Full Text
“Co-stability” of Conservative Difference Schemes for a Differential Equation and a System of Differential Equations of the Fourth Order with Discontinuous Coefficients
A. K. BOYARCHUK
The definition of a coefficient-stable (co-stable) difference scheme was given by A. N. Tikhonov and A. A. Samarskii in [1]. It arose in connection with the fact that, when solving a differential equation by the finite-difference method, the information about the coefficients of this equation may be insufficiently complete (for example, when these coefficients are determined from experiment or with the aid of some computational algorithm), as a result of which errors will be admitted even in the exact computation of the coefficients of the difference equation. In addition, the coefficients of the difference equation themselves may be computed approximately. Therefore, the question of studying difference schemes with perturbed coefficients is important for practice.
In the present paper we consider a fourth-order differential equation and a system of fourth-order differential equations, which are of great applied importance in mechanics. All arguments are carried out for one boundary-value problem; however, the results remain valid also for other types of boundary-value problems considered for the given equation. As the initial family of difference schemes we take homogeneous conservative difference schemes, introduced into the literature under this name by A. N. Tikhonov and A. A. Samarskii and studied in [2]. The results of §§ 1–4 of the present paper are analogous to the corresponding results of [1]. In § 5 the case of a local perturbation of the coefficients of the differential equation on one mesh interval containing a discontinuity point of the first kind of these coefficients is considered. In § 6 a system of differential equations is considered and the corresponding results are formulated. In defining co-stability we assume that the coefficients of the perturbed problem are taken from the same functional class as the coefficients of the original problem, because the perturbed coefficients can be chosen so that the problem will not be solvable.
1°. Boundary-Value Problem for a Differential Equation
Consider the differential equation
\[ L^{(k,p,q,f)}U \equiv \frac{d^2}{dx^2}\left[k(x)\frac{d^2U}{dx^2}\right] -\frac{d}{dx}\left[p(x)\frac{dU}{dx}\right] +q(x)U-f(x)=0 \tag{1} \]
\((0 < x < 1)\) with boundary conditions
\[ U(0)=U'(0)=U(1)=U'(1)=0. \tag{2} \]
We shall assume that the coefficients of equation (1) on the interval \([0,1]\) satisfy the conditions
\[ 0<c_1\leq k(x)\leq c_2,\quad 0\leq p(x)\leq C_3,\quad 0\leq q(x)\leq c_4,\quad |f(x)|\leq c_5, \tag{3} \]
where \(c_j\) \((j=1,2,3,4,5)\) are certain positive constants. If the functions \(k(x)\) and \(p(x)\) have discontinuities of the first kind at some point of the interval \(0<x<1\), then at this point the function \(U(x)\) must satisfy the following conjugation conditions:
\[ [U]=0,\quad [U']=0,\quad [kU'']=0,\quad [(kU'')'-pU']=0, \tag{4} \]
where \([\varphi]=\varphi_{\mathrm{r}}-\varphi_{\mathrm{l}},\ \varphi_{\mathrm{r}}=\varphi(\xi+0),\ \varphi_{\mathrm{l}}=\varphi(\xi-0)\) \((x=\xi\) is the point of discontinuity of the coefficients of equation (1)). We shall call problem (1)—(4) Problem I.
