MATHEMATICS

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

ON HOMOGENEOUS DIFFERENCE SCHEMES

In the paper \((^{1})\) the problem was posed of finding difference schemes suitable for a uniform solution of differential equations in as broad a class of coefficients as possible. The present work is a further development of \((^{1})\).

§ 1. Consider the equation

\[ Lu=-f(x) \qquad (0<x<1), \tag{1} \]

where \(L\) is some linear differential operator.

Let \(S_N(x_0=0,\ x_1=h,\ldots,x_i=ih,\ldots,x_N=Nh=1)\) be a difference grid;

\[ L_h y_i^h=-F_i^h \tag{2} \]

is the difference equation corresponding to equation (1).

The linear difference operator \(L_h\) is defined by means of the matrix of coefficients \(a_{ij}^{h}\) of the system of linear equations (2), which are functions of the step of the difference grid \(h=\frac{1}{N}\). To obtain the difference equations (2), it is also necessary to specify the functionals \(F_i^h[f]\), defined in some class \(\{f(x)\}\), and the boundary conditions.

§ 2. Consider the class of equations

\[ L^{(p(x))}u=-f(x). \tag{3} \]

We shall call the functions \((p(x))=\{p_1(x),p_2(x),\ldots,p_m(x)\}\) the coefficients of equation (3).

The class of differential equations (3) will be defined if the type of the operator \(L\) is fixed and the class to which the coefficients \((p(x))\) belong is specified.

Let \(L_h^{(p)}\) denote the class of difference operators \(L_h\) whose matrix elements \(a_{ij}^{h}\) are functionals defined in the class of coefficients \((p(x))\) under consideration and depending on the parameter \(h\). Such a functional matrix \(L_h^{(p)}=(a_{ij}^{h}[p(x)])\) will be called a difference scheme.

§ 3. We introduce the definitions needed in what follows:

1. We shall call \(C_m^\gamma(f)\) the class of functions \(f(x)\) having an \(m\)-th derivative satisfying on the interval \([0,1]\) a Hölder condition of order \(\gamma>0\). If the \(m\)-th derivative is continuous, then we shall denote the corresponding class of functions by \(C_m(f)\). In particular, \(C_0(f)\) is the class of continuous functions.

2. We shall say that \(f(x)\) belongs to the class \(Q_m(f)\) if \(f(x)\) and its \(m\) derivatives are piecewise continuous on \((0,1)\). If, in addition, the \(m\)-th derivative in each of the intervals of continuity satisfies a Hölder condition of order \(\gamma\), then the corresponding class will be called \(Q_m^\gamma\). In particular, \(Q_0\) is the class of piecewise-continuous functions.

1. Let \(u(x)\) be some solution of the equation \(Lu=-f\); let \(y_i^h\) be the corresponding solution of the equation \(L_h y_i^h=-F_i^h\); \(z_i^h=y_i^h-u(x_i)\); \(z(x,h)\) a function equal to \(z_i^h\) for \(x=x_i\) and linear between neighboring nodal points of the grid. We shall say that the difference operator \(L_h\) converges to the differential operator \(L\) if the function \(z(x,h)\) tends uniformly to zero as \(h\to 0\) for an arbitrary function \(f(x)\) from some class, i.e.

\[ |z(x,h)|<\rho(h), \qquad \text{where } \rho(h)\to 0 \text{ as } h\to 0. \]

If \(z(x,h)=O(h^n)\) or \(|z(x,h)|<Mh^n\), where \(M\) is a positive constant depending on the choice of the function \(f(x)\), then we shall say that \(L_h\) has \(n\)-th (integral) order of accuracy relative to \(L\).

1. The difference operator \(L_h\) has \(n\)-th order of approximation relative to the operator \(L\) if there is an \(m\) such that, for any function \(y(x)\) of class \(C_m\), for all values of \(N\) and at all points of the difference grid we have

\[ |L_h y_i-(Ly)_i|<Mh^n, \]

where \(M\) is a positive constant depending on the choice of \(y(x)\). Analogously, one may speak of the order of approximation on some interval \([a,b]\subset[0,1]\).

