MATHEMATICS

L. I. KAMYNIN

ON THE STABILITY OF DIFFERENCE PARABOLIC EQUATIONS

(Presented by Academician S. L. Sobolev on 16 IX 1960)

This note considers the question of solving, by the method of lines and by the method of finite differences, the first boundary-value problem for the parabolic equation

\[ Lu \equiv \frac{\partial u}{\partial t} - a(x,t)\frac{\partial^2 u}{\partial x^2} - b(x,t)\frac{\partial u}{\partial x} + c(x,t)u = F\left(x,t,u\frac{\partial u}{\partial x}\right); \tag{1} \]

\[ u(x,0)=\varphi(x), \qquad 0\leq x\leq l; \tag{2} \]

\[ u(0,t)=u(l,t)=0, \qquad 0\leq t\leq T; \tag{3} \]

\[ a(x,t)\geq a_0>0, \qquad c(x,t)>0, \]

in the rectangular domain \(\bar G\) \((0\leq x\leq l,\ 0\leq t\leq T)\). The well-known investigations of Lax and Richtmyer \((^{1,2})\) make it possible to reduce the question of convergence of the method to the study of the stability of the corresponding differential-difference and finite-difference equations. In this note, certain energy inequalities for solutions of difference equations are indicated, and with their help the unconditional stability (in the sense of John \((^{3})\)) of these equations is established.

Let, in (1), the functions \(a\), \(b\), \(c\), \(\partial a/\partial x\), and \(\partial a/\partial t\) be continuous in \(\bar G\), and let
\[ |a|\leq A_1,\quad |b|\leq A_2,\quad |c|\leq A_3,\quad |\partial a/\partial x|\leq A_4,\quad |\partial a/\partial t|\leq A_5. \]

Method of lines. On the set

\[ D_h:\quad \{(x,t)\in G,\ 0\leq t\leq T,\ x=nh,\ n=1,2,\ldots,N-1,\ Nh=l\} \]

consider the differential-difference operator

\[ \bar R_h u(x,t)\equiv \frac{\partial u(x,t)}{\partial t} - a(x,t)u_{\bar x x}(x,t) - b(x,t)u_{\bar x}(x,t) + c(x,t)u(x,t), \]

where the difference quotients with forward and backward step are defined as usual:

\[ u_{\bar x}=\frac{u(x,t)-u(x-h,t)}{h},\qquad u_x=\frac{u(x+h,t)-u(x,t)}{h},\qquad u_{\mathring x}=\frac{1}{2}\bigl(u_x+u_{\bar x}\bigr). \]

Let

\[ \|u\|_t^2=h\sum_{n=1}^{N}u^2(nh,t),\qquad \|\varphi\|_0^2=h\sum_{n=1}^{N}\varphi^2(nh). \]

If the continuous function \(F(x,t,y,z)\) has uniformly bounded derivatives \(\partial F/\partial y\) and \(\partial F/\partial z\) in \(\overline G\),

\[ \left|\frac{\partial F}{\partial y}\right| \leq B_1,\qquad \left|\frac{\partial F}{\partial z}\right| \leq B_2, \tag{4} \]

then for the solution of the quasilinear differential-difference equation on \(D_h\)

\[ \overline R_h u(x,t)=F\left(x,t,u(x,t),u_{\bar x}(x,t)\right) \tag{5} \]

with the initial condition

\[ u(x,0)=\varphi(x),\qquad x=nh,\quad n=0,1,2,\ldots,N, \tag{6} \]

and the boundary condition (3), the energy inequality holds

\[ \|u\|_t^2+\|u_{\bar x}\|_t^2 \leq C_1\left(\|\varphi\|_0^2+\|\varphi_{\bar x}\|_0^2+ \int_0^t \|F(x,\tau,0,0)\|_\tau^2\,d\tau\right). \]

Definition 1 (cf. John [3]). The nonlinear differential-difference equation (5) is called stable with respect to a given solution \(u(x,t)\) of the problem (5), (6), (3) if, for fixed \(T\) and for the chosen solution, there exists a constant \(B\), independent of \(h\), such that for any \(t_0,t\) \((0\leq t_0<t\leq T)\) and \(h\) on \(D_h\), the inequality

\[ \|u\|_t^2+\|u_{\bar x}\|_t^2 \leq B\left(\|u\|_{t_0}^2+\|u_{\bar x}\|_{t_0}^2\right). \tag{7} \]

holds.

