THE HYPERPLANE METHOD IN NON-STATIONARY PROBLEMS WITH A SELF-ADJOINT OPERATOR

O. A. LISKOVETS

The article proves convergence and gives a mean-square error estimate for solving problems of mathematical physics by the hyperplane method (a multidimensional analogue of the method of lines). In doing so, the theory of self-adjoint operators in Hilbert space [1] is used, in particular representations by means of Hellinger integrals. In Sec. 1 regular problems are studied; in Sec. 2 the singular case is considered and an example is given. The results obtained for one-dimensional problems clearly overlap with the results of [2], and for multidimensional problems generalize them.

1. Let, in the finite cylinder \(V \times [0,T]\) (\(V\) is a domain of Euclidean space \(E_s\), \(s \geqslant 1\), with boundary \(\Gamma\)), the equation

\[ u_t = l(u) + f(x,t) \quad (x \in V) \tag{1} \]

be solved under the conditions

\[ u(x,0)=\varphi(x) \quad (x \in \overline V), \tag{2} \]

\[ D(u)\big|_{\Gamma}=\psi(x,t)\big|_{\Gamma} \quad (0 \leqslant t \leqslant T), \tag{3} \]

where \(l(u)\) is a prescribed regular self-adjoint differential operation in \(V\), the linear (matrix) expression \(D(u)\) also does not depend on \(t\), and in the Hilbert space \(L_2(V)\) the operator \(L\), defined by the operation \(l(u)\) and the conditions \(D(u)|_{\Gamma}=0\), is self-adjoint and bounded above:
\[ (Lu,u) \leqslant \Lambda (u,u) \]
(\(\Lambda < \infty\) is the upper bound of the spectrum). Suppose that this problem has a unique solution, possessing a derivative \(\partial^{Q+1}u/\partial t^{Q+1}\) continuous in \(t\) and belonging to \(L_2(V)\) (\(Q \geqslant 1\)).

For an approximate solution on the hyperplanes \(t=t_n=nh\), \(h>0\), \(n \geqslant 0\), we shall solve the boundary-value problems

\[ \left. \begin{gathered} \sum_{i=0}^{p} A_i u_{n+i}(x) = h \sum_{i=0}^{p} B_i \bigl[l(u_{n+i})+f(x,t_{n+i})\bigr] \quad (x \in V), \\ D(u_{n+p})\big|_{\Gamma} = \psi(x,t_{n+p})\big|_{\Gamma}, \end{gathered} \right\} \tag{4} \]

where \(A_i, B_i\) satisfy the same conditions as in [2]. Then the error \(\varepsilon_n(x)=u(x,t_n)-u_n(x)\) is the solution of the problem

\[ \sum_{i=0}^{p} A_i \varepsilon_{n+i} = h \sum_{i=0}^{p} B_i l(\varepsilon_{n+i}) + h r_n \quad (x \in V), \qquad D(\varepsilon_{n+p})\big|_{\Gamma}=0,\quad n \geqslant 0, \tag{5} \]

where the local error \(r_n(x)\) is expressed in terms of \(\partial^{Q+1}u/\partial t^{Q+1}\) (see [2]) and belongs to \(L_2(V)\). We shall assume that on the initial \(p\) layers \(\varepsilon_i,\, l(\varepsilon_i)\in L_2(V)\) and \(D(\varepsilon_i)|_{\Gamma}=0\).

Let us consider the spectral function of the operator \(L\)—the family of projectors \(P_\lambda\) \((P_\lambda = I\) for \(\lambda \geqslant \Lambda)\). In \(L_2(V)\) one can choose a (finite or countable) set of elements \(y_k\) such that the orthogonal sum of the closed linear spans of the elements \(P_\lambda y_k\) \((-\infty < \lambda \leqslant \Lambda)\) coincides with \(L_2(V)\). In this case, for any \(\varepsilon \in L_2(V)\), the formulas hold (we restrict ourselves to real problems)

