{j\lambda} = \left\{ \frac{1}{\lambda!}\frac{\omega_{n+k}(x)}{(x+1)^{\nu-\lambda}} \sum_{i=0}^{\nu-\lambda-1}\frac{1}{i!} \left[\frac{(x+1)^\nu}{\omega_{n+k}(x)}\right]^{(i)}_{x=-1} (x+1)^i \right\}^{(s)}_{x=x_j}, \]

\[ C_{j\lambda} = \left\{ \frac{1}{\gamma!}\frac{\omega_{n+k}(x)}{(x-1)^{\mu-\gamma}} \sum_{i=0}^{\mu-\gamma-1}\frac{1}{i!} \left[\frac{(x-1)^\mu}{\omega_{n+k}(x)}\right]^{(i)}_{x=1} (x-1)^i \right\}^{(s)}_{x=x_j}. \]

The convergence and stability conditions for difference schemes on optimal grids have not been investigated. An estimate of the practical convergence and accuracy of the solution can be carried out by some of the methods set forth in \((^4)\).

In the examples given below, the maximum absolute errors \(\Delta\) at the grid nodes are rounded to powers of 10. The boundary nodes are \(\pm1\). The interior optimal nodes \(x_j\) for \(n=2\), \(k=3\): \((1\pm1/\sqrt{3})/3\); for \(n=2\), \(k=4\): \(\pm1/\sqrt{7}\). The roots of \(P_n^{(1,1)}(x)\) (\(x_j\) for \(k=2\)) are given in \((^5)\).

1. \[ \sum_{\alpha=1}^{p}\frac{\partial^2 u}{\partial x_\alpha^2} = p\left(\frac{\pi}{2}\right)^2 \sum_{\alpha=1}^{p}\cos\frac{\pi}{2}x_\alpha; \qquad u\big|_{x_\alpha=\pm1}=0; \qquad p=1,\ n=2,4,6; \]

\[ p=2,\ n=2,4;\qquad p=3,\ n=2;\qquad \Delta=10^{-n}. \]

1. \[ \frac{\partial^k u}{\partial t^k} = \frac{\partial^2 u}{\partial x^2}; \qquad u\big|_{x=\pm1}=0; \qquad k=1,2; \qquad u\big|_{t=0}=\cos\frac{\pi}{2}x; \]

for \(k=2\)

\[ \frac{\partial u}{\partial t}\bigg|_{t=0}=0; \qquad n=2,4,6; \]

only \(\partial^2 u/\partial x^2\) is approximated; the resulting equations are solved analytically; the errors in computing the period (for \(k=2\)) and the damping decrement (for \(k=1\)) are \(10^{-n}\).

1. \[ \frac{\partial^4 u}{\partial x_1^4} + 2\frac{\partial^4 u}{\partial x_1^2\partial x_2^2} + \frac{\partial^4 u}{\partial x_2^4} = \frac{\pi^4}{4}\cos\frac{\pi}{2}x_1\cos\frac{\pi}{2}x_2. \]

A grid is used for \(k=4,\ n=2\);

\[ u\big|_{x_{1,2}=\pm1}=0; \]

\[ \frac{\partial u}{\partial x_{1,2}}\bigg|_{x_{1,2}=\pm1} = \mp\frac{\pi}{2}\cos\frac{\pi}{2}x_{2,1}; \qquad \Delta=10^{-3}. \]

1. \[ \frac{\partial^2 v}{\partial x_1^2} + \frac{\partial^2 v}{\partial x_2^2} = \frac{\pi^4}{4}\cos\frac{\pi}{2}x_1\cos\frac{\pi}{2}x_2; \qquad \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} = v; \]

\[ u\big|_{x_{1,2}=\pm1} = v\big|_{x_{1,2}=\pm1} = 0; \qquad n=2,4; \qquad \Delta=10^{-n}. \]

1. \[ u^{(k)}=u; \qquad n=2; \qquad u(\pm1)=e^{\pm1}; \]

for \(k=2\), \(\Delta=10^{-3}\); for \(k=3\),

\[ u'(-1)=e^{-1}, \qquad \Delta=10^{-3}; \]

for \(k=4\),

\[ u'(\pm1)=e^{\pm1}, \qquad \Delta=10^{-4}. \]

Irkutsk State University
named after A. A. Zhdanov

Received
19 X 1966

REFERENCES

\(^1\) A. S. Kronrod, Nodes and Weights of Quadrature Formulas, Moscow, 1964.
\(^2\) I. S. Berezin, N. P. Zhidkov, Methods of Computation, 1, Moscow, 1959.
\(^3\) I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Moscow, 1962.
\(^4\) O. M. Belotserkovskii, P. I. Chushkin, Journal of Computational Mathematics and Mathematical Physics, 2, 5, 731 (1962).
\(^5\) I. K. Daugavet, E. F. Domanovskaya, Methods of Computation, issue 1, Leningrad, 1963.