Theorem 1. Let \(F(x,t,y,z)\) satisfy (4). Then:

a) the quasilinear differential-difference equation (5) is stable for any solution of the problem (5), (6), (3);

b) if \(F\) has continuous \(\partial^2F/\partial y^2\), \(\partial^2F/\partial y\,\partial z\), \(\partial^2F/\partial z^2\), then there exists a unique solution of the problem (5), (6), (3);

c) if \(u(x,t)\) is a solution of (1)—(3), and \(\partial u/\partial t\), \(\partial u/\partial x\), \(\partial^2u/\partial x^2\) are uniformly continuous on \(\overline G\), then

\[ \lim_{h\to 0}\ \sup_{(x,t)\in \overline D_h} \left|u(x,t)-u(x,t;h)\right|=0, \tag{8} \]

where \(u(x,t;h)\) is the solution of the problem (5), (6), (3);

d) if, in c), \(u(x,t)\) has uniformly continuous \(\partial^2u/\partial t^2\), \(\partial^3u/\partial x^3\), \(\partial^4u/\partial x^4\) on \(\overline G\), then

\[ \sup_{(x,t)\in \overline D_h} \left|u(x,t)-u(x,t;h)\right|=O(h^2). \tag{9} \]

Let now \(F(x,t,y,z)\) satisfy the growth restriction in \(y\) and \(z\)

\[ |F(x,t,y,z)|\leq f(x,t)+K\left(|y|+|z|\right)^\alpha, \tag{10} \]

where \(K>0\) and \(\alpha>0\) are constants. If \(\varphi(x)\) and \(F(x,t,y,z)\) are such that

\[ 2C_1(\alpha-1)2^{1+\alpha} \left(\|\varphi\|_0^2+\|\varphi_{\bar x}\|_0^2+ \int_0^T \|f\|_t^2\,dt\right)^{\alpha-1} K^2T<1, \tag{11} \]

then for the solution of the nonlinear differential-difference equation (5), the energy inequality holds

\[ \|u\|_{t}^{2}+\|u_x\|_{t}^{2}\leq P\left[1-(\alpha-1)P^{\alpha-1}2^{1+\alpha}K^{2}C_1t\right]^{\frac{1}{1-\alpha}}, \tag{12} \]

where

\[ P=2C_1\left(\|\varphi\|_{0}^{2}+\|\varphi_x\|_{0}^{2}+\int_{0}^{T}\|f\|_{t}^{2}\,dt\right). \]

Theorem 2. If \(\varphi(x)\) and \(F(x,t,y,z)\) satisfy (12), then the nonlinear differential-difference equation (5) is stable for any solution of problem (5), (6), (3).

The finite-difference method. On the grid

\[ G_h:\{(x,t)\in G,\ x=nh,\ n=1,2,\ldots,N-1,\ t=mk,\ m=1,2,\ldots,M\} \]

\((l=Nh,\ T=Mk,\ k/h^2=\lambda\) is constant) consider the inhomogeneous difference equation

\[ R_hu(x,t)\equiv u_{\bar t}(x,t)-a(x,t)u_{\bar x x}(x,t)-b(x,t)u_{\bar x}(x,t)+ \]
\[ +c(x,t)u(x,t)=f(x,t) \tag{13} \]

with conditions (6) and

\[ u(0,t)=u(l,t)=0,\qquad t=mk,\quad m=0,1,2,\ldots,M. \tag{14} \]

For the solution of problem (13), (6), (14), for sufficiently small \(k\), the following energy inequality holds (cf. Lees [4]):

\[ \|u\|_{t}^{2}+\|u_{\bar x}\|_{t}^{2}\leq C_2\left(\|\varphi\|_{0}^{2}+\|\varphi_{\bar x}\|_{0}^{2} +k\sum_{s=1}^{m}\|f\|_{s k}^{2}\right),\qquad t=mk. \]

Definition 2 (cf. John [3]). The linear equation (13) is called stable if, for the given \(T\), there exists a constant \(B\), independent of \(h\), such that for every function \(v(x,t)\) satisfying the homogeneous equation (13) \((f(x,t)\equiv0)\) with condition (3), for any \(t_0,t\) \((0\leq t_0<t\leq T)\) and \(h\) on \(\bar G_h\) \((k=\lambda h^2)\), inequality (7) holds.

Theorem 3.
a) The linear difference equation (13) is unconditionally stable (i.e., for any ratio of the steps in \(x\) and \(t\); \(\lambda=k/h^2\));

b) on \(G_h\) there exists a unique solution of problem (13), (6), (14);

c) if \(u(x,t)\) is a solution of (1) \((F\equiv f(x,t))\), (2), (3), and \(\partial u/\partial t\), \(\partial u/\partial x\), \(\partial^2u/\partial x^2\) are uniformly continuous on \(\bar G\), then (9) holds, where \(D_h\) is replaced by \(G_h\), and \(u(x,t;h)\) is the solution of (13), (6), (14);

d) if in c), for \(u(x,t)\), there exist and are uniformly continuous on \(\bar G\) the derivatives \(\partial^2u/\partial t^2\), \(\partial^3u/\partial x^3\), \(\partial^4u/\partial x^4\), then (cf. Lees [4])

\[ \sup_{(x,t)\in\bar G_h}|u(x,t)-u(x,t;h)|=O(k+h^2). \tag{15} \]

