UDC 518.1

MATHEMATICS

A. A. SAMARSKII

ON THE ADDITIVITY PRINCIPLE FOR CONSTRUCTING ECONOMICAL DIFFERENCE SCHEMES

(Presented by Academician M. V. Keldysh, 19 IV 1965)

In all works \((^{1-7})\) and others devoted to economical (in the number of operations) methods for the numerical solution of multidimensional problems of mathematical physics, the representability of a multidimensional elliptic operator in the form of a sum of one-dimensional operators is used in one form or another. In \((^{5-7})\), for equations and systems of equations of parabolic and hyperbolic types, locally one-dimensional schemes (l.o.s.) were studied, constructed on the basis of the principle of modeling a multidimensional differential equation by means of a system of one-dimensional equations, with subsequent application, for the solution of each of the one-dimensional equations, of the simplest difference schemes. Below this principle of constructing locally one-dimensional schemes is applied to an abstract Cauchy problem in a Banach space (the additivity principle).

1. Let a linear (unbounded) operator \(A(t)\), depending on the parameter \(t \in [0,T]\), be defined in a Banach space \(H\), with a domain of definition \(D(A)\) dense in \(H\) and independent of \(t\), and with range \(\Delta(A) \subset H\). Consider the abstract Cauchy problem

\[ du/dt + A(t)u = f(t), \qquad 0 \leq t \leq T, \qquad u(0)=u_0,\quad u_0 \in D(A), \tag{1} \]

where the derivative is understood in the strong sense; \(u=u(t)\in H\) (the desired function) and \(f=f(t)\in H\) (the prescribed function) are defined on the interval \(0 \leq t \leq T\). The boundary conditions are taken into account by the requirement \(u=u(t)\in D(A)\). We shall be interested in the case when \(A\) is a sum of linear operators \(A_\alpha\) with the same domains of definition and ranges as the operator \(A\):

\[ A(t)=\sum_{\alpha=1}^{m} A_\alpha(t). \tag{2} \]

We shall agree to call \(A_\alpha(t)\) one-dimensional operators, and the corresponding Cauchy problems \((1_\alpha)\) with \(A=A_\alpha\) one-dimensional problems. We shall assume that problem (1) is uniformly well posed.

1. Introduce the mesh \(\omega_\tau=\{t^j=j\tau;\ j=0,1,\ldots,[T/\tau]\}\) with step \(\tau\). The “multidimensional” problem (1) will be modeled (approximated) by means of \(m\) one-dimensional problems.

On each interval \([t^j,t^{j+1}]\), instead of (1), we shall successively solve the system of one-dimensional differential equations

\[ dv_{(\alpha)}/dt + A_\alpha(t)v_{(\alpha)}(t)=f_\alpha(t), \]

\[ t\in (t^j,t^{j+1}], \qquad j=0,1,\ldots;\qquad \alpha=1,2,\ldots,m, \tag{3} \]

with initial conditions

\[ v_{(\alpha)}(t^j)=v_{(\alpha-1)}(t^{j+1}),\quad \alpha>1;\qquad v_{(1)}(t^j)=v_{(m)}(t^j)=v(t^j); \]

\[ j \geq 0,\qquad v(0)=u_0. \tag{4} \]

By the solution of problem (3)—(4) on the mesh \(\omega_\tau\) we shall mean the function

\[ v=v^{j+1}=v_{(m)}(t^{j+1}). \]

The functions \(f_\alpha=f_\alpha(t,\tau)\) belong to the domain of definition of \(f(t)\) in \(H\), depend, generally speaking, on \(\tau\), and satisfy the normalization condition

\[ \left\|\sum_{\alpha=1}^{m} f_\alpha - f(t)\right\|=O(\tau^{k_0}),\quad k_0>0, \]

where \(\|\cdot\|\) is the norm in \(H\). In (5) we assumed that

\[ \sum_{\alpha=1}^{m} f_\alpha=f. \]

The functions \(v_{(1)}^{j+1},\ldots,v_{(m)}^{j+1}\) may be interpreted as successive approximations to the solution of problem (1), the number of which \(m\) is equal to the number of terms in the sum (2).

