UDC 517.941.92 : 517.949.22

HOMOGENEOUS DIFFERENCE SCHEMES

ON IRREGULAR NETS FOR A SYSTEM

OF FOURTH-ORDER DIFFERENTIAL EQUATIONS

WITH DISCONTINUOUS COEFFICIENTS

A. K. BOYARCHUK

The paper investigates the numerical solution of a system of ordinary differential equations of the fourth order, having applied significance in mechanics, by means of homogeneous difference schemes on irregular nets. Conservative difference schemes are considered, described in detail and introduced into the literature under this name in [1].

The question of the accuracy of homogeneous difference schemes on irregular nets was first investigated in the work of A. N. Tikhonov and A. A. Samarskii [2]. The method of investigation which A. N. Tikhonov and A. A. Samarskii applied in [2], based on a special way of representing the approximation error, is used essentially in the present work. This method was applied in papers [3] and [4]. In our paper the results of [4] are generalized to the case of a system of fourth-order differential equations with discontinuous coefficients. For convenience of notation and simplicity we use the index-free notation introduced by A. A. Samarskii.

§ 1. DIFFERENCE BOUNDARY-VALUE PROBLEM

1°. Boundary-value problem for a system of differential equations.

Consider the system of differential equations

\[ \sum_{s=1}^{r} \left\{ \frac{d^{2}}{dx^{2}} \left[ k_{ms}(x)\frac{d^{2}U_s}{dx^{2}} \right] - \frac{d}{dx} \left[ p_{ms}(x)\frac{dU_s}{dx} \right] + q_{ms}(x)U_s - f_m(x) \right\} =0 \tag{1.1} \]

\[ (m=1,2,\ldots,r) \]

with homogeneous boundary conditions

\[ U_1(0)=U_2(0)=\ldots=U_r(0)=U'_1(0)=U'_2(0)=\ldots=U'_r(0)=0, \]

\[ U_1(1)=U_2(1)=\ldots=U_r(1)=U'_1(1)=U'_2(1)=\ldots=U'_r(1)=0. \tag{1.2} \]

We write the boundary-value problem (1.1), (1.2) in vector form

\[ L^{(k,p,q,f)}U = \frac{d^{2}}{dx^{2}} \left[ k(x)\frac{d^{2}U}{dx^{2}} \right] - \frac{d}{dx} \left[ p(x)\frac{dU}{dx} \right] + q(x)U-f(x)=0; \tag{1.3} \]

\[ U(0)=U'(0)=U(1)=U'(1)=0, \tag{1.4} \]

where \(k(x), p(x), q(x)\) are symmetric matrices of order \(r\); \(U(x), f(x)\) are \(r\)-dimensional vectors. We assume that the rank of the matrix \(k(x)\), for any fixed \(x \in (0,1)\), is equal to \(r\). We shall say that a matrix (vector) function has a discontinuity at a certain point if the elements of the matrix (components of the vector) are discontinuous at this point. If the functions \(k, p, q\), and \(f\) have a discontinuity of the first kind at some point \(\xi \in (0,1)\), then at this point the vector function \(U(x)\) must satisfy the following conjugation conditions:

\[ [U]=0,\qquad [U']=0,\qquad [kU'']=0,\qquad [(kU'')'-pU']=0, \tag{1.5} \]

where \([\varphi]=\varphi_{\mathrm{r}}-\varphi_{\mathrm{l}}\), \(\varphi_{\mathrm{r}}=\varphi(\xi+0)\), \(\varphi_{\mathrm{l}}=\varphi(\xi-0)\). The boundary-value problem determined by conditions (1.3)—(1.5) will be called problem I.

2°. Notation adopted in the paper:

\[ \begin{gathered} 1)\ \omega_h=\left\{x_0=0,\ x_1=h_1,\ldots,\ x_i=\sum_{s=1}^{i}h_s,\ldots,\ x_N=\sum_{s=1}^{N}h_s=1\right\},\\ \\ h_i=x_i-x_{i-1}=h,\qquad h_{i-1}=h^{(-1)},\qquad h_{i+1}=h^{(+1)},\\ h_{i+2}=h^{(+2)},\qquad h^*=\max h,\\ \overline h=0.5\,(h+h^{(+1)}),\qquad \overline h^{(-1)}=0.5\,(h^{(-1)}+h),\\ \overline h^{(+1)}=0.5\,(h^{(+1)}+h^{(+2)}),\qquad \overline{s h}^{\,*}=h+h^{(+1)}+\cdots+h^{(+s-1)}. \end{gathered} \]

