UDC 518:517.944/.947

MATHEMATICS

E. G. DYAKONOV

DIFFERENCE SCHEMES WITH A SPLITTING OPERATOR FOR SYSTEMS OF EQUATIONS OF THE FORM \(L_0 \dfrac{\partial u}{\partial t} + L_1u = f\)

(Presented by Academician A. N. Tikhonov on November 22, 1966)

1. Let \(\Omega\) be a domain in the space \(x'=(x_1,x_2,\ldots,x_p)\), composed of a finite number of parallelepipeds with faces parallel to the coordinate planes, and let in \(Q_T \equiv \Omega \times [0 \le x_0 \le T]\) one seek the solution of the system of equations

\[ L_0D_0u+L_1u=f(x), \tag{1} \]

satisfying the initial and boundary conditions

\[ u|_{x_0=0}=\varphi(x');\qquad D^\alpha u=0 \text{ for } x'\in\Gamma \text{ and } |\alpha|\le m=\max(m_0,m_1), \tag{2} \]

where \(u\equiv(u_1(x),\ldots,u_N(x));\ x=(x_0,x');\ D_s=\partial/\partial x_s;\ D^\alpha=D_1^{\alpha_1}D_2^{\alpha_2}\ldots D_p^{\alpha_p};\ \alpha=(\alpha_1,\alpha_2,\ldots,\alpha_p);\ |\alpha|=\sum_{s=1}^p \alpha_s;\ a_k^{\alpha\beta}(x), a_k^\alpha(x)\) are bounded matrices of order \(N\); \(\Gamma\) is the boundary of \(\Omega\);

\[ L_k \equiv \sum_{\substack{|\alpha|\le m_k\\ |\beta|\le m_k}} D^\alpha\bigl(a_k^{\alpha\beta}(x)D^\beta\bigr) + \sum_{|\alpha|\le m_k} a_k^\alpha(x)D^\alpha \quad (k=0,1) \tag{3} \]

are strongly elliptic operators (1).

Let \(h_s>0,\ \tau>0\) be the mesh steps with respect to \(x_s\) \((s=1,2,\ldots,p)\) and \(x_0\), respectively: \(i=(i_1,i_2,\ldots,i_p);\ x_i=ih;\ v_i^n=v(n\tau,x_i);\ \bar\Omega_h\equiv\{x_i:x_i\in\bar\Omega\};\ \Omega_h\equiv\{x_i:x_{i+j}\in\bar\Omega \text{ for } |j|\le m\};\ \Gamma_h=\bar\Omega_h\setminus\Omega_h;\ \Delta^\alpha v,\ \Delta v,\ v_{x_0}^n,\ [v,w],\ \|v\|,\ \|v\|_{W_2^m}\) are the notations from [2]; \(A_{k,s}\equiv\sum_{r\le m_k}(-1)^r\Delta_s^r(b_{k,s,r}(x_s)\cdot \Delta_s^r),\ s=1,2,\ldots,p;\ k=0,1;\ b_{k,s,r}(x_s)\) are diagonal nonnegative matrices of order \(N\); \(A_k\) is the difference approximation obtained from (3) by replacing \(D^\alpha\) by \(\Delta^\alpha\) and \(D^\beta\) by \(\Delta^\beta\);

\[ \hat\Lambda_0\equiv\prod_{s=1}^p \Lambda_{0,s}; \qquad \hat\Lambda_1\equiv \sum_{r=1}^p \Lambda_{0,1}\ldots\Lambda_{0,r-1}\Lambda_{1,r}\Lambda_{0,r+1}\ldots\Lambda_{0,p}. \]

Consider the difference scheme*

\[ \prod_{s=1}^p(\Lambda_{0,s}+\tau\Lambda_{1,s})v_{x_0}^n +(\Lambda_0-\hat\Lambda_0)v_{x_0}^{\,n-1} +\Lambda_1v^n=f^n,\qquad x_i\in\Omega_h; \tag{4} \]

\[ v^0=\varphi;\qquad v^1=\varphi^1;\qquad v_i^n=0 \text{ for } x_i\in\Gamma_h, \tag{5} \]

