MATHEMATICS

O. A. OLEINIK

ON THE CONVERGENCE OF CERTAIN DIFFERENCE SCHEMES

(Presented by Academician I. G. Petrovskii on 28 X 1960)

In proving the convergence of difference schemes one usually uses either the existence, for the corresponding difference scheme, of a smooth solution of the problem for the differential equation, or the uniqueness of the solution of this problem. In the first case a direct estimate is made of the deviation of the solution of the difference scheme from the exact solution of the problem for the differential equation; in the second case, on the basis of a priori estimates of the solutions of the difference scheme, compactness of these solutions is established, and then from the uniqueness of the limiting function the convergence of the solutions of the difference scheme follows as the mesh sizes tend to zero.

In the present note the proof of convergence of solutions of difference schemes is carried out directly on the basis of one-sided a priori estimates, without using the theorem on existence or uniqueness of the solution of the problem for the differential equation. In this process a monotone sequence is constructed from the solutions of the difference scheme, and from its convergence the convergence of the difference scheme follows. Below we shall prove in this way the convergence of certain difference schemes. In the examples considered, the solutions of the corresponding problems for the differential equation are discontinuous or nonsmooth functions.

1. Lax’s scheme (1) for a quasilinear equation. We shall prove the convergence of the Lax difference scheme for the solution of the Cauchy problem for \(t \geqslant 0\) for the simplest quasilinear equation

\[ \partial u/\partial t+\partial \varphi(u)/\partial x=0 \tag{1} \]

with the condition

\[ u(0,x)=u_0(x),\quad -\infty<x<+\infty,\quad \text{where } |u_0(x)|\leqslant M. \tag{2} \]

The convergence of this scheme was first established in \((^2)\). The Lax difference scheme has the form:

\[ u_n^{k+1}=\tfrac12\bigl(u_{n+1}^k+u_{n-1}^k\bigr)-\bigl[\varphi(u_{n+1}^k)-\varphi(u_{n-1}^k)\bigr]h/2l, \tag{3} \]

where \(u_n^k=u_{hl}(kh,nl)\), \(h\) and \(l\) are some positive numbers; \(n=\pm1,\pm3,\ldots,\pm(2m+1),\ldots\) for \(k=1,3,5,\ldots\), and \(n=0,\pm2,\pm4,\ldots,\pm2m,\ldots\) for \(k=0,2,4,\ldots\), \(u_n^0=u_0(nl)\).

If \(\varphi(u)\) is an arbitrary smooth function, then the proof is carried out for any bounded monotone function \(u_0(x)\). In the case when \(\varphi''\) preserves its sign, we shall prove the convergence of the scheme also for any bounded \(u_0(x)\), if \(|u_0'(x)|\leqslant M_1\). Using Lemma 5 from \((^3)\), it is then easy to obtain convergence of the scheme also for any bounded measurable \(u_0(x)\).

Writing the difference scheme (3) in the form
\[ u_n^{k+1}=\left(\tfrac12-\varphi' h/2l\right)u_{n+1}^k+ \left(\tfrac12+\varphi' h/2l\right)u_{n-1}^k \]
and assuming that \(|\varphi'(u)|\,h/l<1\) for \(|u|\leqslant M\), we easily obtain that \(|u_n^k|\leqslant M\) and that \(u_{n+1}^k-u_{n-1}^k\leqslant0\) \((\geqslant0)\), if \(u_{n+1}^0-u_{n-1}^0\leqslant0\) \((\geqslant0)\), \(k=1,2,\ldots\). In the case \(\varphi''\geqslant0\) \((\leqslant0)\) it is easy to prove the inequality (see \((^3)\))

\[ z_n^k\equiv (u_n^k-u_{n-2}^k)/2l\leqslant M_1\;(\geqslant M_1). \tag{4} \]

