MATHEMATICS

LE DINH THINH

A STUDY, BY SPECTRAL CRITERIA, OF POWER STABILITY OF DIFFERENCE APPROXIMATIONS OF PROBLEMS WITH TIME

(Presented by Academician A. N. Tikhonov, 29 IV 1969)

Any approximation of a mixed problem for a system of differential equations in two variables \((t, x)\) with constant coefficients can be written in the form \((^{1})\)

\[ \sum_{k=0}^{1} A_k^1 u_{n+k}^{m+1} + \sum_{k=0}^{1} A_k^0 u_{n+k}^{m} =0, \tag{1} \]

\[ \sum_{i=0}^{1} B_i u_0^{m+i}=0, \tag{2} \]

\[ \sum_{i=0}^{1} C_i u_N^{m+i}=0. \tag{3} \]

Here \(A_k^i,\ i,k=0,1\), are square matrices of dimension \(l\times l\); \(B_i,\ i=0,1\), are rectangular matrices of dimension \(l_1\times l\); \(C_i,\ i=0,1\), are rectangular matrices of dimension \(l_2\times l\). By \(u_n^m\) is denoted the value of the solution of the grid problem at the node \((m,n)\), the point \((m\tau,nh)\), where \(\tau\) and \(h\) are the mesh sizes in \(t\) and \(x\), respectively.

In what follows, without further stipulation, we assume that \(l_1+l_2=l\); it is possible here that either \(l_1=0\) or \(l_2=0\).

Under certain natural solvability conditions, problems (1)—(3) can be represented \((^{2,3})\) in the form \(\mathbf{u}^{m+1}=R_N\mathbf{u}^m\); here \(R_N\) is a certain linear operator, \(\mathbf{u}^m=\{u_0^m,\ldots,u_N^m\}\).

Suppose that in the space \(U\) of vectors \(\mathbf{u}^m\) of this form a norm is introduced. We shall assume that it is one of the \(l_p\) norms:

\[ \|\mathbf{u}^m\| = \left( \frac{1}{N+1} \sum_{n=0}^{N} \|u_n^m\|_{(2)}^p \right)^{1/p}, \]

where, in turn, \(\|u_n^m\|_{(2)}\) is the Euclidean norm in the space of vectors of dimension \(l\).

In \((^{2,3})\) the notion of the spectrum of a family of operators \(\{R_N\}\) was introduced:

Definition. Let the complex number \(z\) satisfy the condition: for any \(\varepsilon>0\) one can specify \(N_0(\varepsilon,z)\) such that for every \(N>N_0(\varepsilon,z)\) there exists a vector \(u\) satisfying the inequality \(\|R_Nu-zu\|\leqslant \varepsilon\|u\|\). The totality of all such \(z\) will be called the spectrum \(\Lambda\) of the family of operators \(\{R_N\}\).

There it was also indicated that the spectrum has the representation \(\Lambda=\Lambda_x\cup\Lambda_y\cup\Lambda_z\) and an algorithm for constructing the sets \(\Lambda_x,\Lambda_y,\Lambda_z\).

Let \(\Lambda_y\) be the set of those \(z\) for which the grid problem

\[ \sum_{i=0}^{1} (A_i^1 z + A_i^0)\,v_{n+i}=0, \qquad -\infty<n<\infty, \]

has a bounded solution; \(\overline{\Lambda}_x\) is the set of those \(z\) for which the mesh problem has a bounded solution

\[ \sum_{i=0}^{1}(A_i^1z+A_i^0)\mathbf v_{n,i}=0\quad (B_1z+B_0)\mathbf v_0=0,\quad n\geq 0; \]

\(\overline{\Lambda}_z\) is the set of those \(z\) for which the mesh problem has a bounded solution

\[ \sum_{i=0}^{1}(A_i^1z+A_i^0)\mathbf v_{n+i}=0\quad (C_1z+C_0)\mathbf v_N=0,\quad n<N. \]

By \(\Lambda_x,\Lambda_y,\Lambda_z\) we shall denote the closures of the sets \(\overline{\Lambda}_x,\overline{\Lambda}_y,\overline{\Lambda}_z\). The indicated sets do not depend on \(N\) and consist of a finite number of points and analytic arcs in the complex \(z\)-plane; moreover \(\overline{\Lambda}_y=\Lambda_y\).