1. Let \(U(t,t')\) be the resolving operator of equation (1) for \(0\le t'\le t\le T\); let \(U_\alpha(t,t')\) be the resolving operator of the one-dimensional equation \((3_\alpha)\) with number \(\alpha\). Replacing equation (1) by the locally one-dimensional system (3) means that \(U(t,t')\) is approximated by the product

\[ \widetilde U(t,t')=U_m(t,t')\ldots U_1(t,t'), \]

so that \(U\approx \widetilde U\). Thus the proposed method of one-dimensional modeling of problem (1) corresponds to an approximate factorization, or splitting, of the resolving operator \(U(t,t')\) into one-dimensional operators. If \(\widetilde U=U\), then for \(f=0\) we have \(v^j=w^j\) for all \(j=0,1,\ldots\). If \(j\ne0\), then \(v^j\) coincides with \(w^j\) on the grid \(\omega_t\) only under a special choice of the right-hand sides \(f_\alpha\), namely: \(f_\alpha=0\) for \(\alpha=1,2,\ldots,m-1\),

\[ f_m(\theta)=w_{(m-1)}(t^{j+1},\theta),\qquad t^j\le\theta\le t^{j+1}, \]

where \(w_{m-1}(t,\theta)\) is the solution of the problem

\[ \frac{dw_{(\alpha)}}{dt}(t,\theta)+A_\alpha(t)w_{(\alpha)}(t,\theta)=0,\qquad t^j\le\theta\le t\le t^{j+1}, \]

\[ w_{(\alpha)}(\theta,\theta)=w_{(\alpha-1)}(t^{j+1},\theta),\qquad \alpha=2,\ldots,m-1, \]

\[ w_{(1)}(\theta,\theta)=f(\theta). \]

Let us give the simplest examples when \(\widetilde U=U\): 1) the heat-conduction equation with coefficients allowing separation of variables in a parallelepiped \((0\le x_\alpha\le l_\alpha,\ \alpha=1,2,\ldots,m)\) with homogeneous boundary conditions; 2) a first-order hyperbolic-type equation with constant coefficients, etc. (V. I. Gol’din, N. N. Kalitkin). In general, if \(A_\alpha(t_1)A_\beta(t_2)=A_\beta(t_2)A_\alpha(t_1)\), \(\beta\ne\alpha\), \(t_1,t_2\in[0,T]\), then \(U=\widetilde U\).

1. We shall say that problem (3), (4) is uniformly well-posed if there exist resolving operators \(U_\alpha(t,t')\) for all \(0\le t'\le t\le T\), \(\alpha=1,2,\ldots,m\), strongly continuous in the aggregate \(t,t'\), and

\[ \|U_\alpha(t,t')\|\le 1+M_D(t-t'), \]

where \(M_D=\mathrm{const}>0\) does not depend on \(\tau\).

Theorem 1. Let problem (3), (4) be uniformly well-posed and let the solution \(u=u(t)\) of problem (1) satisfy the conditions: 1)

\[ \left\|(U_\alpha(t,t')-E)\frac{du}{dt}\right\|\le \rho_1(\tau), \]

2)

\[ \|(U_\alpha(t,t')-E)\psi_\alpha\|\le \rho_2(\tau),\qquad \psi_\alpha(t)=f_\alpha(t)-A_\alpha(t)u-\frac{du}{dt}, \]

3)

\[ \Psi_\alpha(t)=\int_{t_j}^{t} U_\alpha(t,\theta)\psi_\alpha(\theta)\,d\theta,\quad t\ge t_j, \]

is the solution of the inhomogeneous equation \((3_\alpha)\) with number \(\alpha\), with right-hand side \(\psi_\alpha(t)\) and initial condition \(\psi_\alpha(t_j)=0\), \(\alpha=1,\ldots,m\). Then the solution of problem (3), (4) converges uniformly in \(t\) to the solution of problem

\[ \|v^j-u^j\|\le \rho_3(\tau),\qquad j=1,2,\ldots \]

(here \(\rho_k(\tau)\to0\), \(k=1,2,3\), uniformly in \(t\)).