\[ \begin{gathered} 2)\ y^{(+1)}=y_{i+1},\qquad y^{(-1)}=y_{i-1},\qquad y_i=y(x)=y,\\ y_{\bar x}=\frac{y-y^{(-1)}}{h},\\ y_x=\frac{y^{(+1)}-y}{h^{(+1)}},\qquad y_{\hat x}=\frac{y^{(+1)}-y}{\overline h},\\ y_{\mathring x}=\frac{y-y^{(-1)}}{\overline h^{(-1)}},\qquad y_{\circ x}=\frac{y^{(+1)}-y}{\overline h^{(+1)}},\\ y_{\cdot x}=\frac{y^{(+1)}-y}{h^{(+2)}},\qquad y_{\sim x}=\frac{y-y^{(-1)}}{h^{(-1)}}. \end{gathered} \]

\[ \begin{gathered} 3)\ ((y,z))=\sum_{i=2}^{N-2}\overline h_i y_i z_i,\qquad ((y,z)=\sum_{i=2}^{N-1}\overline h_i y_i z,\\ (y,z))=\sum_{i=1}^{N-2}\overline h_i y_i z_i,\qquad (y,z)=\sum_{i=1}^{N-1}\overline h_i y_i z_i. \end{gathered} \]

\[ \begin{gathered} 4)\ \|W\|_0=\max_{\omega_h}|W_i|,\\ |W|=\sqrt{\sum_i |W_i|^2},\qquad \|T\|_k=\sqrt{\sum_{i,j}|t_{ij}|^2},\\ \|\Psi\|_1=\left(\left(\frac{|\nabla\eta|}{h},1\right)+|\eta_{\bar x,N}|\right),\qquad \|\Psi\|_2=(|\chi|,1)+|\chi_{N-1}|\,h_N, \end{gathered} \]

where

\[ \nabla \eta = h \sum_{s=1}^{i-1} \bar h_s a_s^{-1}\Psi^{(a)}_s;\qquad \varkappa = \sum_{s=2}^{i-1}\bar h_s\Psi_s^{*}\sum_{k=s+1}^{i}h_k; \]

\(a_s^{-1}\) is the matrix inverse to the matrix \(a_s\);

\[ \|\Psi\|_3 =|\Psi_{n-1}|\bar h_{n-1}h_n\bar h_n +|2\Psi_{n-1}\bar h_{n-1}\bar h_n+\Psi_n\bar h_n h_{n+1}|\bar h_{n+1} \]
\[ \quad +|3\Psi_{n-1}\bar h_{n-1}\bar h_n^{*} +2\Psi_n\bar h_n h_{n+1} +\Psi_{n+1}\bar h_{n+1}h_{n+2}| +|\Psi_{n-1}\bar h_{n-1}+\Psi_n\bar h_n+\Psi_{n+1}\bar h_{n+1} +\Psi_{n+2}\bar h_{n+2}|. \]

5) \(C^{(m)}_{[0,1]}(Q^{(m)}_{[0,1]})\) is the class of functions continuous (piecewise continuous) on \([0,1]\), together with their derivatives up to order \(m\) inclusive.

6) \(Q^{(m,1)}\) is the class of functions from \(Q^{(m)}_{[0,1]}\) whose \(m\)-th derivatives satisfy, on intervals of continuity, a Lipschitz condition; \(C^{(m,1)}\) is the class of functions defined analogously to the class \(Q^{(m,1)}\).

7) \(\rho(h)\) is a quantity tending to zero as \(h\to 0\).

3°. Homogeneous difference schemes for equation (1.3). We consider homogeneous difference schemes of the form

\[ L_h^{(k,p,q,f)}y=L_h^{(k,p,q)}y-\Phi, \tag{1.6} \]

where

\[ L_h^{(k,p,q)}y=(a y_{\bar x\hat x})_{\bar x\hat x}-(b y_{\bar x})_{\hat x}+dy, \quad (a=(a^{ml})_{rr},\ b=(b^{ml})_{rr},\ d=(d^{ml})_{rr} \]

are symmetric matrices of order \(r\); \(y=(y^m)^r\), \(\Phi=(\Phi^m)^r\) are \(r\)-dimensional vectors.