2°. A DIFFERENCE BOUNDARY-VALUE PROBLEM
We consider conservative difference schemes
\[ L_h^{(k,p,q,f)}y=L_h^{(k,p,q)}y-\Phi, \tag{5} \]
where
\[ L_h^{(k,p,q)}y=\left(ay_{\bar{x}x}\right)_{\bar{x}x}-\left(by_{\bar{x}}\right)_x+dy,\quad y_{\bar{x}}=\frac{1}{h}\left(y-y^{(-1)}\right),\quad y_x=\frac{1}{h}\left(y^{(+1)}-y\right), \]
\(y^{(-1)}=y_{i-1},\ y^{(+1)}=y_{i+1};\ y=y(x)=y_i\) is a grid function given on the grid \(\omega_h=\{x_0=0,\ldots,x_i=ih,\ldots,x_N=Nh=1\}\). The coefficients of the difference scheme \(a(x), b(x), d(x)\), and \(\Phi(x)\) are determined by means of template functionals, given in the class \((Q^{(0)})\),
\[ A^h[\varphi(s)],\quad -1\leq s\leq 1, \]
\[ B^h[(\varphi(s))],\quad -1\leq s\leq 0, \]
\[ D^h[\varphi(s)],\quad -\frac{1}{2}\leq s\leq \frac{1}{2}, \]
\[ F^h[\varphi(s)],\quad -\frac{1}{2}\leq s\leq \frac{1}{2}, \]
by the formulas
\[ a=A^{(k,h)}=A^h[\bar{k}(s)],\quad \bar{k}(s)=k(x+sh), \]
\[ b=B^{(p,h)}=B^h[\bar{p}(s)],\quad \bar{p}(s)=p(x+sh), \]
\[ d=D^{(q,h)}=D^h[\bar{q}(s)],\quad \bar{q}(s)=q(x+sh), \]
\[ \Phi=F^{(f,h)}=F^h[\bar{f}(s)],\quad \bar{f}(s)=f(x+sh). \tag{6} \]
To characterize the class of template functionals one must use the concept of the rank of a functional introduced in [1]. This characterization was carried out in [2], and we shall not present it here.
The difference boundary-value problem
\[ L_h^{(k,p,q)}y=\Phi,\quad h<x<1-h, \tag{7} \]
\[ y_1=0,\qquad y_{N-1}=0, \tag{8} \]
\[ y_{x,0}-\frac{h}{2}y_{\bar x x,0}=0,\qquad y_{\bar x,N}+\frac{h}{2}y_{x\bar x,N}=0, \tag{9} \]
corresponding to problem I, will be called problem II. The coefficients of the difference equation (7) are computed by formulas (6).
3°. DEPENDENCE OF THE SOLUTION OF THE DIFFERENTIAL EQUATION ON THE COEFFICIENTS
Consider the boundary-value problem
\[ L^{(k,p,q,f)}U=0,\quad 0<x<1, \tag{10} \]
\[ U(0)=U'(0)=U(1)=U'(1)=0 \tag{11} \]
and compare its solution \(U(x)\) with the solution \(\widetilde U(x)\) of the perturbed problem
\[ L^{(\tilde k,\tilde p,\tilde q,\tilde f)}\widetilde U=0,\quad 0<x<1, \tag{12} \]
\[ \widetilde U(0)=\widetilde U'(0)=\widetilde U(1)=\widetilde U'(1)=0. \tag{13} \]
Lemma 1. If the coefficients of equations (10) and (12) are piecewise continuous and satisfy conditions (3), then the estimate is valid
\[ |U(x)-\widetilde U(x)|\le M_1\int_0^1 |k(x)-\widetilde k(x)|\,dx +M_2\int_0^1 |p(x)-\widetilde p(x)|\,dx+ \]
\[ +M_3\int_0^1 |q(x)-\widetilde q(x)|\,dx +M_4\int_0^1 |f(x)-\widetilde f(x)|\,dx, \tag{14} \]
where \(M_j\) \((j=1,2,3,4)\) are certain positive constants.
Proof. Form the difference \(z(x)=U(x)-\widetilde U(x)\), for which we obtain the boundary-value problem
\[ L^{(k,p,q)}z=\Psi(x),\quad 0<x<1, \tag{15} \]
\[ z(0)=z'(0)=z(1)=z'(1)=0, \tag{16} \]
where
\[ \Psi(x)=\frac{d^2}{dx^2} \left[(\widetilde k(x)-k(x))\frac{d^2\widetilde U}{dx^2}\right] -\frac{d}{dx} \left[(\widetilde p(x)-p(x))\frac{d\widetilde U}{dx}\right] +(\widetilde q(x)- \]
\[ -q(x))\widetilde U+f(x)-\widetilde f(x). \]
Using the Green’s function of problem (10)—(11), we write the solution of problem (15)—(16) in the form
\[ z(x)= \int_0^1(\widetilde k(\xi)-k(\xi))\widetilde U''(\xi)G''(x,\xi)\,d\xi +\int_0^1(\widetilde p(\xi)-p(\xi))\widetilde U'(\xi)G'(x,\xi)\,d\xi+ \]
\[ +\int_0^1(\widetilde q(\xi)-q(\xi))\widetilde U(\xi)G(x,\xi)\,d\xi +\int_0^1(f(\xi)-\widetilde f(\xi))G(x,\xi)\,d\xi. \tag{17} \]
By virtue of the boundedness of \(\widetilde U, \widetilde U', \widetilde U'', G, G', G''\), we obtain estimate (14). Inequality (14) expresses the stability of problem I with respect to a change in the coefficients of the differential equation.