1. For solving problem (3)—(4), depending on the concrete form of \(A_\alpha\), one may use various suitable methods (for example, seek an exact expression for \(v_{(\alpha)}\), use the method of characteristics, the method of straight lines, the method of finite differences, etc.).

Following (8), we shall seek an approximate solution of problem (3), (4) in a Banach space \(H_N\) (for example, in the space of functions defined on a grid \(\omega_N\) in some domain of \(p\)-dimensional space \(x=(x_1,\ldots,x_p)\)). Let \(P_N\) be a linear operator projecting \(H\) onto \(H_N\) (\(P_Nu=u_N\in H_N\), if \(u\in H\)) and \(\lim_{N\to\infty}\|P_Nu\|_N=\|u\|\) for every

\(u \in H\), where \(\|\cdot\|_N\) is the norm in \(H_N\), \(\|\cdot\|\) is the norm in \(H\).

Replacing each one-dimensional equation \((3_\alpha)\) of number \(\alpha\) by a difference scheme in \(H_N\), we obtain a L.o.s. approximating the original problem (1). We shall consider two-level schemes. To equation \((3_\alpha)\) there corresponds in \(H_N\) a two-level scheme
\(R_\alpha^N y_{(\alpha)}=S_\alpha^N y_{(\alpha)}+\tau\varphi_\alpha\), where \(y_{(\alpha)}=y_{(\alpha)}^{j+1}\), \(y_{(\alpha)}=y_{(\alpha)}^j\), and \(R_\alpha^N\) and \(S_\alpha^N\) are linear operators depending on \(t,\tau,N\) and mapping \(H_N\) into itself (see (8)). Taking into account the initial conditions \(y_{(\alpha)}=y_{(\alpha-1)}\) for \(\alpha>1\) and \(y_{(1)}=y_{(0)}y=y^j\), we obtain the L.o.s. in the form:

\[ R_\alpha^N y_{(\alpha)}=S_\alpha^N y_{(\alpha-1)}+\tau\varphi_\alpha,\quad \alpha=1,2,\ldots,m,\ t\in\omega_\tau,\quad y_{(0)}=y, \]
\[ y(0)=P_Nu_0. \tag{5} \]

The solution of this problem is the grid function \(y=y^{j+1}=y_{(m)}^{j+1}\).

1. Problem (5) is posed correctly (the L.o.s. (5) is correct) if, for sufficiently large \(N\ge N_0\) and sufficiently small \(\tau\le\tau_0\), its solution \(y=y_{(m)}(N,\tau;t)\) for \(t\in\omega_\tau\) exists for arbitrary \(\varphi_\alpha\) and \(y(0)\) from \(H_N\) and depends uniformly with respect to \(N,\tau\) continuously on \(\varphi_\alpha\) and \(y(0)\), \(\alpha=1,2,\ldots,m\) (see (8)), so that, for example,

\[ \max_{\omega_\tau}\|y(t)\|_N\le M_1\|y(0)\|_N+M_2\max_{\alpha,\omega_\tau}\|\varphi_\alpha(t)\|_N. \tag{6} \]

Here and below \(M_k\) \((k=1,2,\ldots)\) are positive constants independent of \(N\) and \(\tau\). For correctness of the L.o.s. (5) it is sufficient that the inverse operator \(C_\alpha=(R_\alpha^N)^{-1}\) exist and that the conditions

\[ \|B_m(t)\ldots B_1(t)\|_N\le 1+M_3\tau,\quad \|B_m(t)\ldots B_{s+1}(t)C_s(t)\|\le M_4, \]
\[ \|C_m(t)\|_N\le M_5,\quad t\in\omega_\tau, \tag{7} \]

be satisfied, where \(B_\alpha=(R_\alpha^N)^{-1}S_\alpha^N;\ s=1,2,\ldots,m-1\).