We shall assume that (see [5])

\[ \sum_{m,l} a^{ml}\xi_m\xi_l \ge \beta\sum_m \xi_m^2,\qquad \beta>0; \]

\[ \sum_{m,l} b^{ml}\xi_m\xi_l \ge \gamma\sum_m \xi_m^2,\qquad \gamma\ge 0;\qquad \sum_{m,l} d^{ml}\xi_m\xi_l \ge \delta\sum_m \xi_m^2,\qquad \delta\ge 0, \]

where \(\{\xi_m\}\) are arbitrary real numbers; \(\beta,\gamma\), and \(\delta\) are real constants independent of the mesh. The elements of the matrices \(a,b,d\) and the components of the vector \(\Phi\) are determined by means of template functionals given in the class \(Q^{(0)}\)

\[ \bar A^h[\varphi(s)],\qquad -1\le s\le 1,\qquad B^h[\varphi(s)],\qquad -1\le s\le 0, \]

\[ \bar D^h[\varphi(s)],\qquad -\frac12\le s\le \frac12,\qquad F^h[\varphi(s)],\qquad -\frac12\le s\le \frac12, \]

according to the formulas

\[ a^{ml}=A^{(k_{ml};\bar h)}=\bar A^h(\bar k_{ml}(s)),\qquad \bar k_{ml}(s)=k_{ml}(x+s\bar h), \]

\[ b^{ml}=B^{(p_{ml};h)}=B^h[\bar p_{ml}(s)],\qquad \bar p_{ml}(s)=p_{ml}(x+sh), \tag{1.7} \]

\[ d^{ml}=D^{(q_{ml};h)}=\bar D^h[\bar q_{ml}(s)],\qquad \bar q_{ml}(s)=q_{ml}(x+s\bar h), \]

\[ \Phi^m=F^{(f_m;\bar h)}=\bar F^h[\bar f_m(s)],\qquad \bar f_m(s)=f_m(x+s\bar h) \]

\[ (m,l=1,2,\ldots,r). \]

To characterize the class of template functionals one must use the notion, introduced in [1], of the rank of a functional. The functionals \(\bar A^h[\varphi(s)]\), \(B^h[\varphi(s)]\), \(\bar D^h[\varphi(s)]\), and \(\bar F^h[\varphi(s)]\) belong to the class of template functionals considered in detail in [1], and therefore we shall not give their characteristics here.

4°. The approximation error of the scheme. The difference scheme is characterized by the vector approximation error on the solution of equation (1.3)

\[ \Psi=L^{(k,p,q,f)}U-L_h^{(k,p,q,f)}U. \tag{1.8} \]

Theorem 1. The approximation error of any scheme \(L_h^{(k,p,q,f)}\) for \(k, p, q, f \in C^{(2,1)}\) can be represented in the form

\[ \Psi = \Psi_{\widehat{x}\widehat{x}}^{(a)} + \Psi_x^{(b)} + \Psi^*, \tag{1.9} \]

where

\[ \Psi^{(a)} = kU'' - aU_{\widehat{x}\widehat{x}};\qquad \Psi^{(b)} = bU_x - \overline{pU'};\qquad \Psi^* = (q-d)U+ \]

\[ + \Phi - f + \frac{h^{(+1)}-h}{3}\bigl[(pU')' - qU + f\bigr] + \frac{h^{(+1)}-h}{4}(pU')'' + \gamma, \]

\[ \overline{pU'} = pU'\bigm|_{x=\overline{x}},\qquad \overline{x}=x-\frac{h}{2},\qquad \|\gamma\|^0 = O(\overline{h}^{\,2}),\quad aU=U(x) \]

is the solution of the vector equation \(L^{(k,p,q,f)}U=0\).

The proof is omitted.

Let \(L_h^{(k,p,q,f)}\) be a second-order scheme. Expanding \(a^{ml}\) in powers of \(\overline{h}\), we obtain

\[ a^{ml}=k_{ml}+\overline{h}\,k'_{ml}A_1^{(0)ml}[s]+O(\overline{h}^{\,2}), \]

\[ a=k+\overline{h}\,(k_{ml}A_1^{(0)ml}[s])_{rr}+\delta^{(1)},\qquad \|\delta^{(1)}\|_k=O(\overline{h}^{\,2}), \]

\[ \Psi^{(a)} = -\,\frac{h^{(+1)}-h}{3}\,kU''' -\overline{h}\,(k'_{ml}A_1^{(0)ml}[s])_{rr}U'' +\delta^{(2)},\qquad \|\delta^{(2)}\|_0=O(\overline{h}^{\,2}). \tag{1.10} \]

Similarly,

\[ \Psi^{(b)} =\overline{h}\left(\overline{p}'_{ml}\left(B_1^{(0)ml}[s]+\frac12\right)\right)_{rr}\overline{U}' +\delta^{(3)},\qquad \|\delta^{(3)}\|_0=O(\overline{h}^{\,2}); \tag{1.11} \]

\[ \Psi^* =-\overline{h}\,(q_{ml}'D_1^{(0)ml}[s])_{rr}U +\overline{h}\,(f'_mF_1^{(0)m}[s])^{r} +\varepsilon+\delta_1^{(3)}, \]

\[ \|\varepsilon\|_0=O(h^{(+1)}-h),\qquad \|\delta_1^{(3)}\|_0=O(\overline{h}^{\,2}). \tag{1.12} \]

