Full Text
MATHEMATICS
Corresponding Member of the Academy of Sciences of the USSR A. N. TIKHONOV, A. A. SAMARSKII
ON THE STABILITY OF DIFFERENCE SCHEMES
It has repeatedly been conjectured that if a difference scheme is stable in the class of constant coefficients, then it is also stable in the class of variable coefficients. In this note an example is given showing that this conjecture is false if, as the class of variable coefficients, one takes piecewise-discontinuous and piecewise-differentiable functions.*
- Consider the implicit difference scheme:
\[ \frac{y_i^j-y_i^{j-1}}{\tau} = \frac{1}{h^2}\left[b_i\left(y_{i+1}^j-y_i^j\right)-a_i\left(y_i^j-y_{i-1}^j\right)\right] = L_h y_i^j,\quad 0<i<N,\ j>0, \tag{1} \]
and the first boundary-value problem corresponding to it:
\[ y_0^j=y_N^j=0,\quad j\geqslant 0; \tag{2} \]
\[ y_i^0=\varphi_i,\quad 0<i<N, \tag{3} \]
where \(\varphi_i\) are prescribed initial values, and
\[ a_i=k_i-\frac{1}{4}(k_{i+1}-k_{i-1}),\qquad b_i=k_i+\frac{1}{4}(k_{i+1}-k_{i-1}), \tag{4} \]
\(h\) and \(\tau\) are the steps of the difference grid \((x_i=ih,\ t_j=j\tau,\ i=0,1,\ldots,N;\ j=0,\ldots,m,\ h=1/N,\ \tau=T/m)\).
This scheme corresponds to the equation
\[ \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \left(k(x)\frac{\partial u}{\partial x}\right), \quad 0<x<1,\ 0<t\leq T,\ k(x)>0, \tag{5} \]
as is easily verified by rewriting it in the form
\[ \frac{y_i^j-y_i^{j-1}}{\tau} = k_i\frac{y_{i-1}^j-2y_i^j+y_{i+1}^j}{h^2} + \frac{k_{i+1}-k_{i-1}}{2h}\, \frac{y_{i+1}^j-y_{i-1}^j}{2h}. \]
The difference scheme (1) with conditions (2) and (3) is stable in the class of continuous coefficients for sufficiently small \(h\) and arbitrary \(\tau\), since in this case \(a_i>0,\ b_i>0\), and the maximum principle is valid, whence uniform correctness with respect to \(h\) follows.
- We shall seek a solution of the difference equations (1) with conditions (2) in the form \(y_i^j=s^j v_i\). For the function \(v_i\) we obtain the eigenvalue problem
\[ L_h v_i=-\hat{\lambda}v_i,\quad 0<i<N,\qquad v_0=v_N=0 \]
or
\[ L_h h v_i=\lambda v_i,\qquad v_0=v_N=0\quad (\lambda=-\hat{\lambda}), \tag{6} \]
where \(\lambda=(s-1)/s\tau\), so that \(s=1/(1-\lambda\tau)\).
Let \(k(x)>0\) be a piecewise-constant function
\[ k(x)= \begin{cases} k_1, & \text{for } x<\xi=x_n+\theta h,\ 0<\theta<1,\\ k_2, & \text{for } x>\xi, \end{cases} \tag{7} \]
where \(\xi\) is the point of discontinuity of \(k(x)\), irrational.
* The main result of this work was presented in a report at the Fourth All-Union Mathematical Congress in 1961.
It will be shown that if
\[ q_0=\frac{a_n a_{n+1}}{\sqrt{k_1}}+\frac{b_n b_{n+1}}{\sqrt{k_2}} \tag{8} \]
is negative, then for sufficiently small \(h<h_0\) problem (6) has a positive eigenvalue \(\lambda_0\), with \(\lambda_0>z_0/h\), where \(z_0\) is an arbitrary positive constant. It will then follow that, for \(q_0<0\),
\[ s^m=s^{T/\tau}\underset{\tau\to0}{\longrightarrow}e^{\lambda_0 T}\geq e^{z_0T/h} \qquad (m\tau=T). \]
For our scheme
\[ q_0=\frac{k_1^2}{\sqrt{k_2}} \left\{\left[1-\frac14(\varkappa-1)\right]\left[\varkappa-\frac14(\varkappa-1)\right]\sqrt{\varkappa}+ \right. \]
\[ \left. +\left[1+\frac14(\varkappa-1)\right]\left[\varkappa+\frac14(\varkappa-1)\right]\right\}, \]
where \(\varkappa=k_2/k_1\).
If \(k_2>k_1\) \((\varkappa>1)\), then all square brackets except the first are positive; if \(\varkappa>5\), the first bracket is negative, and the first term inside the braces is also negative. Since the degree in \(\varkappa\) of the first term is \(5/2\), while that of the second is \(2\), it is clear that there exists a \(\varkappa_0\) such that \(q_0<0\) for \(\varkappa>\varkappa_0\) (the approximate value is \(\varkappa_0\simeq 11.88\ldots\)).
- It is not difficult to verify that the function
\[ v_i= \begin{cases} v_1\,\operatorname{sh}\alpha x_i/\operatorname{sh}\alpha x_n, & \text{for } i\leq n,\\ v_2\,\operatorname{sh}\beta(1-x_i)/\operatorname{sh}\beta(1-x_{n+1}), & \text{for } i\geq n+1 \end{cases} \]
satisfies the conditions of problem (6) for \(i<n\) and \(i>n+1\), if the conditions
\[ \lambda h^2=4k_1\operatorname{sh}^2\frac{\alpha h}{2} =4k_2\operatorname{sh}^2\frac{\beta h}{2} \tag{9} \]
are satisfied.
