UDC 518.517.944/947

MATHEMATICS

Corresponding Member of the USSR Academy of Sciences A. A. SAMARSKII, A. V. GULIN

ON THE STABILITY OF DIFFERENCE SCHEMES WITH RESPECT TO RIGHT-HAND SIDES

In papers \((^{1,2})\), necessary and sufficient conditions were obtained for stability with respect to the initial data of two-level difference schemes, defined as operator-difference equations on a family of Hilbert spaces \(\{H_h\}\).

In the present paper estimates are constructed for certain energy norms of solutions of two-level difference schemes in terms of the initial data \(y_0 \in H_h\) and the right-hand side \(\varphi(t) \in H_h\), which coincide for \(\varphi=0\) with the estimates of \((^{1,2})\).

To study stability with respect to the right-hand side, the method of separating stationary inhomogeneities is applied (see \((^{3,4})\)), which makes it possible to remove certain restrictions (such, for example, as smallness of the time step \(\tau\)) of paper \((^5)\), where stability was investigated by the method of energy inequalities. In addition, it proved possible to consider the case of non-self-adjoint operators and to establish a number of new estimates. We note that all the estimates obtained hold under the same conditions on the operators of the difference scheme that ensure stability with respect to the initial data.

1. Let \(\{H_h\}\) be a family of Hilbert spaces (real or complex) depending on a parameter \(h\) (\(H_h\) is an analogue of the space of grid functions), \(\omega_\tau=\{t_n=n\tau,\ n=0,1,\ldots\}\) a grid with step \(\tau>0\), and \(A,B: H_h \to H_h\) linear (additive and homogeneous) operators acting in \(H_h\) and depending on \(h\), \(\tau\), and \(t=t_n\).

A two-level difference scheme is defined as the operator equation (see \((^5)\))

\[ B(t_n)y_t + A(t_n)y = \varphi(t_n),\quad n=0,1,\ldots;\qquad y_0 \in H_h, \tag{1} \]

where \(y=y_n=y_{h,\tau}(t_n)\in H_h\), \(y_t=(y_{n+1}-y_n)/\tau\), \(\varphi(t)\in H_h\).

An operator \(A:H\to H\) is called positive \((A>0)\) if \((Ax,x)>0\) for all \(0\ne x\in H\), and nonnegative \((A\ge 0)\) if \((Ax,x)\ge 0\) for all \(x\in H\). If \(A\) is a self-adjoint and positive operator, then one may introduce the energy norm \(\|y\|_A=\sqrt{(Ay,y)}\).

The following lemmas will be needed for the proof of the stability theorems.

Lemma 1 (see \((^{1,2})\)). The inequality

\[ \|S\|\le \rho,\qquad S=E-\tau C, \tag{2} \]

follows from the condition

\[ B_0 \ge \frac{\tau}{1+\rho}A,\qquad B_0=\operatorname{Re} B=\frac{B+B^*}{2}, \tag{3} \]

if \(\rho\ge 1\), \(A^*=A>0\), \(C=A^{1/2}B^{-1}A^{1/2}\), and from the condition

\[ (A^{-1})_0 \ge \frac{\tau}{1+\rho}B^{-1},\qquad (A^{-1})_0=\operatorname{Re} A^{-1}, \tag{4} \]

if \(\rho\ge 1\), \(B^*=B>0\), \(C=B^{-1/2}AB^{-1/2}\).

Remark. In the case of a real space \(H_h\), inequalities (3) and (4) are equivalent respectively to the conditions

\[ B \ge \frac{\tau}{1+\rho}A;\qquad A^{-1}\ge \frac{\tau}{1+\rho}B^{-1}. \]

We shall say (see (5)) that the self-adjoint operator \(\mathcal D=\mathcal D(t)\) satisfies the Lipschitz condition with respect to \(t\) if there exists a constant \(c_1\ge 0\), independent of \(\tau,h,n\), such that for all \(t\) and \(\tau\) the operator inequalities
\[ (1-c_1\tau)\mathcal D(t-\tau)\le \mathcal D(t)\le (1+c_1\tau)\mathcal D(t-\tau). \tag{5} \]