We shall say that the difference scheme \(L_h^{(k,p,q,f)}\) has approximation order \(h^{(+1)}-h+O(\overline{h}^{\,2})\) if the conditions are satisfied:
\(\Psi^{(a)}=2(h^{(+1)}-h)\varphi+\gamma^{(1)}\),
\(\Psi^{(b)}=\gamma^{(2)}\),
\(\Psi^*=2(h^{(+1)}-h)\omega+\gamma^{(3)}\), where \(\varphi(x)\) and \(\omega(x)\) are some vector-functions differentiable on \((0,1)\);
\(\|\gamma^{(s)}\|_0=O(\overline{h}^{\,2})\) \((s=1,3)\),
\(\|\gamma^{(2)}\|_0=O(\overline{h}^{\,2})\).

Theorem 2. Necessary conditions for approximation order \(h^{(+1)}-h+O(\overline{h}^{\,2})\) for the second-order difference scheme \(L_h^{(k,p,q,f)}\) in the class \(k, p, q, f\in C^{(2,1)}\) have the form

\[ A_1^{(0)ml}[s]=0,\qquad B_1^{(0)ml}[s]+\frac12=0, \]

\[ D_1^{(0)ml}[s]=0,\qquad F_1^{(0)m}[s]=0 \tag{1.13} \]

\[ (m,l=1,2,\ldots,r). \]

5°. Difference boundary-value problem. To boundary-value problem I we put in correspondence the difference boundary-value problem

\[ L_h^{(k,p,q)}y=\Phi,\qquad x_1<x<x_{N-1}; \tag{1.14} \]

\[ y_1=0,\qquad y_{N-1}=0; \tag{1.15} \]

\[ y_{\overline{x},1}-\left(\frac{h}{2}y_{\overline{x}\widehat{x}}\right)_1=0,\qquad y_{\overline{x},N}+\left(\frac{h}{2}y_{\overline{x}\check{x}}\right)_N=0. \tag{1.16} \]

The coefficients of equation (1.14) are computed by formulas (1.7). We shall call problem (1.14)—(1.16) problem II.

6°. Difference boundary-value problem for the error. Let \(U=(U_m)^r\) and \(y=(y^m)^r\) be, respectively, the solutions of problems I and II. Form the vector \(z=y-U\), which we shall call the error vector of the solution. For the vector \(z\) we obtain the difference boundary-value problem

\[ L_h^{(k,p,q)}z=\Psi,\qquad x_1<x<x_{N-1}; \tag{1.17} \]

\[ z_1=\nu_1,\qquad z_{N-1}=\mu_1; \tag{1.18} \]

\[ z_{\bar x,1}-\left(\frac{h}{2}z_{\bar x\hat x}\right)_1=\nu_2,\qquad z_{\bar x,N}+\left(\frac{h}{2}z_{\bar x\hat x}\right)_N=\mu_2, \tag{1.19} \]

where \(\Psi\) is the vector error of approximation on the solution of equation (1.3);

\[ \nu_1=-U_1;\qquad \mu_1=-U_{N-1};\qquad \nu_2=-\left[\,U_{\bar x,1}-\left(\frac{h}{2}U_{\bar x\hat x}\right)_1\,\right];\qquad \mu_2=-\left[\,U_{\bar x,N}+\left(\frac{h}{2}U_{\bar x\hat x}\right)_N\,\right]. \]

7°. Difference Green formulas. Let \(V\) and \(W\) be arbitrary \(r\)-dimensional vectors. We multiply \(V\) scalarly by \(L_h^{(k,p,q)}W\bar h\) and sum over the mesh \(\omega_h\). We obtain the first difference Green formula

\[ \begin{aligned} ((V,L_h^{(k,p,q)}W))={}&(V_{\bar x\hat x},\,aW_{\bar x\hat x}) +\left(\left(\frac{h}{\bar h}V_{\bar x},\,bW_{\bar x}\right)\right)\\ &+\{V^{(-1)}(a^{(-1)}W_{\bar x\hat x})_{\hat x} -V_{\bar x}(a^{(-1)}W_{\bar x\hat x})\}_N\\ &-\{V^{(+1)}(a^{(+1)}W_{x\hat x})_{\dot x} -V_x(a^{(+1)}W_{x\hat x})\}_0\\ &-V_{N-1}(bW_{\bar x})_{N-1} +V_1(bW_{\bar x})_2 . \end{aligned} \tag{1.20} \]