* Scheme (4), (5) was presented in a communication by the author at the International Mathematical Congress in Moscow in August 1966.

where all coefficients in (4) are computed at \(x_0=n\tau\), while \(\varphi^1\approx u^1\) and can be found, for example, by means of two-level difference schemes. To formulate the stability theorem we introduce the notation \(\omega_l\equiv\{\alpha:\alpha_{s_1}, \alpha_{s_2},\ldots,\alpha_{s_l}\leq m_1,\) and the remaining \(\alpha_s\leq m_0\}\), \(\|v\|_l^2\equiv \sum_{\alpha\in\varphi_l}\|\Delta^\alpha v\|^2\), and assume that

\[ \frac12\{[\hat{\Lambda}_0^n v,v]+[\hat{\Lambda}_0^{n-1}w,w]\}+[(\Lambda_0^n-\hat{\Lambda}_0^{n-1})v,w]\geq \delta_0(\|v\|_{W_2^{m_0}}^2+\|w\|_{W_2^{m_0}}^2); \tag{6} \]

\[ \Lambda_1=\bar{\Lambda}_1+\sum_{\substack{|\alpha|\leq m_0\\|\beta|\leq m_1}} \bar{\Delta}^{\alpha}(c^{\alpha\beta}(x)\Delta^\beta), \tag{7} \]

where \(c^{\alpha\beta}(x)\) are bounded, \([\bar{\Lambda}_1 v,w]=[v,\bar{\Lambda}_1 w]\),

\[ [\hat{\Lambda}_1 v,v]\geq \delta_1\|v\|_{W_2^{m_1}}^2,\qquad \left|\left[\frac{\bar{\Lambda}_1^{\,n+1}-\bar{\Lambda}_1^{\,n}}{\tau}v,v\right]\right|\leq \delta\|v\|_{W_2^{m_1}}^2, \]

\[ 2[\hat{\Lambda}_1 v,v]-[\bar{\Lambda}_1 v,v]\geq \delta_2\|v\|_1^2, \tag{8} \]

\(v,w\) are arbitrary grid vector-functions vanishing on \(\Gamma_h\); \(\delta\) are constants independent of the grid; \(\delta_0,\delta_1,\delta_2>0\).

Theorem 1. Let conditions (6)—(8) be satisfied and let the vector-function \(v\) be a solution of (4), (5) with

\[ f\equiv \sum_{|\alpha|\leq m}\bar{\Delta}^{\alpha}f_\alpha +\tau\hat{\Lambda}_0 f_0+\tau\hat{\Lambda}_1 f_1+ \left\{\prod_{s=1}^{p}(\Lambda_{0,s}+\tau\Lambda_{1,s})-\hat{\Lambda}_0-\tau\hat{\Lambda}_1\right\}f_2. \tag{9} \]

Then for \(k\leq T/\tau\) the estimate holds

\[ V_k^1(v)\equiv \tau\sum_{n=0}^{k-1}\|v_{x_0}^n\|_{W_2^{m_0}}^2 +\|v^k\|_{W_2^{m_1}}^2 +\sum_{n=1}^{k-1}\sum_{l=1}^{p}\tau^{l+1}\|v_{x_0}^{\,n}\|_l^2 \leq \]

\[ \leq M\left\{\tau\|v_{x_0}^{0}\|_0^2+\|v^1\|_{W_2^{m_1}}^2 +\sum_{n=1}^{k-1}\left[\sum_{l=2}^{p}\tau^{l+1}\|f_2^n\|_l^2+\tau^2\|f_1^n\|_1^2\right]+\|f\|_f^2\right\}, \tag{10} \]

where

\[ \|f\|_f^2\equiv \tau\sum_{n=1}^{k-1}\sum_{|\alpha|\leq m_0}\|f_\alpha^n\|^2 +\sum_{m_0\leq|\alpha|\leq m} \left\{\tau\sum_{n=1}^{k-2}\|(f_\alpha^n)_{x_0}\|^2+\|f_\alpha^1\|^2\right\} +\tau\sum_{n=1}^{k-1}\tau\|f_0^n\|_0^2 \quad \text{for } m_0\leq m_1, \]

\[ \|f\|_f^2\equiv \tau\sum_{n=1}^{k-1}\left(\sum_{|\alpha|\leq m_1}\|f_\alpha^n\|^2+\tau^2\|\hat{\Lambda}_0 f_0^n\|^2\right) \quad \text{for } m_0>m_1, \]

and the constant \(M\) does not depend on \(\tau,h_s,k\).