4°. Co-stability of difference schemes \(L_h^{(k,p,q,f)}\)
Consider the difference boundary-value problem
\[ L_h^{(\tilde k,\tilde p,\tilde q)}\tilde y=\tilde\Phi,\qquad h<x<1-h, \tag{18} \]
\[ \tilde y_1=0,\qquad \tilde y_{N-1}=0, \tag{19} \]
\[ \tilde y_{\bar x,0}-\frac{h}{2}\tilde y_{x\bar x,0}=0,\qquad \tilde y_{\bar x,N}+\frac{h}{2}\tilde y_{\bar x x,N}=0, \tag{20} \]
which we shall call problem III.
Assume that the coefficients of the scheme \(L_h^{(\tilde k,\tilde p,\tilde q,\tilde f)}\), \(\tilde a\), \(\tilde b\), \(\tilde d\), and \(\tilde\Phi\) are obtained from the coefficients \(a\), \(b\), \(d\), and \(\Phi\) by means of an arbitrary perturbation. In the general case this perturbation may be caused by distortion of the coefficients of equation (1), distortion of the functionals \(A^h\), \(\dot B^h\), \(D^h\), and \(F^h\), as well as by errors admitted in computing the coefficients of the scheme. To estimate the magnitude of the distortion of the coefficients we shall use the A. N. Tikhonov–A. A. Samarskii norm
\[ \|\Psi\|_1=(1,|\Psi|), \tag{21} \]
where
\[ (y,z)=\sum_{i=1}^{N-1}hy_i z_i . \]
Following the definition in [1], we shall say that the scheme \(L_h^{(k,p,q,f)}\) is co-stable if, in the case of convergence as \(h\to0\) in the norm (21) of the coefficients of any arbitrarily perturbed scheme \(L_h^{(\tilde k,\tilde p,\tilde q,\tilde f)}\) to the coefficients of the scheme \(L_h^{(k,p,q,f)}\), the solution of problem III converges uniformly to the solution of problem I, provided that the coefficients \(k\), \(p\), \(q\), \(f\) belong to some class \(Q^{(m)}\) \((m\geqslant0)\).
By definition, the scheme \(L_h^{(k,p,q,f)}\) is co-stable if from the conditions
\[ \|a-\tilde a\|_1=\rho(h),\qquad \|b-\tilde b\|_1=\rho(h),\qquad \|d-\tilde d\|_1= \]
\[ =\rho(h),\qquad \|\Phi-\tilde\Phi\|_1=\rho(h) \tag{22} \]
it follows that
\[ \|z_h\|_0=\max_{\omega_h}|\tilde y-U|=\rho(h), \tag{23} \]
where \(\rho(h)\) is a quantity tending to zero as \(h\to0\). As in [1], we also specialize the type of perturbation of the coefficients of the difference scheme, assuming that it is caused by perturbation of the coefficients of the differential equation (1). In this case we suppose that the functions \(\tilde k\), \(\tilde p\), \(\tilde q\), and \(\tilde f\) belong to the same functional class as \(k\), \(p\), \(q\), \(f\). Thus, the coefficients of the difference scheme \(L_h^{(\tilde k,\tilde p,\tilde q,\tilde f)}\) are equal to:
\[ \tilde a=A^{(\tilde k,h)}=A^h[\tilde k(x+sh)],\qquad \tilde b=B^{(\tilde p,h)}=B^h[\tilde p(x+sh)], \]
\[ \tilde d=D^{(\tilde q,h)}=D^h[\tilde q(x+sh)],\qquad \tilde\Phi=F^{(\tilde f,h)}=F^h[\tilde f(x+sh)]. \]
Lemma 2. If the coefficients of the difference schemes \(L_h^{(k,p,q,f)}\) and \(L_h^{(\tilde k,\tilde p,\tilde q,\tilde f)}\) satisfy conditions (3), then the inequality
\[ \|\tilde y_h-y_h\|_0 \leq M'_1\|a-\tilde a\|_1+M'_2\|b-\tilde b\|_1+M'_3\|d-\tilde d\|_1+M'_4\|\Phi-\tilde\Phi\|_1, \tag{24} \]
holds, where \(M'_j\) \((j=1,2,3,4)\) are constants independent of \(h\).