From formula (1.20) we obtain the second difference Green formula

\[ \begin{aligned} ((V,L_h^{(k,p,q)}W))-((W,L_h^{(k,p,q)}V))={}& \{[V^{(-1)}(a^{(-1)}W_{\bar x\hat x})_{\hat x} -V_{\bar x}(a^{(-1)}W_{\bar x\hat x})]\\ &\quad-[W^{(-1)}(a^{(-1)}V_{\bar x\hat x})_{\hat x} -W_{\bar x}(a^{(-1)}V_{\bar x\hat x})]\}_N\\ &-\{[V^{(+1)}(a^{(+1)}W_{x\hat x})_{\dot x} -V_x(a^{(+1)}W_{x\hat x})]\\ &\quad-[W^{(+1)}(a^{(+1)}V_{x\hat x})_{\dot x} -W_x(a^{(+1)}V_{x\hat x})]\}_0\\ &-[V_{N-1}(bW_{\bar x})_{N-1} -W_{N-1}(bV_{\bar x})_{N-1}]\\ &+V_1(bW_{\bar x})_2-W_1(bV_{\bar x})_2 . \end{aligned} \tag{1.21} \]

8°. Difference Green vector-functions. Introduce the \(r\)-dimensional vectors \(l_s\) \((s=1,2,\ldots,r)\), whose \(s\)-th components are equal to \(\delta(x,\xi)\), and all the remaining components are zero, as well as the vectors \(\bar l_s\) \((s=1,2,\ldots,r)\), obtained from \(l_s\) for \(x=\xi\), where

\[ \delta(x,\xi)= \begin{cases} 1, & x=\xi,\\ 0, & x\ne \xi. \end{cases} \]

We introduce the difference Green vector-functions, defined by the conditions:

1) the Green’s functions \(G^s(x,\xi)\) \((s=1,2,\ldots,r)\) satisfy the equations

\[ L_h^{(k,p,q)}G^s=\frac{l_s}{h},\qquad x_1<x<x_{N-1}; \tag{1.22} \]

2) the functions \(G^s(x,\xi)\) \((s=1,2,\ldots,r)\) satisfy the difference boundary conditions

\[ G^s(h_1,\xi)=0,\qquad G^s(1-h_N,\xi)=0; \tag{1.23} \]

\[ G^s_{\bar{x},1}-\left(\frac{h}{2}G^s_{\bar{x}\hat{x}}\right)_1=0,\qquad G^s_{\bar{x},N}+\left(\frac{h}{2}G^s_{\bar{x}\hat{x}}\right)_N=0. \tag{1.24} \]

With the aid of the functions \(G^s\), we write the solution of problem II in the form

\[ y^s=((G^s(x,\xi),\Phi(\xi))). \tag{1.25} \]

We seek the functions \(G^s\) in the form

\[ G^s(x,\xi)= \begin{cases} \overline{G}^{\,s}(x,\xi)=\alpha_s^{(1)}(\xi)V^{(1)s}(x)+\alpha_s^{(2)}(\xi)V^{(2)s}(x), & x<\xi,\\[2mm] \overline{\overline{G}}^{\,s}(x,\xi)=\alpha_s^{(3)}(\xi)V^{(3)s}(x)+\alpha_s^{(4)}(\xi)V^{(4)s}(x), & x>\xi \end{cases} \]

\[ (s=1,2,\ldots,r), \]

where \(\alpha_s^{(j)}\) \((j=1,2,3,4)\) are diagonal matrices of order \(r\); \(V^{(j)s}(x)\) \((j=1,2,3,4)\) are \(r\)-dimensional vectors satisfying the equation \(L_h^{(k,p,q)}\times\)
\(\times V^{(j)s}=0\), and also the conditions:

\[ V_1^{(1)s}=0,\qquad V_1^{(2)s}=0, \]

\[ V_{\bar{x},1}^{(1)s}-\left(\frac{h}{2}V_{\bar{x}\hat{x}}^{(1)s}\right)_1=0,\qquad V_{\bar{x},1}^{(2)s}-\left(\frac{h}{2}V_{\bar{x}\hat{x}}^{(2)s}\right)_1=0, \]

\[ (aV_{\bar{x}\hat{x}}^{(1)s})_1=\overline{l}_s,\qquad (aV_{\bar{x}\hat{x}}^{(2)s})_1=0, \]

\[ [(aV_{\bar{x}\hat{x}}^{(1)s})_{\bar{x}}]_1=0,\qquad [(aV_{\bar{x}\hat{x}}^{(2)s})_{\bar{x}}]_1=\overline{l}_s, \]

\[ V_{N-1}^{(3)s}=0,\qquad V_{N-1}^{(4)s}=0, \]

\[ V_{\bar{x},N}^{(3)s}+\left(\frac{h}{2}V_{\bar{x}\hat{x}}^{(3)s}\right)_N=0,\qquad V_{\bar{x},N}^{(4)s}+\left(\frac{h}{2}V_{\bar{x}\hat{x}}^{(4)s}\right)_N=0, \]

\[ (aV_{\bar{x}\hat{x}}^{(3)s})_{N-1}=\overline{l}_s,\qquad (aV_{\bar{x}\hat{x}}^{(4)s})_{N-1}=0, \]

\[ [(aV_{\bar{x}\hat{x}}^{(3)s})_{\bar{x}}]_{N-1}=0,\qquad [(aV_{\bar{x}\hat{x}}^{(4)s})_{\bar{x}}]_{N-1}=\overline{l}_s. \]