1. If, for any choice of the coefficients \((p(x))\) from a given functional class, the difference scheme gives a difference operator \(L_h\) converging to the operator \(L\) which corresponds to the chosen coefficients \((p(x))\), then the difference scheme \(L_h^{(p)}\) will be called convergent in the given class of coefficients. Similarly, we shall say that the difference scheme \(L_h^{(p)}\) has \(n\)-th integral order of accuracy (or \(n\)-th order of approximation) in the given class of coefficients if, for any functions \((p(x))\) from this class, the difference operator \(L_h\) has \(n\)-th integral order of accuracy (\(n\)-th order of approximation).

2. The difference schemes \(L_h^{(p)}\) and \(\overline{L}_h^{(p)}\) are equivalent in the sense of convergence in some class of coefficients \((p(x))\) if, for any functions from this class, the difference \(y(x,h)-\overline{y}(x,h)\) tends uniformly to zero as \(h\to 0\).

If \(y(x,h)-\overline{y}(x,h)=O(h^n)\) (or \(L_h^{(p)}y_i-\overline{L}_h^{(p)}y_i=O(h^n)\)) for any function \(p(x)\) from the given class, then the difference schemes \(L_h^{(p)}\) and \(\overline{L}_h^{(p)}\) have \(n\)-th integral (or local) order of equivalence.

It is obvious that:

If \(L_h^{(p)}\) and \(\overline{L}_h^{(p)}\) have \(n\)-th order of accuracy, then they have \(n\)-th integral order of equivalence.

If \(L_h^{(p)}\) and \(\overline{L}_h^{(p)}\) have \(n\)-th integral (or local) order of equivalence and \(L_h^{(p)}\) has \(n\)-th order of accuracy (or \(n\)-th order of approximation), then \(\overline{L}_h^{(p)}\) possesses the same property.

1. We shall call a difference scheme

\[ L_h^{(p)}=\bigl(a_{ij}^h[p(x)]\bigr) \]

a symmetric scheme if the difference operator \(L_h\) remains unchanged under a change in the direction of the \(x\)-axis. The symmetry conditions have the form:

1) \(a_{ij}^h[p(x)]=a_{i,\,2i-j}^h[p(2x_i+x)]\) \((x_i=ih)\);

2) \(0\le (2i-j)h\le 1\).

1. The difference scheme \(L_h^{(p)}\) is called a homogeneous scheme if the elements \(a_{ij}^h\) of the matrix \(L_h\) at all points \(i\) are determined uniformly for all functions \((p(x))\), i.e. are functionals of the form

\[ a_{ij}^h[p(x)]=\overline{a}_{j-i}^h[\overline{p}(s)], \qquad \overline{p}(s)=p(x_i+sh), \qquad -n_1\le j-i\le n_2. \]

If a homogeneous scheme is symmetric, then

1) \(n_1=n_2\);

2) \(a^h_{j-i}[p(x_i+sh)]=a^h_{i-j}[p(x_i-sh)]\).

§ 4. Consider on the interval \(0\le x\le 1\) the first boundary-value problem for the class of equations

\[ L^{(p)}u=\frac{d}{dx}\left[\frac{1}{p(x)}\,\frac{du}{dx}\right]=-f(x)\qquad (0<M_1\le p(x)\le M_2). \tag{4} \]

Let

\[ L_h^{(p)}y_i=\frac{1}{h^2}\left[\frac{1}{A_i^h}\,y_{i-1}+\frac{1}{C_i^h}\,y_i+\frac{1}{B_i^h}\,y_{i+1}\right] \tag{5} \]

be a three-point homogeneous difference scheme whose coefficients are

\[ A_i^h=A^h[p(x_i+sh)],\qquad B_i^h=B^h[p(x_i+sh)],\qquad C_i^h=C^h[p(x_i+sh)], \]

where \(A^h[\bar p(s)]\), \(B^h[\bar p(s)]\), \(C^h[\bar p(s)]\) are functionals of the function \(\bar p(s)\), specified for \(-1<s<1\).