We shall assume that

\[ \int_{-\infty}^{x} |u_0(x)-u_-|\,dx<\infty \]

and that \(u_0(x)\) tends monotonically to \(u_-\) as \(x\to-\infty\). Then there exists

\[ \bar u_n^k=\sum_{m=-\infty}^{n}(u_m^k-u_-)\,2l \]

for every point \((t,x)=(kh,nl)\), and \(\bar u_n^k\) are uniformly bounded for \(x<x_0\), \(0\leq t\leq t_0\). Indeed, the existence of \(\bar u_n^k\) successively for \(k=1,2,\ldots\) follows from the equality

\[ \bar u_n^{k+1}=\frac12(\bar u_{n+1}^k+\bar u_{n-1}^k)-h[\varphi(u_{n+1}^k)-\varphi(u_-)]. \]

This equality can be written in the form

\[ \bar u_n^{k+1}=(1/2-\tilde\varphi' h/2l)\bar u_{n+1}^k+(1/2+\tilde\varphi' h/2l)\bar u_{n-1}^k . \]

From the last expression for \(\bar u_n^{k+1}\) it follows that

\[ |\bar u_n^{k+1}|\leq \max_m |u_0(ml)|, \]

where \(n-(k+1)\leq m\leq n+(k+1)\).

Let \(u_n^k\) be a solution of (3) with steps \(h\) and \(l\) in \(t\) and \(x\), respectively, and let \(v_n^k\) be a solution of the same scheme with steps \(h/2\) and \(l/2\). For convenience we shall use half-integer indices for \(v_n^k\): \(n=0,\pm1,\pm2,\ldots\) for \(k=0,1,2,\ldots\), and \(n=\pm1/2,\pm1^{1}/2,\ldots\) for \(k=1/2,1^{1}/2,\ldots\). We assume that \(\varphi(u_-)=0\). We have

\[ v_n^{k+1}=\frac12[v_{n+1/2}^{k+1/2}+v_{n-1/2}^{k+1/2}] -\varphi(v_{n+1/2}^{k+1/2})-[\varphi(v_{n-1/2}^{k+1/2})]\,[h/2l]. \]

For

\[ \bar v_n^k=\sum_{m=-\infty}^{n}[v_m^k-u_-]\,l \]

we obtain

\[ \begin{aligned} \bar v_n^{k+1} &=\frac12(\bar v_{n+1/2}^{k+1/2}+\bar v_{n-1/2}^{k+1/2}) -\varphi(v_{n+1/2}^{k+1/2})\,h/2 \\ &=\frac12\left(\frac12\bar v_{n+1}^k+\frac12\bar v_{n-1}^k\right) +\frac12\left(\frac12\bar v_{n-1}^k+\frac12\bar v_n^k\right) -\varphi(v_{n+1}^k)\,h/4-\varphi(v_n^k)\,h/4 -\varphi(v_{n+1/2}^{k+1/2})\,h/2 . \end{aligned} \]

Let

\[ W_n^k=\bar u_n^k-\bar v_n^k . \]

For \(W_n^k\) we obtain the equation

\[ \begin{aligned} W_n^{k+1} &=\frac12(W_{n+1}^k+W_{n-1}^k)+\frac14\bar v_{n+1}^k+\frac14\bar v_{n-1}^k-\frac12\bar v_n^k \\ &\quad -h\varphi(u_{n+1}^k)+h\varphi(v_{n+1}^k) -\frac34 h\varphi(v_{n+1}^k)+\frac14 h\varphi(v_n^k) +\frac12 h\varphi(v_{n+1/2}^{k+1/2}) . \end{aligned} \]

Transforming the terms on the right-hand side with the aid of the mean value theorem, we obtain

\[ \begin{aligned} W_n^{k+1} &=(1/2-\tilde\varphi' h/2l)W_{n+1}^k +(1/2+\tilde\varphi' h/2l)W_{n-1}^k \\ &\quad +(v_{n+1}^k-v_n^k)\,l \left[1/4+\varphi_1h/2l-\varphi_2h/4l-\varphi_3h/4l-\varphi_4\varphi_5h^2/4l^2\right], \end{aligned} \]

where \(\tilde\varphi'\) and \(\varphi_i\) \((i=1,\ldots,5)\) denote \(\varphi'(u)\) for certain values of \(u\) not exceeding \(M\) in absolute value. Let \(h/l\) be so small that the expression in the last brackets is nonnegative and less than some number \(M_2\). Suppose that \(\varphi''\geq0\) (the other cases are considered analogously). Using (4), we obtain