Estimate (10) is related to the results \((^3,^4)\) and remains valid when the operators \(\Lambda_{k,s}\) are chosen from a more general class (see \((^2,^3)\)).

Condition (6) is satisfied, for example, when

\[ \Lambda_0^n=\bar{\Lambda}_0^n+R_0^n,\qquad [\bar{\Lambda}_0^n v,w]=[v,\bar{\Lambda}_0^n w],\qquad [\bar{\Lambda}_0^n v,v]\geq 2\delta_0\|v\|_{W_2^{m_0}}^2, \tag{6'} \]

\[ |[R_0^n v,w]|\leq \delta_0(\|v\|_{W_2^{m_0}}^2+\|w\|_{W_2^{m_0}}^2),\qquad [\bar{\Lambda}_0^n v,v]\leq [\hat{\Lambda}_0 v,v]. \]

An important example of (1), for which Theorem 1 is valid, is the equation

\[ -\Delta D_0u+\nu\Delta^2u=f,\qquad \nu>0. \]

It is not difficult to see that the theorem is easily generalized also to the case when, instead of \(f(x)\), one has \(f(x,D^\alpha u)\), \(|\alpha|\le m\), and the nonlinearity in \(D^\alpha u\) is weak (see, for example, \((^3)\)).

1. Let \(a(x)\in H_\alpha(0)\) mean that in \(Q_T\), \(a(x)\) has continuous derivatives \(D^\beta a\) \((\beta\le \alpha)\) satisfying the Lipschitz condition in all \(x_s\) whose indices coincide with the indices of the nonzero components of \(\alpha\). We also assume that \(h_s=h\chi_s\), where \(0<\underline{\chi}\le \chi_s\le \overline{\chi}\) and \(\underline{\chi},\overline{\chi}\) do not depend on the mesh.

Theorem 2. Let the conditions of Theorem 1 be satisfied, \(a_k^{\alpha\beta}\in H_\alpha(0)\), \(b_{k,s,r}\in H_s(r)\) (see (2)), \(m_0\le m_1\), and let the solution of (1), (2) exist and possess in \(Q_T\) continuous derivatives \(D^\alpha u\), \(|\alpha|\le 2m+1\); \(D^\alpha D_0^{\alpha_0}u\), \(\alpha_0\le 2\), \(\alpha\in \omega_0\), \(|\alpha|\le m_0(2p-1)\); \(D^\alpha D_0^{\alpha_0}u\), \(\alpha_0\le 1\), \(\alpha\in\omega_1\); \(D^\alpha D_0^{\alpha_0}u\), \(\alpha_0\le 1\), \(\alpha\in\omega_1\), \(|\alpha|\le m_1 l+m_0(p-l-1)\), \(l=2,\ldots,p\); \(D^\alpha D_0u\), \(|\alpha|\le mp\). Then for \(z\equiv u-v\) the estimate is valid

\[ \{V_h^k(z)\}^{1/2}\le M\left\{\tau+h^{1/2}+\tau^{1/2}\|z_{x_0}^0\|_0+\|z^1\|_{W_2^{m_1}}\right\}; \tag{11} \]

moreover, if \(m\le 1\), \(\Gamma_h\in\Gamma\), then in (11) one may replace \(h^{1/2}\) by \(h\).

Remark 1. Estimate (11) remains valid also in the case when there are discontinuities of \(a^{\alpha\alpha}\), \(|\alpha|=\alpha_s\), on a finite number of smooth surfaces \(x_s=x_s(x_1,\ldots,x_{s-1},x_{s+1},\ldots,x_p)\), under the condition that between the indicated surfaces \(a^{\alpha\alpha}\) has the same derivatives as were required throughout \(Q_T\) by the notation \(a^{\alpha\alpha}\in H_\alpha(0)\). Naturally, in this case, on the surfaces of discontinuity it is required, instead of (1), that \(D^\beta u\), \(\beta\le\alpha\), and \(a^{\alpha\alpha}D^\alpha u\) be continuous.