If \(F(x,t,y,z)\) satisfies (4), then for the solution of the quasilinear difference equation on \(G_h\)

\[ R_hu(x,t)=F\left(x,t,u(x,t),u_{\bar x}(x,t)\right) \tag{16} \]

with conditions (6) and (14), for sufficiently small \(k\), the following energy inequality holds:

\[ \|u\|_{t}^{2}+\|u_{\bar x}\|_{t}^{2}\leq C_3\left(\|\varphi\|_{0}^{2}+\|\varphi_{\bar x}\|_{0}^{2} +k\sum_{s=1}^{m}\|F(x,t,0,0)\|_{s k}^{2}\right). \]

Theorem 4. Let \(F(x,t,y,z)\) satisfy (4).

Then:

a) the quasilinear difference equation (16) is unconditionally stable for any solution of the problem (16), (6), (13) (in the sense of Definition 1);

b) if \(F(x,t,y,z)\) has continuous \(\partial^2 F/\partial y^2\), \(\partial^2 F/\partial y\,\partial z\), \(\partial^2 F/\partial z^2\), then there exists a unique solution of the problem (16), (6), (14).

Assertions c) and d) of Theorem 1 hold with \(D_h\) replaced by \(G_h\) and (9) by (15).

Finally, suppose that \(F(x,t,y,z)\) satisfies (10). If \(\varphi(x)\) and \(F(x,t,y,z)\) are such that

\[ 2C_2(\alpha-1)2^{1+\alpha}\left(\|\varphi\|_0^2+\|\varphi_{\bar x}\|_0^2 +k\sum_{m=1}^{M}\|f\|_{mk}^{2}\right)^{\alpha-1}K^2T<1, \tag{17} \]

then for the nonlinear difference equation

\[ R_h u(x,t)=F\left(x,t,u(x,t-k),u_{\bar x}(x,t-k)\right) \tag{18} \]

for sufficiently small \(k\) the energy inequality holds

\[ \|u\|_t^2+\|u_{\bar x}\|_t^2 \le P\left[1-(\alpha-1)P^{\alpha-1}2^{1+\alpha}K^2C_2(t-k)\right]^{\frac{1}{1-\alpha}}, \]

where

\[ P=2C_2\left(\|\varphi\|_0^2+\|\varphi_{\bar x}\|_0^2 +k\sum_{m=1}^{M}\|f\|_{mk}^{2}\right). \]

Theorem 5. If \(\varphi(x)\) and \(F(x,t,y,z)\) satisfy (17), then the nonlinear difference equation (18) is unconditionally stable (in the sense of Definition 1) for any solution of the problem (18), (6), (14).

Theorems 1–5 are proved by means of the indicated energy inequalities, obtained by applying difference analogues of Gronwall-type lemmas (see \((^7,^4,^5)\)), the Schauder principle, and the embedding theorems of S. L. Sobolev \((^6)\). (\(C_1, C_2, C_3\) are constants depending only on \(a_0, T\) and \(A_i\), \(i=1,2,3,4,5\).)

All the arguments remain valid also for nonhomogeneous boundary conditions

\[ u(0,t)=\psi_1(t),\qquad u(l,t)=\psi_2(t), \]

if one observes that the substitution

\[ v(x,t)=u(x,t)-\left(1-\frac{x}{l}\right)\psi_1(t)-\frac{x}{l}\psi_2(t) \]

reduces the problem (1), (2) to homogeneous boundary conditions (3).

For a hyperbolic equation of the form

\[ \frac{\partial^2 u}{\partial t^2} -a(x,t)\frac{\partial^2 u}{\partial x^2} = F\left(x,t,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial t}\right) \]

analogous results were obtained by M. Lees \((^5)\).

Moscow State University
named after M. V. Lomonosov

Received
16 IX 1960

REFERENCES

1. P. D. Lax, R. D. Richtmyer, Comm. Pure and Appl. Math., 9, 267 (1956).
2. R. D. Richtmyer, Difference Methods for Initial-Value Problems, IL, 1960.
3. F. John, Comm. Pure and Appl. Math., 5, 155 (1952).
4. M. Lees, J. Soc. Indust. and Appl. Math., 7, No. 2, 167 (1959).
5. M. Lees, Trans. Am. Math. Soc., 94, No. 1, 58 (1960).
6. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
7. R. Bellman, Stability Theory of Differential Equations, IL, 1954.