\[ W_n^{k+1}-[1/2-\tilde\varphi' h/2l]W_{n+1}^k -[1/2+\tilde\varphi' h/2l]W_{n-1}^k \leq hM_1M_2l^2/h \]

and

\[ |W_n^0|\leq M_3l, \]

if \(u_0(x)\) has bounded variation for \(x\leq x_0\) (\(M_3\) depends on \(x_0\)). For \(nl<x_0-(k+1)l\) it follows from this that

\[ W_n^{k+1}\leq M_3l+(k+1)hM_1M_2l^2/h \]

or

\[ W_n^k\equiv W(kh,nl)\leq C_1l+C_2l^2/h \]

for \(t\leq t_0\) and \(x\leq x_0-t_0l/h\), where \(C_1\) and \(C_2\) are certain constants.

Consequently,

\[ [\bar u_n^k-C_2\,2l-\tfrac12 C_2(2l)^2/h] - [\bar v_n^k-C_1l-\tfrac12 C_2 2l^2/h]\leq0, \]

and the function

\[ Y_n^k\equiv \bar u_n^k-C_1\,2l-\tfrac12 C_2(2l)^2/h, \]

which depends on the mesh steps \(l\) and \(h\), increases monotonically when \(l\) and \(h\) tend to zero as \(l_0/2^s\) and \(h_0/2^s\) as \(s\to+\infty\). Since \(Y_n^k\) are uniformly bounded for \(kh=t\leq t_0\), \(nl=x\leq x_0-t_0h/l\) for arbitrary \(t_0\geq0\) and \(x_0\), it follows that \(Y_n^k\) converge when \(h\) and \(l\) tend to zero. Hence the convergence of \(\bar u_n^k\) follows as \(h\to0\) and \(l\to0\).

Extend \(u_{hl}\) for all \(x\) and \(t\geq0\), putting \(u_{hl}(t,x)=u_n^k\) for \((k-1)h<t\leq kh\), \((n-2)l<x\leq nl\). From the convergence of \(\bar u_n^k\) follows the weak convergence of \(u_{hl}\) on all straight lines \(t=\mathrm{const}\) containing mesh points. Writing (3) in the form

\[ (u_n^{k+1}-u_n^k)/h+[\varphi(u_{n+1}^k)-\varphi(u_{n-1}^k)]/2l =(u_{n+1}^k-2u_n^k+u_{n-1}^k)/2h, \]

it is not difficult

obtain that

\[ \sum_{n=n_1}^{n_2} (u_n^k-u_n^{k-m})\,2l \leqslant C_3mh. \]

Using this, we obtain that the weak convergence of \(u_{hl}\) takes place on all straight lines \(t=\mathrm{const}\geqslant 0\). From the monotonicity with respect to \(n\), or from (4) in the case \(\varphi''\ne0\), compactness of \(u_{hl}\) follows in the sense of convergence almost everywhere. Therefore \(u_{hl}\) converge almost everywhere for \(t\geqslant0\), when \(h,l\to0\).

1. The Rothe scheme for a quasilinear equation. Here, as in § 1, we restrict ourselves to consideration of the simplest equation (1). The differential-difference Rothe scheme for solving the boundary-value problem

\[ u\big|_{t=0}=u_0(x),\qquad u\big|_{x=0}=u_1(t) \tag{5} \]

in the domain \(\{t\geqslant0,\ x\geqslant0\}\) for equation (1) is specified by the equations

\[ du^{n+1}/dt+[\varphi(u^{n+1})-\varphi(u^n)]/l=0, \tag{6} \]

where \(n=0,1,2,\ldots;\ u^n(t)=u_l(t,nl);\ u^0(t)=u_1(t);\ u^n(0)=u_0(nl)\). Let \(|u_0(x)|\leqslant M\), \(|u_1(t)|\leqslant M\), and \(\varphi'(u)>0\) for \(|u|\leqslant M+\varepsilon\), \(\varepsilon>0\). Considering successively \(n=1,2,\ldots\), it is easy to prove that \(|u^n(t)|\leqslant M\) for all \(n\).