Remark 2. The quantities \(\tau^{1/2}\|z_{x_0}^0\|_0\) and \(\|z^1\|_{W_2^{m_1}}\) in (11) may be replaced by \(R_0\equiv \tau^{1/2}\|\hat{\Lambda}_0D_0u|_{x=0}-\psi(x')\|\), if \(v^1\equiv\varphi^1\) is found from

\[ \prod_{s=1}^{p}(\Lambda_{0,s}+\tau\Lambda_{1,s})v_{x_0}^0 +(\Lambda_0-\hat{\Lambda}_0)\psi(x')+\Lambda_1v^0=f^0,\qquad x_i\in\Omega_h. \tag{4'} \]

Here, for \(\hat{\Lambda}_0D_0u|_{x_0=0}\), one can sometimes find an approximation \(\psi(x')\) such that \(R_0=O(\tau^{1/2}h)\). This can be done, for example, when from (1) for \(x_0=0\) and (2) one can obtain a boundary-value problem for \(w(x')\equiv \tilde{L}_0D_0u|_{x_0=0}\); the solution of this problem by the mesh method may be taken as \(\psi(x')\), if \(w(x')\) is sufficiently smooth.

Remark 3. For the stationary problem \(L_1\varphi=f(x')\); \(D^\alpha\varphi|_\Gamma=0\), \(|\alpha|\le m_1\), under the condition \((\Lambda_1v,v)\ge \delta\|v\|_{W_2^{m_1}}^2\), \(\delta>0\), it follows from the proof of Theorem 2, based on representing the approximation error of (1) and (2) in the form (10), that \(\|\varphi-v\|_{W_2^{m_1}}=O(h^{1/2})\). A similar result for one equation with constant coefficients was obtained in \((^5)\).

1. For schemes (4), (5), in the case when \(\Omega\) is a parallelepiped, one can also obtain a priori estimates of the solution, similar to the estimates from \((^6,^7)\). For example, multiplying (4) by \(\tau\tilde{\Lambda}_1v_{x_0}^n\equiv \tau(-1)^{m_1}\sum_{s=1}^p\Delta_s^{m_1}\Delta_s^{m_2}v_{x_0}^n\) and summing over \(n\) from 1 to \(k-1\), one can obtain the estimate

\[ \tau\sum_{n=1}^{k-1}\|v_{x_0}^n\|_{\Lambda_0,\tilde{\Lambda}_1}^{2} +\|v^k\|_{\tilde{\Lambda}_1,\hat{\Lambda}_1}^{2} +\sum_{l=1}^{p}\tau^{l+1}\sum_{n+1}^{k-1}\|v_{x_0}^n\|_{l}^{2} \le \]

\[ \le M\left\{\tau\|v_{x_0}^0\|_{\Lambda_0,\tilde{\Lambda}_1}^{2} +\|v^1\|_{\hat{\Lambda}_1,\tilde{\Lambda}_1}^{2} +\tau\sum_{n=1}^{k-1}\|f^n\|^{2}\right\}, \tag{12} \]

where \(\|v\|_{\Lambda_0,\tilde{\Lambda}_1}^{2}\) denotes \(\sum_{\alpha,\beta,\gamma}\|\Delta^{\alpha+\beta+\gamma}v\|^{2}\), the vectors \(\alpha,\beta,\gamma\) being such that