The values \(v_1\) and \(v_2\) are determined from equations (6) for \(i=n\) and \(i=n+1\), and \(\lambda\) from the solvability condition for these equations with respect to \(v_1\) and \(v_2\).
Introduce the notation
\[ \mu_1=(\operatorname{sh}\alpha x_n-\operatorname{sh}\alpha x_{n-1})/\operatorname{sh}\alpha x_n; \]
\[ \mu_2=[\operatorname{sh}\beta(1-x_{n+1})-\operatorname{sh}\beta(1-x_{n+2})]/ \operatorname{sh}\beta(1-x_{n+1}) \]
and write equations (6) for \(i=n,n+1\) in the form
\[ (\lambda h^2+b_n+a_n\mu_1)v_1-b_nv_2=0, \]
\[ -a_{n+1}v_1+(\lambda h^2+a_{n+1}+b_{n+1}\mu_2)v_2=0. \]
Equating the determinant of this system to zero, we obtain an equation for \(\lambda\):
\[ F(\lambda,h)=\lambda^2h^4+p\lambda h^2+q=0, \]
where
\[ p=b_n+a_{n+1}+a_n\mu_1+b_{n+1}\mu_2, \]
\[ q=a_na_{n+1}\mu_1+b_nb_{n+1}\mu_2+a_nb_{n+1}\mu_1\mu_2. \tag{10} \]
It is required to prove that, when the condition \(q_0<0\) is fulfilled, for any \(z_0\) there exists \(\lambda_0\), a root of the equation \(F(\lambda,h)=0\), with \(\lambda_0>z_0/h\).
- It is not difficult to see that \(0<\mu_1<1\), \(0<\mu_2<1\). Hence \(p\) and \(q\) are bounded for given \(k_1\) and \(k_2\). Consider arbitrary numbers \(z_1\) and \(z_0\) such that \(z_1<z_0\). Replace \(\lambda\) by \(z\), putting \(z=\lambda h\); then the function \(F(\lambda,h)\) is transformed into the function
\[ \overline{F}(z,h)=F(\lambda,h)=z^2h^2+pzh+q. \]
In Sec. 5 it will be shown that
\[ q=q_0\sqrt{zh}\,(1+O(h))+O(h), \tag{11} \]
where \(q_0\) is determined by formula (8). If \(q_0<0\) \((\varkappa>\varkappa_0)\), then, for sufficiently small \(h<h_0\),
\[ \overline F(z,h)<0\quad \text{for } z_1\leq z\leq z_0 . \]
On the other hand, for any \(h\), by virtue of the boundedness of \(p,q\),
\[ \overline F(z,h)>0\quad \text{for } z>Z(h)>z_0, \]
whence it follows that for \(q_0<0\) there is a root of the equation \(\overline F(z,h)=0\) satisfying the condition \(z>z_0\), or a root of the equation \(F(\lambda,h)=0\), for which \(\lambda>z/h\), which proves the instability of scheme (1) for \(\varkappa>\varkappa_0\).
- Let us prove the asymptotic equality (11). It is obvious that, for the values of \(\alpha\) and \(\beta\) determined by the equalities
\[ zh=4k_1\operatorname{sh}^2\frac{\alpha h}{2} =4k_2\operatorname{sh}^2\frac{\beta h}{2} \tag{9'} \]
in the interval \(z_1\leq z\leq z_0\) the following asymptotic equalities in \(h\) hold:
\[ \alpha h=\sqrt{\frac{zh}{k_1}}\,(1+O(h)),\qquad \beta h=\sqrt{\frac{zh}{k_2}}\,(1+O(h)). \tag{12} \]
Moreover, for \(z_1\leq z\leq z_0\),
\[ \mu_1=2\operatorname{sh}\frac{\alpha h}{2}\operatorname{ch}\alpha(x_n-0.5h)/\operatorname{sh}\alpha x_n= \]
\[ =\alpha h\,[1+O(\alpha^2h^2)]=\alpha h\,[1+O(h)] \tag{13} \]
and, analogously,
\[ \mu_2=\beta h\,(1+O(h)). \tag{13'} \]
Substituting equalities (13), (13′) into expression (10) for \(q\), we obtain
\[ q=\left(\frac{a_na_{n+1}}{\sqrt{k_1}}+ \frac{b_nb_{n+1}}{\sqrt{k_2}}\right)\sqrt{zh}\,(1+O(h))+O(h) =q_0\sqrt{zh}\,(1+O(h))+O(h), \]
where \(q_0\) is determined by formula (8).
Thus the instability of scheme (1) for \(\varkappa=k_2/k_1>\varkappa_0\) and sufficiently small \(h<h_0\) has been proved.
It is not difficult to see (cf. (1′)) that, in those cases in which scheme (7) for piecewise constant coefficients \(k(x)\) converges, the limiting function for \(y_i^j\) will differ from the solution of the corresponding boundary-value problem for the differential equation (5). For convergence in the class of discontinuous coefficients, the scheme
\[ \frac{y_i^j-y_i^{j-1}}{\tau}=L_hy_i^j \]
must necessarily and sufficiently have the operator \(L_hy_i\) be conservative (self-adjoint), i.e., one must have \(b_i=a_{i+1}\) (see (1′)).
Received
29 XII 1962
REFERENCES
- A. N. Tikhonov, A. A. Samarskii, Zhurn. vychisl. matem. i matem. fiz., 1, No. 1, 5 (1961).