We shall assume that \(\varphi''\geqslant0\). We shall prove that in this case

\[ z^n\equiv (u^n-u^{n-1})/l\leqslant M_1\quad\text{for } n=1,2,\ldots \tag{7} \]

Define \(z^n\) for \(n=0\). For this, define \(u^{-1}(t)\) from the equation

\[ \varphi(u^{-1})=\varphi(u_1(t))+l\,du_1/dt. \tag{8} \]

It is obvious that for sufficiently small \(l\) this equation has a solution with respect to \(u^{-1}\), and \(|u^{-1}|\leqslant M+\varepsilon/2\). It is easy to see that \(z^0=u_1'(t)/\varphi'\leqslant M_1\), if \(u_1'(t)\leqslant M_2\) (\(M_1\) and \(M_2\) are certain constants). The function \(z^n\) satisfies the equation

\[ dz^n/dt+\varphi'(u^{n-1})(z^n-z^{n-1})/l +\frac12\widetilde{\varphi''}(z^n)^2 +\frac12\widetilde{\widetilde{\varphi''}}(z^{n-1})^2=0. \tag{9} \]

It follows from this that \(z^n\leqslant M_1\), if \(z^n(0)\leqslant M_1\) and \(z^0(t)\leqslant M_1\), since otherwise we would obtain a contradiction with equation (9) at a maximum point of \(z^n(t)\) for \(t\leqslant t_0\) and \(ln\leqslant x_0\).

Let \(v^n\) be solutions of equations (6) with step \(l/2\) \((n=0,\tfrac12,1,1\tfrac12,\ldots)\) and with the conditions \(v^n(0)=u_0(nl)\), \(v^0(t)=u_1(t)\). Introduce the notation:

\[ \overline{u}^{\,n}=\sum_{m=0,1,2,\ldots}^{n} u^m l,\qquad \overline{v}^{\,n}=\sum_{m=0,\frac12,1,\ldots}^{n} v^m l/2. \]

The function \(W^{n+1}=\overline{u}^{\,n+1}-\overline{v}^{\,n+1}\) satisfies the equation

\[ dW^{n+1}/dt+\widetilde{\varphi'}[W^{n+1}-W^n]/l -\frac12\widetilde{\varphi'}(v^{n+1}-v^{n+1/2}) -[\varphi(u^{-1})-\varphi(v^{-1/2})]. \]

Using (7) and (8), we find that

\[ dW^{n+1}/dt+\widetilde{\varphi'}(W^{n+1}-W^n)/l\leqslant M_3l, \]

where \(M_3\) is a certain constant, \(W^0(t)=0\), \(W^n(0)\leqslant M_4l\) for \(x\leqslant x_0\), if \(|u_0'(x)|\leqslant M_5\). Hence it follows that \(W^n(t)\leqslant M_6l\), and \(\overline{u}^{\,n}-2M_6l\leqslant\overline{v}^{\,n}-M_6l\). Thus, the function \(\overline{u}^{\,n}-2M_6l\) decreases monotonically as \(l\) tends to zero, like \(l_0/2^s\) as \(s\to\infty\). Hence follows the weak convergence of the functions \(u_l\), and also, by virtue of (7), the convergence of \(u_l\) almost everywhere as \(l\to0\). Here we put \(u_l(t,x)=u^n(t)\) for \((n-1)l<x\leqslant nl\).

In an analogous manner one can prove the convergence of the Rothe scheme for problem (1), (5) of the form

\[ [u^{k+1}-u^k]/h+d\varphi(u^{k+1})/dx=0,\qquad u^0(x)=u_0(x),\qquad u^k(0)=u_1(kh). \]

1. A difference scheme for the filtration equation. Consider the first boundary-value problem for the equation

\[ \partial^2u/\partial x^2=\partial\Phi(u)/\partial t \tag{10} \]

in the rectangle \(Q\{0\leqslant t\leqslant T,\ 0\leqslant x\leqslant1\}\) with the conditions \(u(0,x)=u_0(x)\), \(u(t,0)=u_1(t)\), \(u(t,1)=u_2(t)\), where \(\Phi(u)\) is continuous for \(u\geqslant0\) and