Proof. Form the difference \(z=\tilde y-y\). For \(z\) we obtain the difference boundary-value problem
\[ L_h^{(\tilde k,\tilde p,\tilde q)}z=\varphi,\qquad h<x<1-h, \tag{25} \]
\[ z_1=0,\qquad z_{N-1}=0, \tag{26} \]
\[ z_{x,0}-\frac{h}{2}z_{xx,0}=0,\qquad z_{\bar x,N}+\frac{h}{2}z_{\bar x x,N}=0, \tag{27} \]
where
\[ \varphi=[(a-\tilde a)y_{\bar x x}]_{xx}-[(b-\tilde b)y_{\bar x}]_x+(d-\tilde d)y+\tilde\Phi-\Phi . \]
We shall call problem (25)—(27) problem IV. Let \(\tilde G^h(x,\xi)\) be the difference Green’s function of problem III, defined by the conditions
\[ L_h^{(\tilde k,\tilde p,\tilde q)}\tilde G^h=\frac{\delta(x,\xi)}{h},\qquad h<x<1-h, \tag{28} \]
\[ \tilde G^h(h,\xi)=0,\qquad \tilde G^h(1-h,\xi)=0, \tag{29} \]
\[ \tilde G^h_{x,0}-\frac{h}{2}\tilde G^h_{xx,0}=0,\qquad \tilde G^h_{\bar x,N}+\frac{h}{2}\tilde G^h_{\bar x x,N}=0, \tag{30} \]
where
\[ \delta(x,\xi)= \begin{cases} 1 & \text{for } x=\xi,\\ 0 & \text{for } x\ne \xi . \end{cases} \]
The function \(\tilde G^h\) was constructed and studied in [2], where it was proved that if \(L_h^{(\tilde k,\tilde p,\tilde q)}\) is a scheme of zero rank, then for \(k,p,q\in Q^{(0)}_{[0,1]}\) the estimates
\[ |\tilde G^h(x,\xi)|\leq M,\qquad |\tilde G^h_x(x,\xi)|\leq M,\qquad |\tilde G^h_{xx}(x,\xi)|\leq M,\qquad |(a\tilde G^h_x)_x|\leq M, \]
hold, where \(M>0\) is a constant independent of \(h\). With the aid of the function \(\tilde G^h\), the solution of problem IV is written in the form
\[ z=((\tilde G^h,\varphi)), \tag{31} \]
where
\[ ((y,z))=\sum_{i=2}^{N-2}hy_iz_i . \]
Substituting \(\varphi\) into (31) and applying the known summation-by-parts formulas, we obtain
\[ \begin{aligned} z={}&\left(\tilde G^h_{xx},(a-\tilde a)y_{\bar x x}\right) +\left(\left(\tilde G^h_x,(b-\tilde b)y_{\bar x}\right)\right)\\ &+\left(\left(\tilde G^h,(d-\tilde d)y\right)\right) +\left(\left(\tilde G^h,(\tilde\Phi-\Phi)\right)\right) +\left[\tilde G^h_x(a-\tilde a)y_{\bar x x}\right]_1 \\ &-\left[\tilde G^h_x(a-\tilde a)y_{\bar x x}\right]_{N-1}, \end{aligned} \tag{32} \]
where
\[ ((y,z))=\sum_{i=2}^{N-1}hy_iz_i . \]
Taking into account the boundedness of \(\tilde G^h\), \(\tilde G^h_x\), \(\tilde G^h_{xx}\), \(y\), \(y_{\bar x}\), and \(y_{\bar x x}\),
we obtain inequality (24). It follows from Lemma 2 that
\[ \| \tilde y_h-y_h\|_0=\rho(h), \tag{33} \]
if conditions (22) are satisfied.