Lemma 2. If \(\mathcal D(t)\) is a self-adjoint positive operator, then the Lipschitz condition (5) is equivalent to the estimate
\[ \left\|\mathcal D^{1/2}(t)\bigl(\mathcal D^{-1}(t)-\mathcal D^{-1}(t-\tau)\bigr)\mathcal D^{1/2}(t)\right\|\le c_1\tau. \tag{6} \]

Proof. Passing in (5) to inverse operators, we obtain
\[ -c_1\tau \mathcal D^{-1}\le \mathcal D^{-1}-\widetilde{\mathcal D}^{-1}\le c_1\tau \mathcal D^{-1}, \]
where \(\mathcal D=\mathcal D(t)\), \(\widetilde{\mathcal D}=\mathcal D(t-\tau)\). Hence the inequality
\[ -c_1\tau E\le \mathcal D^{1/2}\bigl(\mathcal D^{-1}-\widetilde{\mathcal D}^{-1}\bigr)\mathcal D^{1/2}\le c_1\tau E, \]
equivalent to (6), follows.

2. Theorem 1. Let the operator \(A(t)\) satisfy the Lipschitz condition and let \(A^*(t)=A(t)>0\). Then, if the stability condition (3) is fulfilled with constant \(\rho=e^{c_0\tau}\), \(c_0\ge 0\), for the solution of problem (1) the estimate
\[ \|y_{n+1}\|_{A(t_n)} \le \widetilde\rho^{\,n+1}\|y_0\|_{A(0)} + \sum_{n'=0}^{n}\tau \widetilde\rho^{\,n-n'} \|B^{-1}\varphi_{n'}\|_{A(t_{n'})}, \qquad n=0,1,\ldots, \tag{7} \]
is valid, where \(\widetilde\rho=e^{\widetilde c_0\tau}\), \(\widetilde c_0=c_0+c_1/2\).

Proof. Represent equation (1) in the form of an equivalent explicit scheme (see (1))
\[ x_{n+1}=S\bar x_n+\tau A^{1/2}B^{-1}\varphi_n, \tag{8} \]
where \(x_{n+1}=A^{1/2}(t_n)y_{n+1}\), \(\bar x_n=A^{1/2}(t_n)y_n\), \(S=E-\tau C\), \(C=A^{1/2}B^{-1}A^{1/2}\). From the stability condition (3), by Lemma 1, the estimate (2) of the norm of the transition operator \(S\) follows, and from the Lipschitz condition—the obvious estimate
\[ \|\bar x_n\|\le \sqrt{1+c_1\tau}\,\|x_n\|\le e^{c_1\tau/2}\|x_n\|. \]
Therefore from (8) we obtain
\[ \|x_{n+1}\|\le \widetilde\rho\|x_n\|+\tau\|A^{1/2}B^{-1}\varphi_n\|, \]
whence (7) follows.

Similarly one proves

Theorem 2. Let the operator \(B(t)\) satisfy the Lipschitz condition and let \(B^*(t)=B(t)>0\). Then, if the stability condition (4) is fulfilled with constant \(\rho=e^{c_0\tau}\), \(c_0\ge 0\), for the solution of problem (1) the estimate
\[ \|y_{n+1}\|_{B(t_n)} \le \widetilde\rho^{\,n+1}\|y_0\|_{B(0)} + \sum_{n'=0}^{n}\tau \rho^{\,n-n'} \|\varphi(t_{n'})\|_{B^{-1}(t_{n'})}, \tag{9} \]
where \(\widetilde\rho=e^{\widetilde c_0\tau}\), \(\widetilde c_0=c_0+c_1/2\).

We note that if the operator \(A(t)\) is also self-adjoint, then for estimate (9) condition (3) is sufficient.

3. Estimates different from (7) can be obtained by using the method of separating stationary inhomogeneities, which consists in the following.

