MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. N. TIKHONOV and A. A. SAMARSKII

ON CANONICAL HOMOGENEOUS DIFFERENCE SCHEMES

§ 1. In papers ((^{1-3})) homogeneous three-point difference schemes were considered for solving a class of boundary-value problems

[
L^{(k,q,f)}u=\frac{d}{dx}\left[k(x)\frac{du}{dx}\right]-q(x)u+f(x)=0,\qquad 0<x<1,
]

[
u(0)=\mu_1,\qquad u(1)=\mu_2,
\tag{1}
]

[
0<K_1\leq k(x)\leq K_2,\qquad 0\leq q(x)\leq K_2,\qquad |f(x)|\leq K_2,
]

depending on the choice of the coefficients (k(x)), (q(x)), (f(x)) from some functional family.

Let (S_N={x_0=0,\ x_1=h,\ldots,\ x_i=ih,\ldots,\ x_N=Nh=1}) be a uniform difference grid. We shall consider homogeneous three-point difference schemes of the form

[
L_h^{(k,q,f)}y_i=L_h^{(k)}y_i-D_i^{(h,q)}y_i+F_i^{(h,f)},
\tag{2}
]

[
L_h^{(k)}y_i=\frac{1}{h^2}\left[B_i^{(h,k)}(y_{i+1}-y_i)-A_i^{(h,k)}(y_i-y_{i-1})\right].
]

Homogeneity of the scheme means that its coefficients have the form

[
A_i^{(h,k)}=A^h[\bar k_i(s)],\qquad
B_i^{(h,k)}=B^h[\bar k_i(s)],\qquad
\bar k_i(s)=k(x_i+sh),\quad -1<s<1,
]

[
D_i^{(h,q)}=D^h[q(x_i+sh)],\qquad
F_i^{(h,f)}=F^h[f(x_i+sh)],\quad -0.5<s<0.5,
]

where (A^h), (B^h), (D^h), and (F^h) are certain, generally speaking nonlinear, functionals defined on the set of piecewise-continuous functions (Q_0) and depending parametrically on the mesh step (h).

The initial family of difference schemes is determined by specifying the class of functionals by means of which the coefficients of the scheme are computed.

§ 2. We shall specify the class of functionals (A^h[\psi(s)]) by means of the following conditions.

((A_1)). The functional (A^h[\psi]) has a third-order differential with respect to (h), so that

[
A^h[\psi]=A^{(0)}[\psi]+hA^{(1)}[\psi]+h^2A^{(2)}[\psi]+h^3A^{(3)}[\psi]+h^3\rho(h),
]

where (\rho(h)\to0) as (h\to0), and the functional (A^{(m)}[\psi]) has a differential of order (3-m) ((m=0,1,2,3)), so that, for example, for (m=1) one may write

[
A^{(1)}[f+\delta\varphi]=A^{(1)}[f]+\delta A_1^{(1)}[f,\varphi]+\delta^2A_2^{(1)}[f,\varphi]+\delta^2\rho(\delta),
]

where (\rho(\delta)\to0) as (\delta\to0).

((A_2)) (A^h[\psi]) and all the functionals (A^{(m)}[\psi]) ((m=0,1,2,3)) are homogeneous functionals of degree 1:

[
A^h[c\psi]=cA^h[\psi],\quad \text{where } c>0 \text{ is a constant;} \qquad
A^{(m)}[c\psi]=cA^{(m)}[\psi],
]

and (A^h[1]=1).

((A_3)) (A^h[\psi]) and all the functionals (A^{(m)}[\psi]) ((m=0,1,2,3)) are monotonically nondecreasing, i.e.

[
A^h[\psi_2]\geq A^h[\psi_1],\qquad \text{if } \psi_2\geq\psi_1.
]

We shall assume that the functionals (A^h), (B^h), (D^h), and (F^h) satisfy conditions ((A_1)), ((A_2)), ((A_3)), with (D^h) and (F^h) linear.

