Full Text
UDC 518:517.91/94
MATHEMATICS
T. S. ZVERKINA
A NEW CLASS OF FINITE-DIFFERENCE OPERATORS
(Presented by Academician Yu. N. Rabotnov on 24 I 1966)
It is known that in the class of finite-difference methods for integrating systems of ordinary differential equations using one or several formulas of the form
\[ L_n y_k \equiv \sum_{i=0}^{n} a_i y_{k-i} - h \sum_{i=0}^{n} b_i y'_{k-i} = 0, \tag{1} \]
there are no stable methods of degree \(p>n+2\) (see (1)). In the present paper a new class of operators \(L_{n\nu}(\alpha)\), depending on the real parameter \(\alpha\), is studied; it contains the ordinary operators \(L_n\) for \(\alpha=0\) and \(\nu=n\). The extension of the class of operators makes it possible to study finite-difference integration methods more deeply and to construct stable methods of higher degree (in particular, methods of degree \(2n\)).
- Consider the family of operators
\[ L_{n\nu}(\alpha)=\sum_{i=0}^{\nu} a_i(\alpha)T^{-i} -h\sum_{i=0}^{n} b_i(\alpha)T^{-i-\alpha}D, \tag{2} \]
where \(D\) is the differentiation operator: \(Dy(x)=y'(x)\); \(T^s\) is the shift operator by \(sh\): \(T^s y(x)=y(x+sh)\); \(a_i(\alpha)\) and \(b_i(\alpha)\) are real functions of the parameter \(\alpha\), defined on the entire number axis with the exception of at most a finite number of points and satisfying the conditions
\[ a_0(\alpha)\equiv 1,\qquad a_\nu(\alpha)\ne 0,\qquad b_0(\alpha)\ne 0,\qquad b_n(\alpha)\ne 0. \]
Fixing the value of the parameter \(\alpha\), we single out from the family (2) a particular operator. By the degree of the operator \(L_{n\nu}(\alpha)\) from the family (2) we shall mean the greatest integer \(p(\alpha)\) such that
\[ L_{n\nu}(\alpha)x^r \equiv 0,\qquad r=0,1,\ldots,p(\alpha). \]
The number \(p=\min_\alpha p(\alpha)\) will be called the degree of the family of operators \(L_{n\nu}(\alpha)\). We shall be interested only in families of degree \(p\ge n+1\). It can be proved that such families are uniquely determined by specifying the functions \(a_0(\alpha),\ldots,a_\nu(\alpha)\) and can be represented in the form
\[ L_{n\nu}(\alpha)=\sum_{i=1}^{\nu} A_i(\alpha)(E-T^{-1})^i -h\sum_{i=0}^{n} B_i(\alpha)(E-T^{-1})^i T^{-\alpha}D, \tag{3} \]
where
\[ A_i(\alpha)=(-1)^i\sum_{j=i}^{n} C_j^i a_j(\alpha),\qquad i=1,\ldots,\nu, \]
\[ B_i(\alpha)=(-1)^i\sum_{j=i}^{n} C_j^i b_j(\alpha),\qquad i=0,1,\ldots,n. \]
The simplest family in the class of operators under consideration is the family of operators \(L_{n1}(\alpha)\) of degree \(n+1\)
\[ L_{n1}(\alpha) \equiv L_n(\alpha)=E-T^{-1}-h\sum_{i=0}^{n}P_i(\alpha)(E-T^{-1})^iT^{-\alpha}D, \]
where
\[ P_i(\alpha)=\frac{1}{i!}\int_{\alpha-1}^{\alpha}(t)_i\,dt,\qquad i=0,1,\ldots, \]
\[ (t)_i=t(t+1)\ldots(t+i-1),\qquad i=1,2,\ldots,\qquad (t)_0\equiv 1. \]
The operators \(L_n(\alpha)\), which we shall call elementary operators, can be used to construct general finite-difference operators \(L_{n\nu}(\alpha)\) of degree \(p\ge n+1\). Namely, the following holds.