Denote by \(Q^{(m)}_{[0,1]}\) the class of functions that are piecewise continuous on \([0,1]\), together with their derivatives up to order \(m\) inclusive, and by \(Q^{(m,1)}_{[0,1]}\) the class of functions from \(Q^{(m)}_{[0,1]}\) whose \(m\)-th derivatives satisfy a Lipschitz condition on the intervals of their continuity.
Theorem. Any conservative difference scheme \(L_h^{(k,p,q,f)}\) of second rank, satisfying the necessary conditions for second-order approximation (see [2]), is co-stable in the class \(Q^{(2,1)}_{[0,1]}\).
Proof. From the obvious inequality
\[ \| \tilde y_h-U_h\|_0 \leq \|y_h-U_h\|_0+\|\tilde y_h-y_h\|_0 \tag{34} \]
we obtain the inequality
\[ \|\tilde y_h-U_h\|_0 \leq Mh+\rho(h), \tag{35} \]
where \(M>0\) is a constant independent of \(h\). Indeed, it follows from [2] that, when the conditions of the theorem are fulfilled, \(\|y_h-U_h\|\leq Mh\), while \(\|\tilde y_h-y_h\|_0=\rho(h)\) when conditions (22) are fulfilled. The theorem is proved.
5°. The case of a local perturbation of \(k, p, q, f\) on one mesh interval
Let \(k, p, q, f \in Q^{(2,1)}_{[0,1]}\). We consider schemes \(L_h^{(k,p,q,f)}\) of second rank satisfying the necessary conditions for second-order approximation [2]. Let the discontinuity point of the coefficients of equation (1), \(x=\zeta\), belong to some mesh interval \((x_n,x_{n+1})\). Consider functions \(\tilde k,\tilde p,\tilde q,\tilde f\in Q^{(2,1)}_{[0,1]}\), coinciding with \(k,p,q,f\) everywhere except the interval \((x_n,x_{n+1})\). In the case under consideration, using the difference Green’s functions \(G^h\) and \(\tilde G^h\) of problems II and III, one can obtain, by the methods of [2], estimates for the errors of the solutions \(z=y-U\), \(\tilde z=\tilde y-\tilde U\), where \(y\) and \(\tilde y\) are, respectively, the solutions of problems II and III, and \(U\) and \(\tilde U\) are, respectively, the solutions of problems I and (12)—(13):
\[ \|z_h\|_0=O(h), \qquad \|\tilde z_h\|_0=O(h). \tag{36} \]
Taking equalities (36) into account, we obtain
\[ O(h)\geq \|z_h-\tilde z_h\|_0 =\|(y_h-U_h)-(\tilde y_h-\tilde U_h)\|_0= \]
\[ =\|(y_h-\tilde y_h)-(U_h-\tilde U_h)\|_0 \geq \|y_h-\tilde y_h\|_0-\|U_h-\tilde U_h\|_0, \tag{37} \]
whence
\[ \|y_h-\tilde y_h\|_0 \leq \|U_h-\tilde U_h\|_0+O(h). \tag{38} \]
With the aid of inequality (38) one can determine how the solution of the difference boundary-value problem changes as a function of the differences \(k-\tilde k\), \(p-\tilde p\), \(q-\tilde q\), \(f-\tilde f\).
By the distance between two functions \(\Psi(x)\) and \(\eta(x)\) we shall understand, as usual, the quantity \(\sup_{a<x<b}|\Psi(x)-\eta(x)|\) \(((a,b)\) is the interval on which \(\Psi(x)\) and \(\eta(x)\) are considered). The relation between the change in the solution of the difference boundary-value problem and the above-mentioned differences is easily
is considered in examining inequality (14). If in this inequality the integrands are replaced by distances on the interval \((x_n, x_{n+1})\), then it becomes obvious that the solution of difference boundary-value problem I will change by the magnitude of the largest of the distances between the coefficients of equation (1) and the coefficients of equation (12) on the interval \((x_n, x_{n+1})\). If these distances are quantities of order \(\rho(h)\) or 1, then the solution of the difference boundary-value problem under perturbation of the coefficients will change, respectively, by \(h\rho(h)\) or \(O(h)\). Under perturbation of the coefficients of the equation on the whole mesh, one can estimate the magnitude of the change in the solution of the difference boundary-value problem by means of the distance between the coefficients of the original and perturbed problems on the segment \([0,1]\).