In \(({}^2,{}^3)\) a necessary condition for uniform boundedness of the powers of the operators \(R_N\) was obtained: in order that the quantities \(\|(R_N)^m\|\) be bounded uniformly in \(m\) and \(N\), it is necessary that the condition \(\Lambda\subseteq K_1\) be satisfied; here and below \(K_1\) is the disk \(|z|\leq 1\), \(S_\rho\) is the circle \(|z|=\rho\). This condition is at present regarded as a necessary condition for the practical suitability of a mesh approximation.

Below we construct an example showing that the condition \(\Lambda\subseteq K_1\) is not a sufficient condition for stability. We shall consider a notion weaker than stability, namely the notion of power stability.

Definition. The mesh problem (1)—(3) has power stability for \(m\leq \varphi(N)\) if, for some \(c,\gamma,r\), when \(m\leq \varphi(N)\) the relation
\[ \|u^m\|\leq cm^\gamma N^r\|u^0\| \]
holds.

The function
\[ \mathbf v_n(z)=\sum_{m=0}^{\infty}\mathbf u_n^m z^{-m} \]
is analytic in the domain \(|z|>\|R_N\|\), in consequence of the inequality \(\|u^m\|\leq \|R_N\|^m\|u^0\|\). Continue it analytically for \(|z|\leq \|R_N\|\); we can write the equality

\[ \mathbf u_n^m=\frac{1}{2\pi i}\oint_{\Gamma}\mathbf v_n(z)z^{m-1}\,dz; \tag{4} \]

here \(\Gamma\) is any contour that can be obtained by a continuous deformation of the contour \(|z|=\|R_N\|+\varepsilon\), where \(\varepsilon>0\), without passing through the singular points of the function \(\mathbf v_n(z)\).

The function \(\mathbf v_n(z)\) satisfies the mesh problem

\[ A_0(z)\mathbf v_n+A_1(z)\mathbf v_{n+1}=\mathbf f_n,\quad n=0,\ldots,N-1;\quad B(z)\mathbf v_0=\varphi;\quad C(z)\mathbf v_N=\psi, \]

where \(A_i(z)=A_i^1z+A_i^0,\ i=0,1;\ B(z)=B_1z+B_0;\ C(z)=C_1z+C_0\).

We shall call the equation

\[ |A_0(z)+\lambda A_1(z)|=0. \tag{6} \]

the characteristic equation of the system (1)—(3).

Definition. The roots \(\lambda_j(z)\) of equation (6) will be called generalized eigenvalues; if \(\det A_1(z)=0\), then equation (6) has \(q<l\) roots (counting their multiplicities). In this case we assume that equation (6) has \(l-q\) roots
\[ \lambda_{q+1}(z)=\ldots=\lambda_l(z)=\infty . \]

To each root \(\lambda_k(z)\) we associate a normalized eigenvector or associated vector of the matrix \(A_0(z)+\lambda A_1(z)\), corresponding to its finite generalized eigenvalue. To a generalized eigenvalue \(\lambda_k(z)=\infty\) we associate eigenvectors or associated vectors of the matrix \(A_1(z)\), corresponding to its zero eigenvalue.

We shall establish this correspondence so that the following conditions are satisfied: 1) for any \(z\), each of the eigenvectors and associated vectors

corresponds to some eigenvalue; in other words, for each \(z\) the system of vectors \(\{C_k(z)\}\) is complete; 2) \(\lambda_k(z)\) and the vectors \(C_k(z)\) are analytic functions of \(z\) for \(|z|<\infty\), with the exception of a finite number of points \(P_1,\ldots,P_s\); 3) \(\lambda_j(z)\) are continuous functions of \(z\) at the points \(P_j\). We shall choose the contour of integration so that it does not pass through these points.

If the root \(\lambda(z)\) of multiplicity \(s\) corresponds to \(s\) linearly independent eigenvectors, then we say that this root has a simple structure.

In what follows, unless otherwise stated, we assume \(\Lambda\subseteq K_1\). In this case, for \(|z|>1\), equation (6) has \(l_2\) generalized eigenvalues greater than 1 in modulus and \(l_1\) smaller than 1 in modulus; we renumber the generalized eigenvalues so that, for \(|z|>1\), one has \(|\lambda_1(z)|,\ldots,|\lambda_{l_1}(z)|<1\).

