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MATHEMATICS
Corresponding Member of the Academy of Sciences of the USSR A. N. Tikhonov and A. A. Samarskii
ON THE COEFFICIENT-STABILITY OF DIFFERENCE SCHEMES
The question of the stability of the solution of difference boundary-value problems with respect to the coefficients of difference schemes (of coefficient-stability) is considered. It is shown that a necessary and sufficient condition for the coefficient-stability of a canonical scheme is its conservativeness.
1. Consider on the interval \(0 \leq x \leq 1\) the class of boundary-value problems
\[ L^{(p,q,f)}u \equiv \frac{d}{dx}\left[\frac{1}{p(x)}\frac{du}{dx}\right] - q(x)u + f(x) = 0,\quad 0 < x < 1, \]
\[ u(0)=\mu_1,\qquad u(1)=\mu_2. \tag{1} \]
The coefficients of the equation belong to the class \(Q_0\) of piecewise-continuous functions and satisfy the conditions:
\[ 0<K_1 \leq p(x)\leq K_2,\qquad 0\leq q(x)\leq K_2,\qquad |f(x)|\leq K_2, \tag{2} \]
where \(K_1\) and \(K_2\) are positive constants.
Let \(S_N=\{x_0=0,\ x_1=h,\ldots,\ x_i=ih,\ldots,\ x_N=Nh=1\}\) be a uniform difference grid with step \(h=\frac{1}{N}\), and let \(L_h^{(p,q,f)}y_i\) be a homogeneous three-point difference scheme corresponding to the operator
\[ L_h^{(p,q,f)}y_i = \frac{1}{h^2}\left[(y_{i+1}-y_i)/B_i^h-(y_i-y_{i-1})/A_i^h\right] -D_i^h y_i+F_i^h; \tag{3} \]
\[ A_i^h=A^h[\bar p_i(s)],\qquad B_i^h=B^h[\bar p_i(s)],\qquad -1<s<1,\qquad \bar p_i(s)=p(x_i+sh); \]
\[ D_i^h=D^h[q(x_i+sh)],\qquad F_i^h=F^h[f(x_i+sh)],\qquad -0.5<s<0.5. \]
The functionals \(A^h\), \(B^h\), \(D^h\), and \(F^h\) satisfy the conditions \(A_1, A_2, A_3\) of work \((^1)\), i.e., we consider the same initial class of difference schemes as in work \((^1)\). It is assumed here that \(D^h\) and \(F^h\) are linear functionals.
2. If \(B_i^h=A_{i+1}^h\), then the difference operator \(L_h\) is called conservative. A conservative operator can be written in the self-adjoint form
\[ L_h y_i \equiv \frac{1}{h^2}\Delta(\nabla y_i/A_i^h)-D_i^h y_i+F_i^h,\quad \text{where }\Delta y_i=y_{i+1}-y_i,\ \nabla y_i=y_i-y_{i-1}. \tag{3'} \]
Let us note that the difference scheme (3) can be made conservative by multiplication by the multiplier
\[ \mu_i=\prod_{s=1}^{i-1}(A_{s+1}^h/B_s^h). \tag{4} \]
As a result, we obtain a conservative, generally speaking, nonhomogeneous scheme.
- The difference Green’s function \(G_{i,k}\) is defined by the conditions
\[ L_h^{(p,q)}G_{i,k}=-\frac{\delta_{i,k}}{h},\qquad G_{0k}=G_{Nk}=0,\qquad \delta_{ik}= \begin{cases} 1, & i=k,\\ 0, & i\ne k. \end{cases} \tag{5} \]
The solution of the boundary-value problem
\[ L_h^{(p,q)}z_i=-\varphi_i,\qquad 0<i<N,\qquad z_0=0,\qquad z_N=0 \tag{6} \]
is given by the formula
\[ z_i=\sum_{k=1}^{N-1}G_{ik}\varphi_k h. \tag{7} \]
The Green’s function \(G_{ik}\) satisfies the following “symmetry” condition:
\(\mu_iG_{ik}=\mu_kG_{ki}\), where \(\mu_i\) is given by formula (4). For the conservative operator \(L_h^{(p,q)}\), \(B_s^h=A_{s+1}^h\), \(\mu_i=1\), and we obtain the symmetry condition \(G_{ik}=G_{ki}\).