\[ |\alpha|=|\beta|=m_0\quad \text{and}\quad a_0^{\alpha\beta}\ne 0,\quad |\gamma|=2m_1=\gamma_s,\quad a\|v\|_l^2=\sum_{\alpha\in\widetilde{\omega}_l}\|\Delta^\alpha v\|^2; \]
\(\widetilde{\omega}_l\) differs from \(\omega_l\) only in that one of the components \(\alpha\) is additionally permitted to be \(\le 2m_1\) instead of \(\le m_1\). For (12) to hold when \(m_0>m_1\), it is sufficient that the conditions
\[ {}^{1}\!/_{2}\{[\Lambda_0^n z^n,\widetilde{\Lambda}_1 z^n]+[\Lambda_0^{n-1}z^{n-1},\widetilde{\Lambda}_1 z^{n-1}]\} +[(\Lambda_0^n-\widetilde{\Lambda}_0)z^{n-1},\widehat{\Lambda}_1 z^n]\ge \]
\[ \ge \delta\{\|z^n\|_{\Lambda_0,\widetilde{\Lambda}_1}^2+ \|z^{n-1}\|_{\Lambda_0,\widetilde{\Lambda}_1}^2\}, \tag{13} \]
\[ [(2\widehat{\Lambda}_1-\widetilde{\Lambda}_1)z,\widetilde{\Lambda}_1z]\ge \delta\|z\|_1^2, \tag{14} \]
\[ \left[\left\{\prod_{s=1}^{p}(\Lambda_{0,s}+\tau\Lambda_{1,s}) -\widehat{\Lambda}_0-\tau\widehat{\Lambda}_1\right\}z,\widetilde{\Lambda}_1z\right] \ge \delta\sum_{l=2}^{p}\tau^l\|z\|_l^2. \tag{15} \]

In the case \(m_0\le m_1\), in (12) the term
\[ \tau\sum_{n=1}^{k-1}\|f^n\|^2 \]
should be replaced by
\[ \tau\sum_{n=1}^{k-1}\|f_{x_0}^n\|^2+\|f^1\|^2 \]
and one should assume (14), (15), and also that
\[ \Lambda_1=\overline{\Lambda}_1+R_1, \]
where
\[ [\widetilde{\Lambda}_1z,\widetilde{\Lambda}_1y]=[\widetilde{\Lambda}_1z,\overline{\Lambda}_1y], \quad \left|[((\overline{\Lambda}_1^n-\overline{\Lambda}_1^{n-1})z,\Lambda_1z)]\right| \le M\tau\|z\|_{\widehat{\Lambda}_1,\widetilde{\Lambda}_1}^2, \quad [\overline{\Lambda}_1z,\widetilde{\Lambda}_1z]\ge \]
\[ \ge \delta\|z\|_{\widehat{\Lambda}_1,\widetilde{\Lambda}_1}^2,\quad [(2\widehat{\Lambda}_1-\widetilde{\Lambda}_1)z,\widetilde{\Lambda}_1z]\ge \delta\|z\|_1^2,\quad R_1=\sum_{|\alpha|\le 2m_0} r_\alpha(x)\Delta^\alpha \]
and
\[ |\Delta^q r_\alpha(x)|\le M,\quad q\le 2(m_1-m_0). \]

Let us note, in conclusion, that there exists a whole class of a priori estimates similar to (12), obtained, for example, by multiplying (4) not by \(\tau\widetilde{\Lambda}_1v_{x_0}^n\), but by
\[ \tau\widetilde{\Lambda}_0v_{x_0}^n;\quad \tau\widetilde{\Lambda}_1v_{x_0}^n+ \left\{\prod_{s=1}^{p}(\Lambda_{0,s}+\tau\Lambda_{1,s}) -\widehat{\Lambda}_0-\tau\widehat{\Lambda}_1\right\}v_{x_0}^n, \]
and so on.

Moscow State University
named after M. V. Lomonosov

Received
27 X 1966

CITED LITERATURE

1. M. I. Vishik, Matem. sbornik, 19 (71), 3 (1951).
2. E. G. Dyakonov, DAN, 144, No. 1 (1962).
3. E. G. Dyakonov, in: Computational Methods and Programming, issue III, Moscow, 1965; issue VI, Moscow, 1967.
4. V. I. Lebedev, Izv. AN SSSR, ser. matem., 22, No. 5, 717 (1958).
5. V. Thomée, Contribution to Differential Equations, 3, No. 3 (1964).
6. E. G. Dyakonov, Sibirsk. matem. zhurn., 6, No. 5 (1965).
7. V. B. Andreev, Zhurn. vychislitel’n. matem. i matem. fiz., 6, No. 2 (1966).