\[ \varepsilon=\sum_k \int_{-\infty}^{\Lambda} \frac{d(\varepsilon, P_\lambda y_k)}{d\rho_k(\lambda)}\,dP_\lambda y_k, \qquad (\varepsilon,\varepsilon)=\|\varepsilon\|^2= \sum_k \int_{-\infty}^{\Lambda} \frac{\left[d(\varepsilon,P_\lambda y_k)\right]^2}{d\rho_k(\lambda)}, \tag{6} \]

and in the domain of definition of the operator \(L\),

\[ L\varepsilon=\sum_k \int_{-\infty}^{\Lambda} \lambda\,\frac{d(\varepsilon,P_\lambda y_k)}{d\rho_k(\lambda)}\,dP_\lambda y_k, \qquad \rho_k(\lambda)=\|P_\lambda y_k\|^2, \tag{7} \]

where the Hellinger integrals may be understood as limits of the corresponding integral sums under a regular (in the sense of the general Stieltjes integral) sequence of partitions of the interval \((-\infty,\Lambda]\) into a finite number of subintervals (see [1]).

Using these formulas, let us expand equation (5) and bring the summation over \(i\) under the integral:

\[ \sum_k \int_{-\infty}^{\Lambda} \sum_{i=0}^{p} (A_i-h\lambda B_i)\, \frac{d(\varepsilon_{n+i},P_\lambda y_k)}{d\rho_k(\lambda)}\,dP_\lambda y_k = h\sum_k \int_{-\infty}^{\Lambda} \frac{d(r_n,P_\lambda y_k)}{d\rho_k(\lambda)}\,dP_\lambda y_k . \]

Since \(P_\lambda y_k \perp P_\lambda y_j\) \((j\ne k)\), this equality is equivalent to the fulfillment of the equalities

\[ \int_{-\infty}^{\Lambda} \sum_{i=0}^{p} (A_i-h\lambda B_i)\, \frac{d(\varepsilon_{n+i},P_\lambda y)}{d\rho(\lambda)}\,dP_\lambda y = h\int_{-\infty}^{\Lambda} \frac{d(r_n,P_\lambda y)}{d\rho(\lambda)}\,dP_\lambda y \]

for all \(y=y_k\).

Now take a regular sequence \(\omega_\alpha\) \((\alpha=1,2,\ldots)\) of partitions of the interval \((-\infty,\Lambda]\) and, fixing \(\alpha\), replace the integrals by the corresponding sums:

\[ \sum_{\omega_\alpha}\sum_{i=0}^{p} (A_i-h\lambda_m B_i)\, \frac{\Delta_m(\varepsilon_{n+i},P_\lambda y)}{\Delta_m\rho(\lambda)}\, \Delta_m P_\lambda y = h\sum_{\omega_\alpha} \frac{\Delta_m(r_n,P_\lambda y)}{\Delta_m\rho(\lambda)}\, \Delta_m P_\lambda y +e_{n\alpha}, \]

where \(\lambda_m\in\Delta_m\), and \(e_{n\alpha}\) belongs to the closed linear span of \(P_\lambda y\) and tends to \(0\) as \(\alpha\to\infty\). The subintervals of one and the same partition do not intersect, and, since in this case \(\Delta' P_\lambda y \perp \Delta'' P_\lambda y\), the last equality is equivalent to saying that, for all \(m\),

\[ \sum_{i=0}^{p} (A_i-h\lambda_m B_i)\Delta_m(\varepsilon_{n+i},P_\lambda y) = h\Delta_m(r_n,P_\lambda y)+e_{n\alpha}^{(m)}, \]

where

\[ e_{n\alpha}^{(m)}=(e_{n\alpha},\Delta_mP_\lambda y)\Delta_m\rho(\lambda). \]