The functions \(\overline{G}^{\,s}\) and \(\overline{\overline{G}}^{\,s}\) must satisfy the conditions:

\[ \overline{G}^{\,s}=\overline{\overline{G}}^{\,s}\ \text{at }x=\xi,\qquad \overline{G}^{\,s}_{\bar{x}}=\overline{\overline{G}}^{\,s}_{\bar{x}}\ \text{at }x=\xi,\qquad a\overline{G}^{\,s}_{\bar{x}\hat{x}}=a\overline{\overline{G}}^{\,s}_{\bar{x}\hat{x}}\ \text{at }x=\xi, \]

\[ (a\overline{G}^{\,s}_{\bar{x}\hat{x}})_{\bar{x}}-(a\overline{\overline{G}}^{\,s}_{\bar{x}\hat{x}})_{\bar{x}}=-\overline{l}_s\ \text{at }x=\xi. \]

Thus, to determine the elements of the matrices \(\alpha_s^{(j)}\) \((j=1,2,3,4)\), for each fixed \(s\) we obtain \(r\) systems of algebraic equations of the fourth order. Systems of this type are considered in [4].

Lemma. Suppose a system of difference vector equations is given

\[ V_{\bar x}^{(s)}=\sum_{j=1}^{4} a^{(s)j} V^{(s)}, \qquad x_1 \leq x \leq x_{N-1}\quad (s=1,2,3,4), \tag{1.26} \]

the coefficient matrices of which satisfy the conditions \(\|a^{(s)j}\|_k \leq M\), where \(M\) is some positive constant. There exists one and only one system of solutions of equations (1.26), \(V^{(s)}\) \((s=1,2,3,4)\), taking at \(x=0\) the prescribed initial values \(V_0^{(s)}\) \((s=1,2,3,4)\) and satisfying the conditions \(|V^{(s)}|\leq M_1\), \(h\leq h^*\leq h_0\), where \(M_1>0\) and \(h_0>0\) are constants independent of \(h\).

The proof of the lemma is no different from the proof given in [4].

Applying the lemma, we find estimates, uniform in \(h\), for the difference Green vector-functions \(G^s\) and their difference derivatives:

\[ |G^s|\leq M_s,\qquad |G_{\bar x}^s|\leq M_s,\qquad |aG_{\bar x\hat x}^{\,s}|\leq M_s,\qquad |aG_{\bar x\hat x}^{\,s}|_{\bar x}\leq M_s, \]

where \(M_s>0\) are constants independent of \(h\). With the aid of the second difference Green formula, the symmetry property of the functions \(G^s\) is easily proved:

\[ G^s(x,\xi)=G^s(\xi,x). \]

9°. A priori estimates.

Lemma 1. If \(\Psi\) has the form (1.9), then for the components of the vector \(z\) which is the solution of problem (1.17)—(1.19), the estimates

\[ |z^s|\leq M_s'\left\{\|\Psi^{(a)}\|_1+\|\Psi^*\|_2+\|\Psi^{(b)}\|_3\right\}, \tag{1.27} \]

hold, where

\[ \|\Psi^{(b)}\|_3 \left(\left(\frac{h}{\hbar}|\Psi^{(b)}|,1\right) +|\Psi_{N-1}^{(a)}|h_N+|\Psi_1^{(a)}|h_1 +|\mu_1|+|\mu_2|+|\nu_1|+ |\nu_2|\right); \qquad M_s'>0 \]

are constants independent of \(h\).