Lemma 1. If the coefficients \(p(x), q(x)\) of the class \(Q_0\) satisfy conditions (2), and \(L_h^{(p,q)}\) is the initial difference scheme of the form (3), then the difference Green’s function \(G_{ik}\), defined by conditions (5), and its first difference quotients
\[ (G_{i,k+1}-G_{i,k})/h \qquad (0\le i\le N,\; 0\le k\le N-1); \]
\[ (G_{i+1,k}-G_{i,k})/h \qquad (0\le i\le N-1,\; 0\le k\le N) \]
are bounded in absolute value by a constant depending only on \(K_1, K_2\).
- In solving difference boundary-value problems it may turn out that, for one reason or another, the coefficients of the difference equations are determined inaccurately. However, it is desirable that under a small distortion of the coefficients the solution of the problem should change little.
Let \(y_i\) and \(\widetilde y_i\) be solutions of the difference boundary-value problems
\[ L_h^{(p,q,f)}y_i=0,\qquad 0<i<N,\qquad y_0=\mu_1,\qquad y_N=\mu_2, \tag{8} \]
\[ \widetilde L_h^{(p,q,f)}\widetilde y_i=0,\qquad \widetilde y_0=\mu_1,\qquad \widetilde y_N=\mu_2, \]
\[ \widetilde L_h^{(p,q,f)}\widetilde y_i =h^{-2}\bigl(\Delta y_i\,|\widetilde B_i^h-\nabla y_i\,|\widetilde A_i^h\bigr) -\widetilde D_i^h\widetilde y_i+\widetilde F_i^h. \tag{9} \]
Here the coefficients of the equation are distorted either through distortion of the coefficients of the differential equation, or through inaccuracy in the computation of the functionals \(A^h, B^h, D^h\), and \(F^h\), or, finally, as a result of both of the indicated causes.
We shall say that the difference scheme (3) satisfies the principle of coefficient-stability if, from the conditions
\[ \sum_{i=1}^{N-1}|\widetilde A_i^h-A_i^h|h=\rho(h),\qquad \sum_{i=1}^{N-1}|\widetilde B_i^h-B_i^h|h=\rho(h), \]
\[ \sum_{i=1}^{N-1}|\widetilde D_i^h-D_i^h|h=\rho(h),\qquad \sum_{i=1}^{N-1}|\widetilde F_i^h-F_i^h|h=\rho(h), \tag{10} \]
where \(\rho(h)\to0\) as \(h\to0\), there follows convergence of the solution \(\widetilde y_i\) of the difference boundary-value problem (9) to the solution \(u(x)\) of problem (1), i.e.,
\[ |\widetilde y_i-u(x_i)|\le \rho_0(h)\to0 \qquad \text{as } h\to0. \tag{11} \]
Hence, in particular, it follows that for a coefficient-stable scheme
\[ |y_i-u(x_i)|\leqslant \rho_1(h),\qquad |y_i-\widetilde y_i|\leqslant \rho_2(h), \]
where \(\rho_1(h), \rho_2(h)\to 0\) as \(h\to 0\).
If in conditions (10) and (11) one replaces them by
\[ \rho(h)=O(h^n),\qquad \rho_0(h)=O(h^n), \tag{12} \]
then we obtain the principle of coefficient-stability of \(n\)-th order.