Theorem 1. Every family of operators \(L_{n\nu}(\alpha)\) of degree \(p\ge n+1\) is uniquely representable in the form of a linear combination of elementary operators:
\[ L_{n\nu}(\alpha)=\sum_{j=0}^{\nu-1}\widetilde A_j(\alpha)T^{-j}L_n(\alpha-j), \]
where
\[ \widetilde A_i(\alpha)=(-1)^i\sum_{k=i+1}^{\nu}C_{k-1}^{i}A_k(\alpha). \]
This theorem proves to be very useful in the study of general finite-difference operators. Using it, one can prove that if the vector function \(y(x)\) has \(n+q+2\) continuous derivatives, then
\[ L_{n\nu}(\alpha)y(x)=\sum_{i=1}^{q}B_{n+i}T^{-\alpha}\nabla^i y'(x)+h\int_{\alpha-\nu}^{\alpha}\widetilde A_{[-\alpha-t]}(\alpha)R_{n+q}(t)\,dt, \tag{4} \]
where \(R_{n+q}(t)=O(h^{n+q+1})\) is the remainder term of Newton’s interpolation formula; \(\nabla y(x)\equiv y(x)-y(x-h),\ldots\). It follows from (4) that, in order for the family of operators \(L_{n\nu}(\alpha)\) to have degree \(p=n+q\), it is necessary and sufficient that
\[ B_{n+j}(\alpha)\equiv\sum_{i=0}^{\nu-1}A_{i+1}(\alpha)P_{n+j-i}(\alpha)=0,\qquad j=1,\ldots,q-1. \tag{5} \]
For any \(q\le\nu\), the system (5) has a nontrivial solution; for \(q=\nu\) this solution is unique. From what has been set forth it follows:
Theorem 2. There exists a unique family of operators \(L_{n\nu}(\alpha)\) of degree \(n+\nu\). The degree \(n+\nu\) is the greatest possible degree for families of operators of the class under consideration.
An operator \(L_{n\nu}(\alpha)\) from a family of degree \(p\) will be called stable if its corresponding characteristic equation
\[ \sum_{i=0}^{\nu}a_i(\alpha)z^{\nu-i}=0 \]
satisfies the stability condition (see (1)). The set of values \(\alpha\) for which the corresponding operators from the family under consideration are stable will be called the region of stability of the family of operators \(L_{n\nu}(\alpha)\) and denoted by \(\mathfrak A\).
It is easy to verify that, for stability of the operator \(L_{n\nu}(\alpha)\), it is necessary and sufficient that the roots of the equation
\[ \sum_{i=1}^{\nu} A_i(\alpha)(1-\zeta)^{i-1}=0 \tag{6} \]
satisfy the conditions: \(|\zeta_i|\geq 1\), and if \(|\zeta_i|=1\), then \(\zeta_i\) is a simple root not equal to one. If the degree of the family under consideration is \(p=n+\nu\), then the left-hand side of equation (6) is the determinant of the matrix
\[ P(\alpha,\zeta)= \left( \begin{array}{cccc} P_{n+1}(\alpha) & P_n(\alpha) & \cdots & P_{n+2-\nu}(\alpha)\\ \cdots & \cdots & \cdots & \cdots\\ P_{n+\nu-1}(\alpha) & P_{n+\nu-2}(\alpha) & \cdots & P_n(\alpha)\\ 1 & (1-\zeta) & \cdots & (1-\zeta)^{\nu-1} \end{array} \right), \]
and for sufficiently large values of \(\alpha\), equation (6) is equivalent to the equation
\[ \sum_{k=0}^{\nu-1}(-1)^k\left\{C_{\nu-1}^k+O\left(\frac{1}{\alpha}\right)\right\}\zeta^k=0. \]
Investigation of the behavior of the roots of this equation as \(\alpha\to\infty\) leads us to the following theorem.
Theorem 3. For any \(\nu\geq 1\) there exists a number \(n_0=n_0(\nu)\) and a value of the parameter \(\alpha_0=\alpha_0(\nu)\) such that the stability region of the family of operators \(L_{n\nu}(\alpha)\), \(n\geq n_0\), of degree \(n+\nu\) contains the interval \((-\infty,\alpha_0]\).