Proof. With the aid of the functions \(G^s\), we represent the solution of problem (1.17)—(1.19) in the form

\[ z^s=((G^s(x,\xi),\Psi(\xi)))+\omega, \tag{1.28} \]

where

\[ \omega=\mu_1(aG_{\bar x\hat x}^{\,s})_{\bar x,N-1} -\mu_2(aG_{\bar x\hat x}^{\,s})_{N-1} -\nu_1(aG_{\bar x\hat x}^{\,s})_{\bar x,2} + \]

\[ +\nu_2(aG_{\bar x\hat x}^{\,s})_1 -\mu_1(bG_{\bar x}^{\,s})_{N-1} +\nu_1(bG_{\bar x}^{\,s})_2 . \]

Taking (1.9) into account, we obtain

\[ z^s=(G_{\bar x\hat x}^{\,s},\Psi^{(a)}) +G_{\bar x,1}^{\,s}\Psi_1^{(a)} -G_{\bar x,N}^{\,s}\Psi_{N-1}^{(a)}- \]

\[ -\left(\frac{h}{\hbar}G_{\bar x}^{\,s},\Psi^{(b)}\right) +((G^s,\Psi^*))+\omega . \tag{1.29} \]

Introducing into consideration the difference vector-functions \(\eta\) and \(\varkappa\), defined from the conditions

\[ a\eta_{\bar x\hat x}=\Psi^{(a)}, \qquad x_1\leq x\leq x_{N-1},\qquad \eta_0=\eta_1=0, \tag{1.30} \]

\[ \varkappa_{\bar x\hat x}=\Psi^*, \qquad x_1<x<x_N,\qquad \varkappa_1=\varkappa_2=0, \tag{1.31} \]

and applying the summation-by-parts formulas, we obtain estimate (1.27).

Lemma 2. Let \(z\) be the solution of problem (1.17)—(1.19), and suppose that the vector \(\Psi\) has the form (1.9) at all nodes of the grid, except for \(x_{n-1}, x_n, x_{n+1}, x_{n+2}\). Then, for the components of the vector \(z\) at all nodes of the grid \(\omega_h\), the estimates

\[ |z^s| \leq M_s''\{\, \|\Psi^{(a)}\|_1' + \|\Psi^{(b)}\|_1' + \|\Psi^*\|_1' + \|\Psi\|_3 + |\mu_1|+ \]
\[ +|\mu_2|+|\nu_1|+|\nu_2|\,\}, \tag{1.32} \]

hold, where

\[ \|\Psi^{(a)}\|_1' = \sum_{j=1}^{n-1}|\Psi_j^{(a)}|\overline h_j + \sum_{j=n+2}^{N-1}|\Psi_j^{(a)}|\overline h_j + \]
\[ +|\Psi_{n-1}^{(a)}|+|\Psi_{x,n-1}^{(a)}|+|\Psi_{x,n+3}^{(a)}|, \]

\[ \|\Psi^{(b)}\|_1' = \sum_{j=2}^{n-2}|\Psi_j^{(b)}|h_j + \sum_{j=n+4}^{N-1}|\Psi_j^{(b)}|h_j + |\Psi_{n-1}^{(b)}|+|\Psi_{n+3}^{(b)}|, \]

\[ \|\Psi^*\|_1' = \sum_{j=2}^{n-2}|\Psi_j^*|\overline h_j + \sum_{j=n+3}^{N-2}|\Psi_j^*|\overline h_j . \]

Proof. Representing \(z^s=\overline z^{\,s}+\overline{\overline z}^{\,s}\), where

\[ \overline z^{\,s} = \sum_{\xi=x_2}^{x_{n-2}}\overline h\,G^s(x,\xi)\Psi(\xi) + \sum_{\xi=x_{n+3}}^{x_{N-2}}\overline h\,G^s(x,\xi)\Psi(\xi)+\omega, \]

\[ \overline{\overline z}^{\,s} = \sum_{\xi=x_{n-1}}^{x_{n+2}}\overline h\,G^s(x,\xi)\Psi(\xi), \]

and defining the function \(\varphi\) from the conditions

\[ \varphi_{\hat x\hat x}=\Psi,\quad x=x_{n-1},x_n,x_{n+1},x_{n+2},\quad \varphi(x_{n-2})=\varphi(x_{n-1})=0, \]

with the aid of the summation-by-parts formulas we obtain estimate (1.32).

\[ \bullet \]

§ 2. On the accuracy of the solution in the classes \(C^{(2,1)}\) and \(Q^{(2,1)}\)

1°. Estimate of the error of the solution in the class \(C^{(2,1)}\).

Theorem 3. If \(L_h^{(k,p,q,f)}\) is a scheme of second order, for which the conditions (1.13) are satisfied, then for the error vector \(z\) of the solution at each nodal point of the grid \(\omega_h\), in the class \(k,p,q,f\in C^{(2,1)}\), the estimate

\[ \|z\|_0 \leq M'''(h^*)^2, \tag{2.1} \]

is valid, where \(M'''>0\) is a constant independent of \(h\).

The proof of the theorem follows from Lemma 1.