Definition. The set of points \(z\in S_1\) satisfying the condition
\[ \inf_{\substack{i\leq l_1,\; j>l_1}} \left(|\lambda_j(z)|-|\lambda_i(z)|\right)=\delta(z)>0 \tag{7} \]
will be denoted by \(S'\).

Definition. Denote by \(\overline{\Lambda}_y^\sigma\) the set of points \(z_0\in \Lambda_y\cap S_1\) satisfying the condition: there exist \(\gamma,C_1>0\) such that
\[ |\lambda_j(z)|-|\lambda_i(z)|\geq C_1|z-z_0|^\sigma \]
for \(i\leq l_1<j,\ |z-z_0|\leq\gamma,\ 1<|z|<1+\gamma\).

Definition. Put
\[ \Lambda_y^\sigma=\bigcup_{\sigma'<\sigma}\overline{\Lambda}_y^{\sigma'}. \]
Let \(A_1(z), A_2(z)\) be matrices of dimensions \(l\times l_1,\ l\times l_2\), respectively; the columns of \(A_1(z)\) are the vectors \(C_1(z),\ldots,C_{l_1}(z)\), and the columns of \(A_2(z)\) are the vectors \(C_{l_1+1}(z),\ldots,C_l(z)\). By \(A_1^{ij}(z)\), for \(i\leq l_1\leq j\), we denote the matrix obtained from \(A_1(z)\) by replacing the \(i\)-th column by the vector \(C_j(z)\).

Definition. Denote by \(\Lambda_{xy}^\sigma\) the set of points \(z_0\in\Lambda_y\cap S_1\) satisfying the following conditions: there exist \(C_1,C_2,C_3,\gamma>0\) such that, for \(|z-z_0|\leq\gamma,\ 1<|z|<1+\gamma\):
\[ \text{1) }\quad C_1\left(\inf_{j>l_1}|\lambda_j(z)|-\sup_{i\leq l_1}|\lambda_i(z)|\right)\,|z-z_0|^{-\sigma}\leq C_2, \]
\[ \text{2) }\quad \left|\frac{\det(B(z)A_1^{ij}(z))}{\det(B(z)A_1(z))}\right|\leq C_3 \]
for all \(i\leq l_1,\ j>l_1\) such that
\[ |\lambda_j(z)|-|\lambda_i(z)|=O(|z-z_0|^\sigma). \]
The set \(\Lambda_{yz}^\sigma\) is defined analogously.

The proof of the following theorems is carried out as follows: we write an explicit expression for the Green’s function of the grid problem (5) and an expression for \(v_n\) in terms of the Green’s function. It turns out that the Green’s function is analytic at those points where the determinant
\[ \Delta_N(z)= \left| \begin{array}{c} B(z)\bigl((A_1(z),\,A_2(z))\bigr)\\ C(z)\bigl((A_1(z),\,A_2(z))D(\lambda^N)\bigr) \end{array} \right|, \qquad \text{where }\quad D(\lambda^N)= \begin{pmatrix} \lambda_1^N & 0\\ 0 & \ddots & \lambda_l^N \end{pmatrix}. \]
does not vanish.

Next, depending on the properties of the determinant, it is shown that outside some neighborhood \(S_\rho\), \(\rho_N>1\), one has \(\Delta_N\ne 0\). Then, in (4), the contour of integration \(\Gamma=S_{1+\omega(\rho_N-1)}\) is chosen for some finite \(\omega>1\), and the integral is estimated.

Theorem 1. Suppose: 1) \(\Lambda\subseteq K_1\); 2) \((\Lambda\cap S_1)\subseteq S'\).

Then
\[ \rho_N\geq 1+C_1q^N,\qquad \text{where } C_1>0,\quad 0<q<1, \]
and the stability estimate has the form
\[ \|u^m\|\leq C_2m^\nu N^r\exp(nC_3q^N)\|u^0\|. \]

Theorem 2. Suppose: 1) \(\Lambda\subseteq K_1\); 2)
\[ (\Lambda\cap S_1)\subseteq \bigl(S'\cup(\Lambda_{xy}^\sigma\cap\Lambda_{yz}^\sigma)\cup(\Lambda_{xy}^\sigma\cap\Lambda_y^\sigma)\cup(\Lambda_{yz}^\sigma\cap\Lambda_y^\sigma)\bigr); \]
3) the roots \(\lambda_j(z)\) of equation (6), different from \(\lambda=0\) and \(\lambda=\infty\), have simple structure everywhere except for a finite number of points.