Thus we have obtained a difference equation for the quantities \(\Delta_m(\varepsilon_n,P_\lambda y)\). Similar equations were considered in [2], and we can use the estimate found there:

\[ \Delta_m^2(\varepsilon_n,\,P_\lambda y)\leq N\left\{ h^{2-2q}\sum_{i=0}^{p-1}\Delta_m^2(\varepsilon_i,\,P_\lambda y)+ \right. \]
\[ \left. +\,h^{1-2q}\sum_{i=0}^{n-p}\left[ h^2\Delta_m^2(r_i,\,P_\lambda y)+2he_{n\alpha}^{(i)}\Delta_m(r_i,\,P_\lambda y)+(e_{n\alpha}^{(i)})^2 \right]\right\}. \tag{8} \]

Divide (8) by \(\Delta_m\rho(\lambda)\), sum over all \(m\) in the partition \(\omega_\alpha\), and then let \(\alpha\to\infty\). The sums containing \(e_{n\alpha}^{(i)}\), obviously, are infinitely small, and we obtain

\[ \int_{-\infty}^{\Lambda}\frac{[d(\varepsilon_n,\,P_\lambda y)]^2}{d\rho(\lambda)} \leq N\left[ h^{2-2q}\int_{-\infty}^{\Lambda}\sum_{i=0}^{p-1} \frac{[d(\varepsilon_i,\,P_\lambda y)]^2}{d\rho(\lambda)} +\right. \]
\[ \left. +\,h^{3-2q}\sum_{i=0}^{n-p} \int_{-\infty}^{\Lambda} \frac{[d(r_i,\,P_\lambda y)]^2}{d\rho(\lambda)} \right]. \]

Such inequalities hold for all \(y=y_k\). Summing them over all \(k\) and comparing with (6), we conclude that

\[ \|\varepsilon_n\|^2\leq N\left[ h^{2-2q}\sum_{i=0}^{p-1}\|\varepsilon_i\|^2 + h^{3-2q}\sum_{i=0}^{n-p}\|r_i\|^2 \right]. \tag{9} \]

Exactly the same error estimate was proved in [2], and the remarks made there on convergence and stability of the method are valid. From convergence in the mean there follows pointwise convergence of a certain subsequence of approximations to the exact solution almost everywhere in \(\overline V\).

The result obtained is directly generalized to equations of the form \(\partial^k u/\partial t^k=l(u)+f(x,t)\) in all the directions listed in [2].

1. Assuming the hypotheses of item 1 to be fulfilled, we shall now show that, under certain additional requirements, estimate (9) remains valid also in the singular case (for example, when the domain \(V\) is unbounded). As before, we consider, for instance, problem (1)—(3).

The singularity of the problem may entail, first, the appearance for the operator \(L\) of a continuous spectrum, which is immaterial when the representations (6), (7) are used; secondly, non-uniqueness of the solution of the original and/or approximating problem even when high smoothness is required of it. In this case it is necessary to indicate a uniqueness class corresponding simultaneously to both problems (for example, to restrict the order of growth of solutions in the singular part of the domain).

Suppose we are given such linear spaces of functions \(K=K(h)\) and \(F=F(h)\supseteq K\) that: 1) scheme (4), i.e. the operator determined by the operation \(l(u)-u/hB_p\) and by conditions (3), is uniquely solvable in \(K\), for any \(t\in[0,T]\), with right-hand side from \(F\) (in order not to narrow the classes \(K\) and \(F\), it may be useful to reduce condition (3) to a stationary one, for example, by replacing the unknown function); 2) \(f(x,t)\in F\) \((0\leq t\leq T)\); 3) the desired solution belongs to \(K\) for \(0\leq t\leq T\) and is unique in it. If now the values on the initial layers are chosen so that \(u_i,\ l(u_i)\in F\) \((0\leq i\leq p-1)\), then all solutions of scheme (4) \(u_n\) \((n\geq p)\) belong to \(K\) and

are determined in it uniquely. Under the assumptions of item 1, everything said in it concerning the error, up to the estimate (9), obviously remains valid; moreover, \(\xi_n \in K\).