2°. Sufficient conditions for convergence in the class \(Q^{(2,1)}\). If \(\xi\in(x_n,x_{n+1})\) is a point of discontinuity of the functions \(k,p,q,f\), then \(|\Psi|\) at the nodes \(x_{n-1},x_n,x_{n+1},x_{n+2}\) is a quantity of order \(\dfrac{1}{h^2}\). However, this circumstance does not exclude the possibility of convergence of the solution of the difference boundary-value problem II to the solution of problem I. We have no possibility here to dwell in detail on the determination of sufficient conditions for convergence.

It can be shown, using the properties of stencil functionals, that if the conditions

\[ h_{n-2}=h_{n-1}=h_n,\qquad h_{n+2}=h_{n+3}=h_{n+4} \tag{2.2} \]

are satisfied, then any difference scheme \(L_h^{(k,p,q,f)}\) from the class under consideration ensures convergence of the solution of problem II to the solution of problem I as \(h^*\to 0\). Conditions (2.2) are also sufficient conditions for convergence in the class of piecewise-continuous functions.

The convergence conditions can be weakened. To this end, instead of the grid \(\omega_h\), we construct a quasi-uniform grid (see [3]). On a quasi-uniform grid the conditions (2.2) are satisfied automatically.

3°. On the accuracy in the class of discontinuous functions.

Theorem 4. If \(L_h^{(k,p,q,f)}\) is a second-rank scheme for which the conditions (1.13) are satisfied, and the grid \(\omega_h\) is chosen so that the conditions (2.2) are satisfied, or else it is quasi-uniform, then the solution of the difference boundary-value problem II in the class \(k,p,q,f\in Q^{(2,1)}\) has first order of accuracy:

\[ \|z\|_0\leq \overline{M}h^*, \tag{2.3} \]

where \(\overline{M}>0\) is a constant independent of \(h\).

The proof follows from Lemma 2.

4°. The case of a special choice of the grid. Choose the grid \(\omega_h\) so that one of its nodal points coincides with a point of discontinuity of the functions \(k,p,q\), and \(f\).

Theorem 5. Let \(L_h^{(k,p,q,f)}\) be any second-rank difference scheme for which the conditions (1.13) are satisfied, and let the grid \(\omega_h\) be chosen so that one of its nodal points \(x_n\) coincides with a point of discontinuity of first kind of the functions \(k,p,q\), and \(f\). Then, if the condition
\[ h_{n-2}=h_{n-1}=h_n=h_{n+1}=h_{n+2}=h_{n+3} \]
is satisfied, the solution of the difference boundary-value problem II in the class \(k,p,q,f\in Q^{(2,1)}\) has second order of accuracy:

\[ \|z\|_0\leq \widetilde{M}(h^*)^2, \tag{2.4} \]

where \(\widetilde{M}>0\) is a constant independent of \(h\).

We omit the proof of the theorem. Let
\[ 0<C_1\leq k\leq C_2,\qquad 0\leq p\leq C_3,\qquad 0\leq q\leq C_4,\qquad |f|\leq C_5. \]
The scheme with coefficients

\[ a^{ml}= \left( \int_0^1 \frac{(1-s)\,ds}{k_{ml}(x+s\bar h)} + \int_0^1 \frac{(1-s)\,ds}{k_{ml}(x-s\bar h)} \right)^{-1}, \]

\[ b^{ml}=\int_{-1}^{0} p_{ml}(x+s\bar h)\,ds, \]

\[ d^{ml}=\int_{-\frac12}^{\frac12} q_{ml}(x+s\bar h)\,ds, \qquad \Phi^m=\int_{-\frac12}^{\frac12} f_m(x+s\bar h)\,ds \]

under the conditions (2.2), with the additional condition \(h_{n+1}=h_n\), ensures in the class \(Q^{(2,1)}\) second order of accuracy for any choice of the grid.

In conclusion I express my sincere gratitude to Prof. A. A. Samarskii for the formulation of the problem, valuable advice and attention, without which this work could not have been written.

References

1. A. N. Tikhonov, A. A. Samarskii. Journal of Computational Mathematics and Mathematical Physics, 1, No. 1, 5–63, 1961.
2. A. N. Tikhonov, A. A. Samarskii. Journal of Computational Mathematics and Mathematical Physics, 2, No. 5, 812–832, 1962.
3. A. A. Samarskii. Journal of Computational Mathematics and Mathematical Physics, 3, No. 3, 431–466, 1963.
4. S. L. Hao. Journal of Computational Mathematics and Mathematical Physics, 3, No. 5, 841–860, 1963.
5. A. A. Samarskii. Journal of Computational Mathematics and Mathematical Physics, 1, No. 6, 972–1000, 1961.

Received by the editors
December 1, 1965

Taras Shevchenko Kyiv State University