From conditions ((A_1)) and ((A_2)) it follows that

[
A^h[k(x+sh)]
=
k(x)+hk'(x)A_1^{(0)}[s]
+
h^2\left{
k'(x)A_1^{(1)}[s]
+
\frac{(k'(x))^2}{k(x)}A_2^{(0)}[s]
+
\frac{k''(x)}{2}A_1^{(0)}[s^2]
\right}
+
O(h^3),
\tag{3}
]

where (A^{(m)}[\psi(s)] = A^{(m)}[1,\psi(s)]).

We note that in works (({}^2,{}^3)) a narrower class of functionals was studied and, consequently, a narrower initial family of difference schemes.

Item 3. If the functional (A^h[\psi]) does not depend on the parameter (h), then it is called canonical and is denoted by (A[\psi]). A difference scheme (L_h^{(k,q,f)}), whose coefficients are defined through canonical functionals, is called a canonical scheme.

If the condition

[
B_i^{(h,k)} = A_{i+1}^{(h,k)}, \quad \text{i.e. } \quad
B^h[\psi(s)] = A^h[\psi(s+1)]
\tag{4}
]

is fulfilled for any function from (Q_0), then the difference scheme (L_h^{(k)}) (and (L_h^{(k,q,f)})) is called conservative; it can be written as

[
L_h^{(k)}y_i
=
\frac{1}{h^2}\Delta\left(A_i^{(h,k)}\nabla y_i\right),
\quad
\text{where } \Delta y_i = y_{i+1}-y_i,\quad
\nabla y_i = y_i-y_{i-1}.
\tag{5}
]

From condition (4) it follows that the functional (A^h[\psi(s)]) does not depend on the values of (\psi(s)) for (0<s<1), while (B^h[\psi(s)]) does not depend on the values of (\psi(s)) for (-1<s<0).

Item 4. One of the characteristics of a difference scheme is its integral order of accuracy with respect to (h), i.e., the order of accuracy of the difference (z_i=y_i-u(x_i)) as (h\to 0), where (u(x)) is the solution of problem (1), and (y_i) is the solution of the difference boundary-value problem

[
L_h^{(k,q,f)}y_i=0,\quad 0<i<N,\quad y_0=\mu_1,\quad y_N=\mu_2.
\tag{6}
]

The function (z_i) is determined by the conditions

[
L_h^{(k,q)}z_i=-\varphi_i,\quad z_0=0,\quad z_N=0,
\tag{7}
]

where (\varphi_i=\varphi(x_i,h;u(x_i))).

Recall that the difference (\varphi(x,h;v)=L_h^{(k,q,f)}v-L^{(k,q,f)}v), where (v=v(x)) is any sufficiently smooth function, is called the approximation error of the scheme.

If (v=u(x)) is the solution of the differential equation (1), then we shall speak of the approximation error (\varphi(x,h;u)) on the solution of the differential equation. It may turn out, generally speaking, that the order of the approximation error of the scheme on the solution is higher than the order of approximation on the class of smooth functions (v(x)).

In the present paper we study the order of accuracy of difference schemes of the initial family in the class (C_m) of functions having a continuous derivative of order (m) (({}^1)).

The order of approximation of the scheme (L_h^{(k,q,f)}) is determined by the values of the moments of the functionals (A^h), (B^h), (D^h), and (F^h).

A necessary and sufficient condition for first-order approximation of the scheme (L_h^{(k,q,f)}) has the form

[
B_1^{(0)}[s]-A_1^{(0)}[s]=1.
\tag{8}
]

In order that the scheme have second-order approximation ((\varphi(x,h;v)=O(h^2))), it is necessary and sufficient that the conditions

[
B_1^{(0)}[s] = -A_1^{(0)}[s] = 0.5,\quad
B_1^{(1)}[s]=A_1^{(1)}[s];
\tag{9}
]

[
B_2^{(0)}[s]=A_2^{(0)}[s],\quad
B_1^{(0)}[s^2]=A_1^{(0)}[s^2],\quad
D^{(0)}[s]=F^{(0)}[s]=0.
\tag{10}
]

The conditions (10) for a symmetric scheme are satisfied automatically. Here an essential role is played by normalization conditions, for example (A^h[1]=1), from which it follows that (A^{(0)}[1]=1), (A^{(m)}[1]=0) for (m>0).