Theorem 2. Let the conditions of Theorem 1 be satisfied, let the L.o.s. (5) be correct, and let each of the one-dimensional schemes \((5_\alpha)\) of number \(\alpha\) approximate the corresponding equation \((3_\alpha)\) on its solution \(v_{(\alpha)}(t)\). Then the solution of problem (5) converges uniformly in \(t\) to the solution of problem (1) as \(\tau\to0\) and \(N\to\infty\):
\[ \lim_{N\to\infty,\ \tau\to0}\|y-P_Nu\|_N=0 \]
for \(t\in\omega_\tau\).

We do not dwell here on the question of the order of accuracy of the L.o.s. (see \(^{5-7,2}\)). We note that, by passing successively from (1) to (3), (4), and (5), one can strengthen some estimates \((5\text{–}7)\). If the condition \(\tilde U=U\) is fulfilled and, consequently, \(v^j=u^j\), then for the solution of the one-dimensional problems \((3_\alpha)\) it is expedient to use schemes of order of accuracy higher than \(O(\tau)\), since \(y^j-P_Nu^j=y^j-P_Nv^j\) (for example, a scheme \(O(h^4+\tau^2)\) for the heat-conduction equation with allowance for the method of computing \(f_m(t)\) indicated above, etc.).

1. Let \(H_N\) be a Euclidean space with scalar product \((y,z)_N\) and norm \(\|z\|_N=\sqrt{(z,z)_N}\). Consider the family of two-level L.o.s. of the form

\[ \frac{y_{(\alpha)}-y_{(\alpha-1)}}{\tau} +A_\alpha^N\bigl(\sigma_\alpha y_{(\alpha)}+(1-\sigma_\alpha)y_{(\alpha-1)}\bigr) =\varphi_\alpha, \]
\[ \alpha=1,\ldots,m,\ t\in\omega_\tau,\ y(0)=P_Nu_0, \tag{8} \]

where \(A_\alpha^N\) is a linear operator whose domain of definition and range coincide with \(H_N\). For correctness of the L.o.s. (8) one of three groups of conditions is sufficient for \(t\in\omega_\tau\) (cf. \((8)\)):

Theorem 3. If \(\sigma_\alpha\ge0.5\), \(A_\alpha^N(t)\) is semibounded from below, i.e.
\[ (A_\alpha^N(t)y,y)_N\ge -c_1\|y\|_N^2 \]
for any \(y\in H_N\), and \(\tau\le\tau_0(c_1)\) is sufficiently small, then the L.o.s. is correct.

Theorem 4. If \(\sigma_\alpha\ge0.5\), \((A_\alpha^Ny,y)_N\ge0\), \(y\in H_N\), \(\tau>0\) is arbitrary, then the L.o.s. is correct.

Theorem 5. If $\sigma_\alpha \geqslant \sigma_\varepsilon = 0.5 - (1-\varepsilon)/\tau \|A_\alpha^N\|_N$, $(A_\alpha^N y, y)_N \geqslant 0$, $A_\alpha^N$ is a finite-dimensional self-adjoint operator, i.e. $(A_\alpha^N y, v)_N = (y, A_\alpha^N v)_N$, $\varepsilon \in (0,1]$, then the l.o.s. is correct.

1. Changing $f_\alpha$, the numbering of $A_\alpha$ in (2), the number of terms in (2), etc., we obtain an innumerable set of systems (3)—(4). In particular, putting

\[ A=\sum_{\alpha=1}^{m} A_\alpha=\sum_{\alpha=1}^{m} A_\alpha',\quad A_\alpha'=\tfrac12 A_\alpha,\ \alpha\leqslant m,\quad A_\alpha'=\tfrac12 A_{2m+1-\alpha},\ \alpha>m, \]