The value \(n_0=n_0(\nu)\) is comparatively easy to compute for small values of \(\nu\), and it turns out that \(n_0<\nu\). Thus, \(n_0(1)=\cdots=n_0(5)=0\); \(n_0(6)=1\); \(n_0(7)=2\). Hence, at least for \(n\leq 7\), we can assert that in the class of operators under consideration there exists an infinite number of stable operators of degree \(2n\). The interval \((-\infty,\alpha_0]\) does not exhaust the entire stability region. Thus, for example, when \(\nu=1\) the stability region is the entire real axis; the stability region of the families of operators \(L_{n2}(\alpha)\) of degree \(n+2\) (for any \(n\geq 0\)) is the collection of intervals
\[ \mathfrak{A}_{n,-1}=(-\infty,\alpha_{n,n-1}),\qquad \mathfrak{A}_j=[\alpha_j^*,\alpha_{n,j}),\qquad j=0,1,\ldots,n-2, \]
where \(\alpha_j^*\) are the roots of the equation \(2P_{n+1}(\alpha)-P_n(\alpha)=0\), and \(\alpha_{nj}\) are the roots of the equation \(P_n(\alpha)=0\), with \(\alpha_{n0}>0\).
- Consider predictor–corrector methods using, as corrector formulas,
\[ L_{n\nu}(\alpha)y_k\equiv \sum_{i=0}^{\nu} a_i(\alpha)y_{k-i} - h\sum_{i=0}^{n} b_i(\alpha)y'_{k-\alpha-i}=0, \tag{7} \]
where \(L_{n\nu}(\alpha)\) is a stable operator from the family of degree \(n+\nu\). As predictors, take, for example, the formulas
\[ L_{n\nu}^*(\alpha)y_k\equiv y_{k-\alpha} +\sum_{i=1}^{\nu} a_i^*(\alpha)y_{k-i} - h\sum_{i=0}^{n} b_i^*(\alpha)y'_{k-i-\alpha-1}=0, \tag{8} \]
where \(L_{n\nu}^*(\alpha)\) is an operator of degree \(n+\nu\). The error of the method (7)—(8) in integrating the system of equations
\[ y'=f(x,y),\qquad y(x_0)=y_0, \]
on any finite interval has order \(O(h^{n+\nu})\) (assuming sufficient smoothness of the solution of the integrated system). For it, by the usual method (see (2)) one can obtain an asymptotic expansion of the form
\[ d_k=\varkappa(\alpha)h^{n+\nu}\int_0^{x_k}\Omega(x,\xi)y^{(n+\nu+1)}(\xi)\,d\xi+O(h^{n+\nu+1}), \]
where \(\Omega(x,\xi)\) is a bounded matrix independent of \(\alpha\) and \(h\). Minimizing the coefficient
\[ \varkappa(\alpha)=B_{n+\nu}(\alpha)\left[\sum_{i=0}^{n}b_i(\alpha)\right]^{-1} \]
over the domain \(\mathfrak{A}\) (which presents no difficulty, since \(B_{n+\nu}(\alpha)\) and \(b_i(\alpha)\) are polynomials), we single out from the collection of methods (7)—(8) the optimal methods. It may turn out that in the domain \(\mathfrak{A}\) there are zeros of the polynomial \(B_{n+\nu}(\alpha)\). Then the optimal methods will have degree \(n+\nu+1\). Thus, for example, if \(\nu=1\), then \(\varkappa(\alpha)=P_{n+1}(\alpha)\), and from the easily verified relations
\[ \operatorname{sign}P_{n+1}(j)=(-1)^j,\qquad j=1,0,-1,\ldots,-n+1, \]
\[ P'_{n+1}(\alpha)=(\alpha)_n(n!)^{-1}, \]
it follows that for any \(n\ge 0\) there exist \(n+1\) optimal methods of degree \(n+2\), corresponding to the zeros of the polynomial \(P_{n+1}(\alpha)\). In the class of methods under consideration, the ordinary Adams method, corresponding to \(\alpha=0\) for \(\nu=1\), cannot be optimal, since the coefficient \(\varkappa(\alpha)=P_{n+1}(\alpha)\) has a nonzero local extremum at \(\alpha=0\).
Moscow State University
named after M. V. Lomonosov
Received
6 I 1966
REFERENCES
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- A. N. Tikhonov, A. D. Gorbunov, Zh. vychisl. matem. i matem. fiz., 2, 4, 537 (1962).