Item 5. Theorem 1. In order that the original scheme (L_h^{(k,q,f)}) have (n)-th ((n=1,2)) integral order of accuracy in (C_{m_k,m_q,m_f}={C_{m_k},C_{m_q},C_{m_f}}), (m_k\ge n+1,\ m_q\ge n,\ m_f\ge n), it is necessary and sufficient that it have (n)-th order of approximation.

Theorem 2. Every conservative scheme from the original class has first integral order of accuracy in (C_{m_k,m_q,m_f}) for (m_k\ge2,\ m_q\ge1,\ m_f\ge1).

Theorem 3. Every canonical symmetric conservative scheme has second order of accuracy in (C_{m_k,m_q,m_f}) for (m_k\ge3,\ m_q\ge2,\ m_f\ge2).

Item 6. Along with studying the accuracy of difference schemes for solving a certain class of problems, one must be able to estimate the error permitted when solving, by a given scheme, each individual problem from the class under consideration. Such an accuracy estimate can be achieved by studying the asymptotics of the solution of the difference boundary-value problem as (h\to0). First of all one must find expansions in (h) (asymptotics) of the approximation error (\varphi(x,h;v)=L_hv-Lv).

Here we restrict ourselves to computing the coefficient of the lowest power of (h) for a canonical conservative scheme. Introducing (p(x)=1/k(x)), instead of (L_h^{(k)}) we shall write

[
L_h^{(p)}y_i=\frac{1}{h^2}\Delta\left(\frac{\nabla y_i}{A_i}\right),\qquad
A_i=A[p(x_i+sh)],
]

where (A[\psi]) is the canonical functional. If (L_h^{(p,q,f)}), moreover, is a symmetric scheme, then

[
\varphi(x,h;v)=h^2\Phi(x,v)+O(h^4),\qquad
\Phi(x,v)=\frac{1}{12}\left{\left[\frac{(pL^{(p)}v)'}{p}\right]'\right.
]

[
\left.
{}-12A_2[s]\left[\frac{(p')^2v'}{p^3}\right]'
-6\left[\frac{p''v'}{p^2}\right]'\left(A_1[s^2]-\frac{1}{3}\right)
+6\left(F[s^2]f''-D[s^2]q''v\right)\right},
]

[
L^{(p)}v=\left(\frac{v'}{p}\right)'.
]

In the case of the best canonical scheme (see (3))

[
A[\psi]=\int_{-1}^{0}\psi(s)\,ds,
]

[
D[\psi]=F[\psi]=\int_{-0.5}^{0.5}\psi(s)\,ds
]

we have

[
A_1[s^2]=\frac{1}{3},\qquad A_2[s]=0,\qquad D[s^2]=F[s^2]=\frac{1}{12},
]

and the expression for (\Phi) is greatly simplified.

We shall say that difference schemes (L_h^{(p,q,f)}) and (\overline{L}_h^{(p,q,f)}) are asymptotically equivalent in the sense of approximation if (\Phi(x,v)=\overline{\Phi}(x,v)) and, consequently,

[
\varphi(x,h;v)-\overline{\varphi}(x,h;v)=O(h^4).
]

In our case this means that

[
A_2[s]=\overline{A}_2[s],\qquad
A_1[s^2]=\overline{A}_1[s^2],\qquad
D[s^2]=\overline{D}[s^2],\qquad
F[s^2]=\overline{F}[s^2].
]

In particular, the scheme (\overline{L}_h^{(p)}), for which

[
\overline{A}i=\frac{1}{3}\left(p+p_i\right),}+p_{i-0.5
]

is asymptotically equivalent to the best canonical scheme (L_h^{(p)}), for which

[
A_i=\int_{-1}^{0}p(x_i+sh)\,ds.
]