Let us consider the scheme \(L_h^{(k,p,q,f)}\) with coefficients constructed in [2]:
\[ a=\left(\int_0^1 \frac{(1-s)}{\bar{k}(s)}\,ds+ \int_0^1 \frac{(1-s)}{\bar{k}(-s)}\,ds\right)^{-1}, \qquad \bar{k}(s)=k(x+sh), \]
\[ b=\int_{-1}^{0}\bar{p}(s)\,ds,\qquad \bar{p}(s)=p(x+sh),\qquad d=\int_{-\frac12}^{\frac12}\bar{q}(s)\,ds, \tag{39} \]
\[ \bar{q}(s)=q(x+sh),\qquad \Phi=\int_{-\frac12}^{\frac12}\bar{f}(s)\,ds,\qquad \bar{f}(s)=f(x+sh). \]
In [2] it is shown that the difference scheme \(L_h^{(k,p,q,f)}\) of second rank with coefficients (39) in the class \(k,p,q,f\in Q_{[0,1]}^{(2,1)}\), when the necessary conditions of second-order approximation are satisfied, provides second-order accuracy at all mesh nodes,
\[ \|y_h-U_h\|\le Mh^2, \tag{40} \]
where \(M>0\) is a constant independent of \(h\). If, as the difference scheme under consideration, we take the scheme with coefficients (39), then under a local perturbation of the coefficients on the interval \((x_n,x_{n+1})\), inequality (38) takes the form
\[ \|y_h-\tilde{y}_h\|_0\le \|U_h-\tilde{U}_h\|_0+O(h^2). \tag{41} \]
If the distances between the coefficients of equation (1) and the corresponding perturbed coefficients on the interval \((x_n,x_{n+1})\) are quantities of order \(h\), then the solution of the difference boundary-value problem changes by \(O(h^2)\).
In the next subsection, a system of fourth-order differential equations is considered.
6°. SYSTEM OF FOURTH-ORDER DIFFERENTIAL EQUATIONS
Let, in boundary-value problem I, \(k(x)\), \(p(x)\), and \(q(x)\) be symmetric matrices of order \(r\), with the rank of the matrix \(k\) equal to \(r\), and let \(U(x)\) and \(f(x)\) be \(r\)-dimensional vectors. We assume that all elements of the matrices and components of the vector satisfy the corresponding inequalities (3).
Consider difference schemes \(L_h^{(k,p,q,f)}\), whose coefficients are certain symmetric matrices \(a, b, d\) and a vector \(\Phi\). The elements of the mentioned matrices and the components of the vector \(\Phi\) are determined by means of stencil functionals specified in the class \(Q^{(0)}\) by formulas (6). Let \(L_h^{(k,p,q,f)}\) be a scheme of second rank. From the definition of a scheme of \(n\)-th rank (see [1]) it follows that the elements of the matrices \(a, b, d\) and the components of the vector \(\Phi\) can be represented in the form
\[ a^{lm}(x)=k_{lm}(x)+h k'_{lm}(x)A_1^{(0)lm}[s]+O(h^2), \]
\[ b^{lm}(x)=p_{lm}(\bar{x})+h p'_{lm}(\bar{x})\left(B_1^{(0)lm}[s]+\frac{1}{2}\right)+O(h^2), \]
\[ d^{lm}(x)=q_{lm}(x)+h q'_{lm}(x)D_1^{(0)lm}[s]+O(h^2), \tag{42} \]