5. We formulate a necessary condition for coefficient-stability.
Let \(p(x)\) have a discontinuity at the point \(\xi=x_n+\theta h,\; 0\leqslant \theta\leqslant 1,\; x_n=nh\), so that \(p_\ell=p(\xi-0)\ne p_r=p(\xi+0)\). Introduce the function \(\widetilde p(x,h)\), coinciding with \(p(x)\) everywhere except on the intervals \((x_n,x_{n+1})\) and \((x_{n+1},x_{n+2})\). Then for coefficient-stability of a scheme \(\dot L_h^{(p,q,f)}\) of the form (3), it is necessary that the condition
\[ \widetilde B_n^h\widetilde B_{n+1}^h\widetilde B_{n+2}^h/p_r - \widetilde A_n^h\widetilde A_{n+1}^h\widetilde A_{n+2}^h/p_\ell = \rho(h)\to 0 \quad \text{as } h\to 0 \tag{13} \]
be satisfied.
It is not difficult to notice that the necessary condition for convergence in the class of discontinuous coefficients, obtained earlier in works \((^3,^4)\), is a consequence of the necessary condition of coefficient-stability (13) (for \(\widetilde p\equiv p\)).
6. Lemma 2. Every conservative scheme \(\dot L_h^{(p,q,f)}\) from the original family of schemes satisfies the necessary condition of coefficient-stability (13).
Lemma 3. Let \(y_i,\widetilde y_i\) be solutions of the boundary-value problems
\[ L_h y_i=0,\qquad y_0=\mu_1,\qquad y_N=\mu_2;\qquad \widetilde L_h\widetilde y_i=0,\qquad \widetilde y_0=\mu_1,\qquad \widetilde y_N=\mu_2, \]
where \(L_h,\widetilde L_h\) are conservative difference operators of the form \((3')\), whose coefficients satisfy the conditions
\[ 0<K_1\leqslant A_i^h\leqslant K_2,\qquad 0\leqslant D_i^h\leqslant K_2,\qquad |F_i^h|\leqslant K_2. \tag{2'} \]
Then the inequality holds
\[ |y_i-\widetilde y_i|_{(0\leqslant i\leqslant N)} \leqslant C\left\{ \sum_{k=1}^{N}|\widetilde A_k^h-A_k^h|h + \sum_{k=1}^{N-1}|\widetilde D_k^h-D_k^h|h + \sum_{k=1}^{N-1}|\widetilde F_k^h-F_k^h|h \right\}, \tag{14} \]
where \(C\) is a constant depending only on \(K_1\) and \(K_2\).
An analogous lemma holds for problem (1).
Choosing as \(\widetilde L_h\) the exact scheme \(\widetilde L_h^{(p,q,f)}\) (see (2)), and as \(L_h\) a conservative scheme \(\dot L_h^{(p,q,f)}\) from the original family of schemes and relying on Lemma 3, it is not difficult to prove the theorem (cf. (1)).
Theorem 1. If a conservative scheme \(\dot L_h^{(p,q,f)}\) from the original family has, in some class \(C_{m_k,m_q,m_f}\), \(n\)-th integral order of accuracy, then it has this same \(n\)-th order of accuracy for coefficients from the class \(C_{n-1}^{(1)}\), i.e., for \(p\in C_{n-1}^{(1)},\; q\in C_{n-1}^{(1)},\; f\in C_{n-1}^{(1)}\).
7. We now consider the canonical scheme (see (1))
\[ L_h^{(p,q,f)}y_i = h^{-2}\,[\Delta y_i/B_i-\nabla y_i/A_i] - D_i y_i + F_i, \tag{15} \]
whose functionals do not depend on \(h_3\), and require that it satisfy the necessary condition (13).
Theorem 2. If the canonical difference scheme (15) from the original family of schemes satisfies the necessary condition of coefficient-stability (13), then it is conservative, i.e. \(B_i=A_{i+1}\) or \(B[\psi(s)]=A[\psi(1+s)]\).
\(*\) \(C_m^\gamma\) \((0\leqslant \gamma\leqslant 1)\) is the class of functions having on the segment \([0,1]\) a continuous derivative of order \(m\), satisfying a Hölder condition of order \(\gamma\).