[
\text{* The differentiability conditions on the coefficients } k(x),\ q(x),\text{ and } f(x)
\text{ can be substantially weakened. This question will be considered in other papers.}
]

p. 7. Using the asymptotics for (\varphi(x,h;v)), it is not difficult to obtain the asymptotic expansion in (h) of the solution of the difference boundary-value problem (6) in the form (y_i=u(x_i)+h^2\tilde z(x_i)+O(h^4)), where (\tilde z(x)) is the function determined from the conditions (L^{(p,q)}\tilde z=-\Phi(x,u)), (\tilde z(0)=\tilde z(1)=0).

p. 8. The study of the asymptotics of the approximation error makes it possible to indicate a method for constructing difference schemes of increased accuracy. Considering, for a scheme (L_h) corresponding to the linear differential operator (L), the asymptotics

[
\varphi(x,h;u)=h^n\Phi u+\cdots
]

and replacing the differential operator (\Phi u) by the difference operator (\Phi_h u), we obtain the difference scheme

[
\tilde L_h u=L_h u-h^n\Phi_h u,
]

which has a higher order of approximation (higher than (n)) on the solution (u=u(x)) of the equation (Lu=0) as compared with the scheme (L_h). It is expedient first to transform the operator (\Phi u) to an operator of lower order, using for this purpose the equation (Lu=0). In this case one can ensure that the operator (\tilde L_h) will have the same domain of definition as the operator (L_h).

In our case, application of this method makes it possible to construct various three-point difference schemes (\tilde L_h^{(p,q,f)}) of increased order of accuracy. Thus, for example, if (L_h^{(p,q,f)}) is the best canonical scheme or a scheme asymptotically equivalent to it, then the three-point homogeneous scheme

[
\begin{aligned}
\tilde L_h^{(p,q,f)} y_i
&=L_h^{(p)}\left(1-\frac{h^2}{12}p_iq_i\right)y_i-\tilde D_i y_i+\tilde F_i \
&=L_h^{(p,q,f)}y_i-\frac{h^2}{12}\left[L_h^{(p)}(p_iq_i y_i-p_i f_i)+\frac12\left(L_h^{(1)}f_i-y_iL_h^{(1)}q_i\right)\right]
\end{aligned}
]

has 4th integral order of accuracy in the class of smooth coefficients.

p. 9. Suppose that at (x=0) a boundary condition of the 3rd kind is prescribed,

[
lu=\frac{u'(0)}{p(0)}-\sigma u(0)=\mu_1.
]

For the corresponding difference boundary operator (l), one can consider the approximation error both on any sufficiently smooth function (v=v(x)) and on the solution of the differential equation (L^{(p,q,f)}u=0). By analogy with p. 8 one can construct a two-point difference boundary operator (\tilde l_h) of arbitrary order of approximation on the solution (u=u(x)). For example, the operator

[
l_hy=\frac{1}{hA_1}(y_1-y_0)-y_0(\sigma+0.5hq_0)+0.5hf_0,\qquad
A_1=A[p(x_1+sh)]
]

has 2nd order of approximation on the solution ((\tilde l_hu-lu=O(h^2))) (cf. with ((^4))).

It can be shown that the solution of the difference boundary-value problem

[
\tilde L_h^{(p,q,f)}y_i=0,\qquad \tilde l_hy_i=\mu_1,\qquad y_N=\mu_2
]

has (n)-th order of accuracy, i.e. (y_i-u(x_i)=O(h^n)), if the operators (\tilde l_h) and (\tilde L_h^{(p,q,f)}) have (n)-th order of approximation on the solution (u=u(x)) of the equation (L^{(p,q,f)}u=0).

Received
31 XII 1959

CITED LITERATURE

({}^1) A. N. Tikhonov, A. A. Samarskii, DAN, 122, No. 4 (1958).
({}^2) A. N. Tikhonov, A. A. Samarskii, DAN, 124, No. 3 (1959).
({}^3) A. N. Tikhonov, A. A. Samarskii, DAN, 124, No. 4 (1959).
({}^4) A. N. Tikhonov, A. A. Samarskii, DAN, 131, No. 3 (1960).