Let the operator \(A(t)\) in scheme (1) be self-adjoint and positive. Represent the solution of problem (1) as the sum
\[ y_n=v_n+w_n,\qquad n=0,1,\ldots, \tag{10} \]
where \(w\) is the solution of the equation
\[ A(t_n)w(t_{n+1})=\varphi(t_n),\qquad n=0,1,\ldots . \tag{11} \]

Suppose, moreover, that \(w(0)=w(\tau)\). To determine the function \(v(t)\), we obtain the problem

\[ B(t)v_t+A(t)v=\widetilde{\varphi},\qquad v_0=y_0-w_1, \tag{12} \]

where \(\widetilde{\varphi}=-(B-\tau A)w_t\).

The solution \(v_{n+1}\) of the auxiliary problem (12) is estimated by Theorem 1. After this, using the triangle inequality, from (10) and (11) we obtain an estimate for the solution of problem (1).

Theorem 3. Under the same conditions as in Theorem 1, for the solution of problem (1) the estimate

\[ \begin{aligned} \|y_{n+1}\|_{A(t_n)} \leq {}& \widetilde{\rho}^{\,n+1}\bigl(\|y_0\|_{A(0)}+\|\varphi(0)\|_{A^{-1}(0)}\bigr) +\|\varphi(t_n)\|_{A^{-1}(t_n)} \\ &+\sum_{n'=1}^{n}\tau \widetilde{\rho}^{\,n+1-n'} \bigl[c_1\|\varphi_{n'-1}\|_{A^{-1}(t_{n'})} +\|\varphi_{n',\bar t}\|_{A^{-1}(t_{n'})}\bigr], \end{aligned} \tag{13} \]

holds, where \(\widetilde{\rho}=e^{\widetilde c_0\tau}\), \(\widetilde c_0=c_0+c_1/2\), \(\varphi_{n,\bar t}=(\varphi_n-\varphi_{n-1})/\tau\).

Proof. Using the method of separating stationary inhomogeneities, we obtain the estimate

\[ \|y_{n+1}\|_{A(t_n)}\leq \|v_{n+1}\|_{A(t_n)}+\|\varphi(t_n)\|_{A^{-1}(t_n)}, \tag{1} \]

where \(v_{n+1}\) is the solution of problem (12). Just as in Theorem 1, for (12) the estimate

\[ \|v_{n+1}\|_{A(t_n)}\leq \widetilde{\rho}\|v_n\|_{A(t_{n-1})} +\tau\|B^{-1}\widetilde{\varphi}_n\|_{A(t_n)},\qquad n=1,2,\ldots, \tag{15} \]

can be proved, and

\[ \|v_1\|_{A(0)}\leq \rho\|v_0\|_{A(0)}. \]

To estimate the term \(\|B^{-1}\widetilde{\varphi}_n\|_A\) in (15), note that

\[ \|B^{-1}\widetilde{\varphi}\|_A=\|SA^{1/2}w_t\|, \]

where \(S=E-\tau C,\ C=A^{1/2}B^{-1}A^{1/2}\).

Further, according to (11) we have

\[ w_t=(A^{-1}\varphi)_{\bar t} =A^{-1}\varphi_{\bar t}+(A^{-1}-\check A^{-1})\varphi(t_{n-1})/\tau, \]

so that the estimate

\[ \|A^{1/2}w_t\|\leq \|\varphi_t\|_{A^{-1}(t_n)} +\frac1{\tau}\|A^{1/2}(A^{-1}-\check A^{-1})A^{1/2}\| \|\varphi(t_{n-1})\|_{A^{-1}(t_n)} \]

is valid.

Hence, taking Lemma 2 into account, we obtain

\[ \|B^{-1}\widetilde{\varphi}_n\|_{A(t_n)} \leq \rho\{c_1\|\varphi_{n-1}\|_{A^{-1}(t_n)} +\|\varphi_{n,\bar t}\|_{A^{-1}(t_n)}\}. \]

Thus, from (15) there follows the estimate

\[ \|v_{n+1}\|_{A(t_n)}\leq \widetilde{\rho}^{\,n+1}\|v_0\|_{A(0)} +\sum_{n'=1}^{n}\tau \rho^{\,n+1-n'} \bigl[c_1\|\varphi_{n'-1}\|_{A^{-1}(t_{n'})} +\|\varphi_{n',\bar t}\|_{A^{-1}(t_{n'})}\bigr]. \]

Finally, noting that

\[ \|v_0\|_{A(0)}\leq \|y_0\|_{A(0)}+\|w_1\|_{A(0)} =\|y_0\|_{A(0)}+\|\varphi_0\|_{A(0)}, \]

we obtain (13).