Then
\[ \rho_N\geq 1+C_4N^{-1/\sigma},\qquad C_1>0, \]
and the stability estimate has the form
\[ \|u^m\|\leq C_5m^\nu N^r\exp\left(C_6m/N^{1/\sigma}\right)\|u^0\|. \]

Theorem 3. Suppose: 1) \(\Lambda\subseteq K_1\); 2)
\[ (\Lambda\cap S_1)\subseteq (S'\cup\overline{\Lambda}_y^\sigma). \]
Then
\[ \rho_N\geq 1+C(\ln N/N)^{1/\sigma},\qquad C>0. \]
The stability estimate has the form
\[ \|u^m\|\leq C_6m^\nu N^r \exp\left(C_7m/(N/\ln N)^{1/\sigma}\right)\|u^0\|. \]

Suppose we have an approximation of a mixed problem in the domain \(0 \le t\), \(0 \le x \le 1\) for the hyperbolic system \(\mathbf{u}_t = A\mathbf{u}_x\) with boundary-condition operators independent of time. Then this approximation will have the form (1)—(3), i.e., its coefficients do not depend on \(N\), if, as \(\tau, h \to 0\), the condition \(\tau / h = \varkappa = \mathrm{const}\) is satisfied. Similarly, in the case of an approximation of the parabolic system \(\mathbf{u}_t = A\mathbf{u}_{xx}\), the condition \(\tau / h^2 = \varkappa = \mathrm{const}\) must be satisfied. Therefore, in the first case, integration over the time interval \(0 \le t \le T\) corresponds to \(m \le (T/\varkappa)N\), and in the second case to \(m \le (T/\varkappa)N^2\).

For approximations of parabolic systems in the case \(\sigma = 1/2\), the estimate of Theorem 3 does not ensure power stability when \(\varphi(N)=CN^2\), and consequently is of no interest. For approximations of hyperbolic systems in the case \(\sigma=1\), the estimate of Theorem 3 for \(m=(T/\varkappa)N\) becomes the estimate

\[ \|u^m\| \le C_6(T)N^{C_7+C_8T}\|u^0\|, \tag{8} \]

which also is unlikely to be of practical interest.

Lemma. If \(\Delta_N(z)=0\), then \(\|R_N^m\|>|z|^m\).

It can be shown that for problem (1)—(3), having the form

\[ u_n^{m+1}-u_n^m+\frac{r}{2}\bigl(u_{n+1}^{m+1}-u_{n-1}^{m+1}\bigr)=0,\quad u_1^{m+1}-u_0^{m+1}=0,\quad u_N^{m+1}=0 \]

(note that this is an approximation of the equation \(u_t+u_x=0\)), the conditions of Theorem 3 are satisfied for \(\sigma=1\), and \(\Delta_N(z_0)=0\) for \(|z_0|\ge 1+C\ln N/N\). Hence it follows that, for some initial condition \(u^0\), \(\|u^m\|\ge N^{CT}\|u^0\|\), \(C>0\), i.e., estimate (8) cannot be substantially improved.

This example shows that problems (1)—(3) satisfying the condition \(\Lambda \subseteq K_1\) may have error growth that makes them unsuitable for actual computations; that is, investigation of approximations only by means of this condition, without invoking theorems of type (1)—(3), is insufficient. A more detailed exposition of the main results of the work is given in \({}^{5}\).

I express my deep gratitude to N. S. Bakhvalov for his comprehensive assistance in carrying out this work.

CITED LITERATURE

\({}^{1}\) S. K. Godunov, V. S. Ryabenkii, Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki, 3, No. 2, 211 (1963).
\({}^{2}\) S. K. Godunov, V. S. Ryabenkii, Introduction to the Theory of Difference Schemes, Moscow, 1963.
\({}^{3}\) S. K. Godunov, V. S. Ryabenkii, Uspekhi matematicheskikh nauk, 18, issue 3 (111), 3 (1963).
\({}^{4}\) V. S. Ryabenkii, A. F. Filippov, On the Stability of Difference Equations, Moscow, 1956.
\({}^{5}\) Le Dinh Thinh, Candidate’s Dissertation, Moscow University, 1969.