\(\Phi'(u)>0\) for \(u>0\), and \(u_0(x), u_1(t), u_2(t)\) are nonnegative functions. (The filtration equation (10) has been studied in detail in \({}^{4}\).) We shall consider the differential-difference equations for this problem in the form

\[ \frac{d^2 u^k}{dx^2}=\bigl[\Phi(u^k)-\Phi(u^{k-1})\bigr]/h, \tag{11} \]

where

\[ u^k(x)=u_h(kh,x),\quad k=1,2,\ldots; \]

\[ u^0(x)=u_0(x),\quad u^k(0)=u_1(kh),\quad u^k(1)=u_2(kh). \tag{12} \]

It is easy to show the existence of solutions \(u^k\) of equations (11), (12) and their uniform boundedness with respect to \(h\). Denote by \(v^k\) the solution of the same difference scheme (11), (12) with step \(h/2\). For convenience we shall use half-integer indices for \(v^k\): \(k=0,\frac12,1,1\frac12,\ldots\).

Introduce the notation: \(\bar u^k=\sum_{m=0}^{k} u^m h\), \(\bar v^k=\sum_{m=0,\;1/2,\ldots}^{k} v^m h/2\). We have \(d^2\bar u^k/dx^2=\Phi(u^k)-\Phi(u^0)+u_0''h\), \(d^2\bar v^k/dx^2=\Phi(v^k)-\Phi(v^0)+u_0''h/2\), \(k=1,2,\ldots\). For \(W^k=\bar u^k-\bar v^k\) we obtain the equation

\[ \frac{d^2 W^k}{dx^2} = \tilde{\Phi}'\,(W^k-W^{k-1})/h +\frac12\tilde{\Phi}'\,(v^{k-1/2}-v^k) +u_0''h/2; \]

\[ W^0=0,\quad |W^k(0)|\le M_1h,\quad |W^k(1)|\le M_2h \tag{13} \]

under the assumption that \(u_1(t)\) and \(u_2(t)\) have bounded variation. If the condition

\[ v^k-v^{k-1/2}\le 0, \tag{14} \]

is fulfilled, then

\[ \frac{d^2 W^k}{dx^2}-\tilde{\Phi}'\,(W^k-W^{k-1})/h \ge u_0''(x)h/2\ge M_3h. \tag{15} \]

From (13) and (15) it follows that \(W^k\le M_4h\), \((\bar u^k-2M_4h)-(\bar v^k-2M_4h/2)\le 0\), i.e. \(\bar u^k-2M_4h\) is a monotonically decreasing function of the step \(h\). Hence, as in paragraphs 1 and 2, the weak convergence of \(u_h\) as \(h\to0\) follows. Using (14), we obtain convergence of \(u_h\) almost everywhere in \(Q\). (We put \(u_h(t,x)=u^k(x)\) for \((k-1)h<t\le kh\).)

The proof of convergence of the scheme (11), (12) is carried out analogously in the case when

\[ v^k-v^{k-1/2}\ge 0. \tag{16} \]

It is easy to show that condition (14) is fulfilled if \(u_0''(x)\le0\), \(u_1(t)\) and \(u_2(t)\) are nonincreasing functions of \(t\), while condition (16) is fulfilled if \(u_0''(x)\ge0\), and \(u_1\) and \(u_2\) are nondecreasing functions of \(t\). For this it is necessary to write the equation for \(z^k=u^k-u^{k-1}\) and use the maximum principle.

We note that the indicated method is applicable for proving the convergence of a wide class of difference and differential-difference schemes possessing the maximum principle.

Moscow State University
named after M. V. Lomonosov

Received
27 X 1960

CITED LITERATURE

\({}^{1}\) P. Lax, Comm. on Pure and Appl. Math., 7, 159 (1954).
\({}^{2}\) N. D. Vvedenskaya, DAN, 111, No. 3 (1956).
\({}^{3}\) O. A. Oleinik, UMN, 12, issue 3 (75), 3 (1957).
\({}^{4}\) O. A. Oleinik, A. S. Kalashnikov, and Chzhou Yun-lin, Izv. AN SSSR, ser. matem., 22, 667 (1958).