1. In the case of permutable operators, more general estimates can be obtained.

Theorem 4. Let the operator \(A(t)\) satisfy the Lipschitz condition, \(A^*(t)=A(t)>0\), \(AB=BA\), and \(A(t)A(t-\tau)=A(t-\tau)A(t)\) for all \(t\) and \(\tau\). Then, if the stability condition (3) is fulfilled with \(\rho=e^{c_0\tau}\), \(c_0\geq 0\), then for

for \(l=0,1,2,\ldots\) the following estimates hold for the solution of problem (1):

\[ \left\| A^l(t_n)y_{n+1}\right\| \le \widetilde{\rho}^{\,n+1}\left\| A^l(0)y_0\right\| + \sum_{n'=0}^{n}\tau \widetilde{\rho}^{\,n-n'} \left\| A^lB^{-1}\varphi_{n'}\right\|, \tag{16} \]

\[ \begin{aligned} \left\| A^l(t_n)y_{n+1}\right\| \le{}& \widetilde{\rho}^{\,n+1}\left\| A^l(0)y_0\right\| + \left\| A^{l-1}(t_n)\varphi_n\right\| + \widetilde{\rho}^{\,n+1}\left\| A^{l-1}(0)\varphi_0\right\| \\ &+ \sum_{n'=1}^{n}\tau \widetilde{\rho}^{\,n+1-n'} \left[ c_1\left\| A^{l-1}(t_{n'})\varphi_{n'-1}\right\| + \left\| A^{l-1}(t_{n'})\varphi_{n',\bar t}\right\| \right], \end{aligned} \tag{17} \]

where \(\widetilde{\rho}=e^{\widetilde c_0\tau}\), \(\widetilde c_0=c_0+lc_1\).

Theorem 5. Suppose that the operator \(B(t)\) satisfies the Lipschitz condition, \(B^*(t)=B(t)>0\), \(AB=BA\), \(B(t)B(t-\tau)=B(t-\tau)B(t)\). Then, if the stability condition (4) is fulfilled with \(\rho=e^{c_0\tau}\), \(c_0\ge 0\), then for \(l=0,1,2,\ldots\), for the solution of problem (1) the estimate

\[ \left\| B^l(t_n)y_{n+1}\right\| \le \widetilde{\rho}^{\,n+1}\left\| B^l(0)y_0\right\| + \sum_{n'=0}^{n}\tau \widetilde{\rho}^{\,n-n'} \left\| B^{l-1}(t_{n'})\varphi_{n'}\right\| \tag{18} \]

is valid, where \(\rho=e^{c_0\tau}\), \(\widetilde c_0=c_0+lc_1\).

Remark 1. If the operator \(A(t)\) is also self-adjoint, then (18) holds under condition (3).

Remark 2. Estimates (16), (18) with \(l=0\) also hold without the Lipschitz-continuity requirement.

Institute of Applied Mathematics
Academy of Sciences of the USSR
Moscow

Received
24 XII 1969

References

\(^1\) A. A. Samarskii, DAN, 181, No. 4, 808 (1968).
\(^2\) A. A. Samarskii, A. V. Gulin, DAN, 181, No. 5, 1042 (1968).
\(^3\) A. A. Samarskii, I. V. Fryazinov, Zhurn. vychisl. matem. i matem. fiz., 1, No. 5, 806 (1961).
\(^4\) A. A. Samarskii, ibid., 1, No. 6, 972 (1961).
\(^5\) A. A. Samarskii, ibid., 7, No. 5, 1096 (1967).