Relying on Theorem 2 and Lemmas 2 and 3, one can verify that:
Theorem 3. Every homogeneous conservative scheme from the original family of schemes satisfies the principle of coefficient-stability.
As a result we arrive at the following basic theorem.
Theorem 4. A necessary and sufficient condition for the coefficient-stability of the canonical scheme \(L_h^{(p,q,f)}\) is its conservativeness.
Theorem 5. Every conservative scheme \(L_h^{(p,q,f)}\) has first integral order of accuracy in the class \(Q_1^0\).
§ 8. Let us now require that the difference scheme \(L_h^{(p,q,f)}\) satisfy the necessary conditions of coefficient-stability of rank 2.
Theorem 6. There exists a unique canonical scheme (“the best conservative scheme”) having second integral order of accuracy in \(Q_1^{(1)}\) and satisfying the principle of coefficient-stability of rank 2; this scheme \(L_h^{(p,q,f)}\) is conservative and is determined by means of the functionals
\[ A[\psi]=\int_{-1}^{0}\psi(s)\,ds,\qquad D[\psi]=F[\psi]=\int_{-0.5}^{0.5}\psi(s)\,ds . \tag{16} \]
We note that in the proof of this theorem, in particular, Lemma 1 is used.
§ 9. Replacing the integral defining \(A[\psi]\) in formula (16) by a scheme based on some quadrature formula, we obtain, instead of the best canonical scheme \(L_h^{(p)}\), a noncanonical scheme
\[ \widetilde{L}_h^{(p)}y_i=\frac{1}{h^2}\Delta(\nabla y_i/\widetilde{A}_i^h), \qquad \text{where }\ \widetilde{A}_i^h=\widetilde{A}^{h_1}[p(x_i+sh)], \]
\[ \widetilde{A}^{h_1}[\psi(s)]=\sum_{j=1}^{J}a_j\psi(s_j),\qquad s_j=-1+jh_1,\qquad h_1=1/J. \]
Theorem 7. In order that the noncanonical scheme \(\widetilde{L}_h^{(p)}\) defined above have second integral order of accuracy in the class \(Q_1^{(1)}\), it is necessary and sufficient that \(h_1/h=O(1)\) as \(h\to0\) \((N\to\infty)\).
An analogous theorem also holds for the scheme \(\widetilde{L}_h^{(p,q,f)}y_i=\widetilde{L}_h^{(p)}y_i-\widetilde{D}_i^h y_i+\widetilde{F}_i^h\), whose functionals \(\widetilde{D}^{h_1}\) and \(\widetilde{F}^{h_1}\) are computed by analogy with the functional \(\widetilde{A}^{h_1}\).
§ 10. In work [1] we consider an asymptotic expansion for the solution of a difference boundary-value problem in the case of discontinuous coefficients. If \(p(x)\), \(q(x)\), and \(f(x)\) are functions of the class \(Q_2^{(0)}\), and \(L_h^{(p,q,f)}\) is the best canonical scheme, then the solution of problem (8) can be represented in the form
\[ y_i=u(x_i)+h^2Y(x_i,h)+O(h^4), \]
where \(Y(x,h)=O(1)\) and is a function having no limit as \(h\to0\). Hence it follows that, in the case of discontinuous coefficients, the solution of the difference boundary-value problem (8) does not have asymptotics of second order as \(h\to0\).
Received
31 XII 1959
REFERENCES
- A. N. Tikhonov, A. A. Samarskii, DAN, 131, No. 4 (1960).
- A. N. Tikhonov, A. A. Samarskii, DAN, 131, No. 3 (1960).
- A. N. Tikhonov, A. A. Samarskii, DAN, 108, No. 3 (1956).
- A. N. Tikhonov, A. A. Samarskii, DAN, 122, No. 4 (1959).
* \(Q_m^\gamma\) \((0\le \gamma\le 1)\) is the class of functions having on \([0,1]\) \(m\) piecewise-continuous derivatives, with the \(m\)-th derivative satisfying, in the intervals of its discontinuity, a Hölder condition of order \(\gamma\).