In order that the difference scheme have, in the class \(C_m(p)\) \((m\ge k+1,\ k=1,2)\), \(k\)-th order of approximation, it is necessary and sufficient that the conditions

\[ \frac{1}{A_i^h}+\frac{1}{C_i^h}+\frac{1}{B_i^h}=O(h^{k+2}), \tag{6} \]

\[ \frac{1}{h}\left[\frac{1}{B_i^h}-\frac{1}{A_i^h}\right] =-\frac{p_i'}{p_i^2}+O(h^k),\qquad \frac{1}{2}\left(\frac{1}{B_i^h}+\frac{1}{A_i^h}\right) =\frac{1}{p_i}+O(h^k). \tag{7} \]

Lemma 1. If the difference scheme (5) has \(k\)-th order of approximation, then the scheme

\[ L_h^{(p)}y_i=\frac{1}{h^2}\left[\frac{1}{B_i^h}(y_{i+1}-y_i)-\frac{1}{A_i^h}(y_i-y_{i-1})\right] \tag{8} \]

has the same property.

§ 5. The homogeneous difference scheme (8) will be called \(p\)-linear (or simply linear) if: 1) \(A^h[\bar p]\) and \(B^h[\bar p]\) are linear regular functionals (*); 2) for \(0\le h\le h_0<1\) the representation

\[ \begin{aligned} A^h[\bar p]&=A^{(0)}[\bar p]+hA^{(1)}[\bar p]+h^2A^{(2)}[\bar p]+O_{\bar p}(h^3),\\ B^h[\bar p]&=B^{(0)}[\bar p]+hB^{(1)}[\bar p]+h^2B^{(2)}[\bar p]+O_{\bar p}(h^3), \end{aligned} \tag{9} \]

holds, where \(|O_{\bar p}(h^3)|\le K_{\bar p}h^3\); \(K_{\bar p}\) is a constant depending on the choice of \(\bar p\), and all coefficients of the powers \(h^0\), \(h\), \(h^2\) are linear regular functionals.

The linear difference scheme

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

\[ A_i=A[p(x_i+sh)],\qquad B_i=B[p(x_i+sh)] \tag{10} \]

is called canonical if the functionals \(A[\bar p(s)]\) and \(B[\bar p(s)]\) do not depend on \(h\).

Lemma 2. If a linear difference scheme of the form (8) has \(k\)-th order of approximation \((k=1,2)\), then the corresponding canonical scheme, for which \(A=A^{(0)}\), \(B=B^{(0)}\), also has \(k\)-th order of approximation.

We note that, for a scheme of first order of approximation, the conditions

\[ A[1]=1,\qquad B[1]=1,\qquad B[s]-A[s]=1, \]

must be satisfied, while for a scheme of second order of approximation the conditions are

\[ A[1]=1,\qquad B[1]=1,\qquad A[s]=-0.5,\qquad B[s]=0.5,\qquad A[s^2]=B[s^2]. \]

Lemma 3. If a canonical scheme of first order of approximation is symmetric, then it has second order of approximation.

§ 6. The requirement that \(L_h^{(p)}\) be defined in \(Q_m(p)\) means that \(A_i\ne 0\), \(B_i\ne 0\) at no point of the difference grid for any function \(p\in Q_m\). These conditions will be fulfilled if the functionals \(A\) and \(B\) are positive \(\bigl(A[\bar p]>0,\ B[\bar p]>0\) for \(\bar p(s)>0\bigr)\) (see (2)).

If the canonical scheme \(L_h^{(p)}\) is symmetric and the functionals \(A[\bar p(s)]\), \(B[\bar p(s)]\) are positive, then such a difference scheme is called normal. In what follows we shall consider normal schemes.

The relation between the order of approximation and the order of accuracy is established by the following theorem:

Theorem. Convergence of a normal difference scheme in the sense of approximation is necessary and sufficient for integral convergence; more precisely:

1) If a normal scheme converges in \(C_1'(p)\), then it has first order of approximation in \(C_2(p)\) and, by virtue of symmetry, second order of approximation for \(p(x)\in C_3\).

2) If a normal scheme has second order of approximation in \(C_3(p)\), then it converges in \(C_1'(p)\), has first order of accuracy in \(C_2(p)\), and second order of accuracy in \(C_3(p)\).

Questions concerning convergence and the order of accuracy of normal difference schemes in the class \(Q_m(p)\) will be considered separately.

REFERENCES CITED

¹ A. N. Tikhonov, A. A. Samarskii, DAN, 108, No. 3 (1956).
² A. N. Tikhonov, A. A. Samarskii, DAN, 122, No. 2 (1958).