we obtain the “symmetric” system (3), (4), which in a number of cases has accuracy $O(\tau^2)$ ($\|v^j-u^j\|=O(\tau^2)$). For $m=2$ we have $A_1'=\tfrac12 A_1$, $A_2'=\tfrac12 A_2$, $A_3'=\tfrac12 A_2$, $A_4'=\tfrac12 A_1$ (the scheme $\tfrac12 A_1\to \tfrac12 A_2\to \tfrac12 A_2\to \tfrac12 A_1$). Choosing in (8) $\sigma_1=0$, $\sigma_2=1$, $\sigma_3=0$, $\sigma_4=1$, we obtain a generalization of scheme (1), as may be verified after eliminating $y_{(1)}$, $y_{(3)}$. (This was also pointed out by I. V. Fryazinov.) Thus, scheme (1) is an l.o.s. One may also use an l.o.s. with $\sigma_1=\sigma_2=0$, $\sigma_3=\sigma_4=1$. The indicated symmetrization makes it possible to obtain second-order accuracy in $\tau$.

It is possible to use three-layer schemes (for $m=2$ this is obvious), and also combinations of two-layer and three-layer schemes. Schemes (3) for the homogeneous ($f=0$) heat equation are more naturally interpreted as l.o.s., and not as splitting schemes.

1. In (5) a somewhat different method of one-dimensional modeling of problem (1) was proposed. The system considered was

\[ \frac1m\,\frac{dv_{(\alpha)}}{dt}+A_\alpha(t)v_{(\alpha)}=f_\alpha(t),\quad t\in (t_{\alpha-1}^j,t_\alpha^j),\quad t_\alpha^j=t^j+\frac{\alpha}{m}\tau,\quad \alpha=1,2,\ldots,m, \]

\[ v_{(\alpha)}(t_{\alpha-1}^j)=v_{(\alpha-1)}(t_{\alpha-1}^j),\quad \alpha>1,\quad v_{(1)}(t^j)=v(t^j)=v_{(m)}(t^j), \]

where $v(t^{j+1})=v_{(m)}(t^{j+1})$. If $f_\alpha=0$ and $A_\alpha$ do not depend on $t$, then this problem is equivalent to problem (3), (4). In the general case their solutions differ by $O(\tau)$. An analogue of Theorem 1 holds (see also (10)).

Hence the same l.o.s. as before are obtained (see (9)).

1. The case

\[ A=\sum_{\alpha,\beta=1}^{p} A_{\alpha\beta} \]

has also been studied. System (3) has the form ($m=2p$)

\[ \frac{dv_{(\alpha)}}{dt}+\sum_{\beta=1}^{\alpha} A_{\alpha\beta}^{-}(t)v_{(\beta)}=f_\alpha,\quad \alpha\leqslant p; \qquad \frac{dv_{(\alpha)}}{dt}+\sum_{\beta=p}^{2p+1-\alpha} A_{2p+1-\alpha,\beta}^{+}(t)v_{(\beta)}=f_\alpha, \]

\[ \alpha>p, \]

with the initial conditions (4); here $(A_{\alpha\beta}^{-})$ and $A_{\alpha\beta}^{+}$ are triangular matrix-operators ($A_{\alpha\beta}^{-}=0$ for $\beta>\alpha$, $A_{\alpha\beta}^{+}=0$ for $\beta<\alpha$, $A_{\alpha\beta}^{-}+A_{\alpha\beta}^{+}=A_{\alpha\beta}$). The passage to an l.o.s. is evident. Such a method for constructing l.o.s. was used in (6) for a system of multidimensional parabolic equations with mixed derivatives.

1. Locally one-dimensional schemes were studied for quasilinear equations, and also for equations of second order (see (7)):

\[ \frac{d^2u}{dt^2}+A(t)u=f(t),\quad u(0)=u_0,\quad \frac{du}{dt}(0)=\bar u_0,\quad A=\sum_{\alpha=1}^{m} A_\alpha,\quad A=\sum_{\alpha=1}^{m}(A_\alpha+A_\alpha^*), \]

where $A_\alpha^*$ is the operator adjoint to $A_\alpha$.

Received
30 III 1965

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