To prove the convergence of the scheme it is enough to assume the existence of linear functionals of the functions \(K_0\) and \(F_0 \supset K_0\), satisfying the three conditions of the preceding paragraph for all \(h,\ 0<h \le h_0\) (such classes may be
\[ K_0=\bigcap_{0<h\le h_0}K(h)\quad \text{and}\quad F_0=\bigcap_{0<h\le h_0}F(h)). \]
Since the uniqueness classes of problems (1)—(3) and (4), generally speaking, do not coincide with one another, it is not always possible to find linear functionals \(K(h)\) and \(F(h)\), respectively \(K_0\) and \(F_0\). This “defect of the method of lines” in longitudinal (with preservation of the time derivative) schemes is noted in [3, 5].

The singular case admits the same generalizations as in item 1.

Example. Consider the Cauchy problem
\[ u_t=u_{xx}\quad (-\infty<x<\infty,\quad 0\le t\le T),\qquad u(x,0)=\varphi(x). \]

Its uniqueness classes are, as is known [6], the functions
\[ u(x,t)=o(\exp Cx^2),\ |x|\to\infty\ (C>0,\ 0\le t\le T), \tag{10} \]
provided, of course, that \(\varphi(x)\) belongs to a similar class.

The operator \(d^2/dx^2\) is nonpositive in \(L_2(-\infty,\infty)\) [7], and for the approximation one may take, for example, the Rothe scheme
\[ u_{n+1}''(x)=[u_{n+1}(x)-u_n(x)]/h,\qquad u_0(x)=\varphi(x),\quad -\infty<x<\infty,\quad n\ge 0. \]

The fundamental solutions of this equation are \(\exp(\pm x/\sqrt h)\), and therefore the uniqueness class is exhausted by the functions
\[ u(x)=o(\exp |x|/\sqrt h),\quad |x|\to\infty, \]
which is considerably narrower than (10). Consequently, if \(\varphi(x)\) grows no more slowly than \(\exp |x|/\sqrt h\), but satisfies relation (10) for some \(C<\infty\), then the scheme with the indicated \(h\) cannot be applied, despite the fact that the solution of the original problem exists and is unique in the class (10) (at least for small \(t\)).

If, however, the initial function lies in the class
\[ \varphi(x)=O(\exp C|x|),\quad |x|\to\infty, \tag{11} \]
for some \(C,\ 0\le C<1/\sqrt h\), then, as is not difficult to show, the approximating solution also belongs to this same class; hence, if the desired solution belongs to the same class, it may be taken as \(K(h)\) and \(F(h)\). In this case the solution of the scheme reduces to successive integration:
\[ u_{n+1}(x)=\int_{-\infty}^{\infty} e^{-|x-y|/\sqrt h}\,u_n(y)\,dy/2\sqrt h,\qquad u_0(x)=\varphi(x). \]

In our case, as the step is decreased, the uniqueness class expands. Therefore, if \(\varphi(x)\) satisfies (11) with finite \(C\ge 0\), the scheme is applicable for all \(h<1/C^2\) and, under the corresponding conditions, converges when the step is infinitely small.

It is interesting to note that, for the same original problem, in the longitudinal variant of the scheme the uniqueness class turned out to be somewhat wider [4].

References

1. V. I. Smirnov, A Course of Higher Mathematics, 5, Fizmatgiz, Moscow, 1959.
2. V. I. Krylov and O. A. Liskovets, DAN BSSR, 8, no. 6, 1964, pp. 353–356.
3. L. I. Kamynin, Izv. AN SSSR, Ser. Math., 17, no. 2, 1953, pp. 163–180.
4. L. I. Kamynin, Izv. AN SSSR, Ser. Math., 17, no. 3, 1953, pp. 249–268.
5. L. I. Kamynin, DAN SSSR, 95, no. 1, 1954, pp. 13–16.
6. I. G. Petrovsky, Lectures on Partial Differential Equations, GITTL, Moscow, 1953.
7. M. A. Naimark, Linear Differential Operators, GITTL, Moscow, 1954.

Received by the editors
December 1, 1964

Institute of Mathematics
Academy of Sciences of the BSSR