\[ \Phi^l(x)=f_l(x)+h f'_l(x)F_1^{(0)l}[s]+O(h^2), \]
where \(\bar{x}=x-\dfrac{h}{2}\), \(l,m=1,2,\ldots,r\). The necessary conditions for second order of approximation for the difference scheme \(L_h^{(k,p,q,f)}\) of second rank in the class \(k,p,q,f\in C_{[0,1]}^{(2,1)}\) have the form
\[ A_1^{(0)lm}[s]=0,\qquad B_1^{(0)lm}[s]+\frac{1}{2}=0,\qquad D_1^{(0)lm}[s]=0,\qquad F_1^{(0)l}[s]=0 \tag{43} \]
\[ (l,m=1,2,\ldots,r). \]
Introduce into consideration the \(r\)-dimensional vectors
\[ e_1= \begin{vmatrix} \delta(x,\xi)\\ 0\\ \cdot\\ \cdot\\ 0 \end{vmatrix}, \qquad e_2= \begin{vmatrix} 0\\ \delta(x,\xi)\\ 0\\ \cdot\\ \cdot\\ 0 \end{vmatrix}, \qquad \ldots,\qquad e_r= \begin{vmatrix} 0\\ 0\\ \cdot\\ \cdot\\ \delta(x,\xi) \end{vmatrix}, \tag{44} \]
and also the \(r\)-dimensional vectors \(\bar e_s\) \((s=1,2,\ldots,r)\), obtained from \(e_s\) for \(x=\xi\). We define the system of difference Green vector-functions \(G^{hs}(x,\xi)\) \((s=1,2,\ldots,r)\) from the conditions:
\[ L_h^{(k,p,q)}G^{hs}=\frac{e_s}{h},\qquad h<x<1-h, \tag{45} \]
\[ G^{hs}(h,\xi)=\vec{0},\qquad G^{hs}(1-h,\xi)=\vec{0}, \tag{46} \]
\[ G_{x,0}^{hs}-\frac{h}{2}G_{xx,0}^{hs}=\vec{0},\qquad G_{\bar x,N}^{hs}+\frac{h}{2}G_{\bar x x,N}^{hs}=\vec{0}. \tag{47} \]
We seek the functions \(G^{hs}(x,\xi)\) in the form
\[ G^{hs}(x,\xi)= \begin{cases} \bar G^{hs}(x,\xi)=\alpha_s(\xi)V^{1s}(x)+\beta_s(\xi)V^{2s}(x), & x<\xi,\\ \bar{\bar G}^{hs}(x,\xi)=\gamma_s(\xi)V^{3s}(x)+\delta_s(\xi)V^{4s}(x), & x>\xi. \end{cases} \]
where
\[ \alpha_s(\xi)= \begin{pmatrix} \alpha_s^{11}(\xi) & & 0\\ & \alpha_s^{22}(\xi) & \\ & & \ddots\\ 0 & & \alpha_s^{rr}(\xi) \end{pmatrix}, \qquad \beta_s(\xi)= \begin{pmatrix} \beta_s^{11}(\xi) & & 0\\ & \beta_s^{22}(\xi) & \\ & & \ddots\\ 0 & & \beta_s^{rr}(\xi) \end{pmatrix}, \]
\[ \gamma_s(\xi)= \begin{pmatrix} \gamma_s^{11}(\xi) & & 0\\ & \gamma_s^{22}(\xi) & \\ & & \ddots\\ 0 & & \gamma_s^{rr}(\xi) \end{pmatrix}, \qquad \delta_s(\xi)= \begin{pmatrix} \delta_s^{11}(\xi) & & 0\\ & \delta_s^{22}(\xi) & \\ & & \ddots\\ 0 & & \delta_s^{rr}(\xi) \end{pmatrix}, \]
are diagonal matrices of order \(r\), \(V^{ms}(x)\) \((m=1,2,3,4)\) are \(r\)-dimensional vectors satisfying the difference equation \(L_h^{(k,p,q)}V^{ms}(x)=\overrightarrow{0}\), and also the conditions:
\[ V_1^{1s}=\overrightarrow{0},\qquad V_1^{2s}=\overrightarrow{0}, \]
\[ V_{x,0}^{1s}-\frac{h}{2}V_{xx,0}^{1s}=\overrightarrow{0},\qquad V_{x,0}^{2s}-\frac{h}{2}V_{xx,0}^{2s}=\overrightarrow{0}, \]
\[ (a^{(+1)}V_{xx}^{1s})_0=\bar e_s,\qquad (a^{(+1)}V_{xx}^{2s})_0=\overrightarrow{0}, \]
\[ (a^{(+1)}V_{xx}^{1s})_{x,0}=\overrightarrow{0},\qquad (a^{(+1)}V_{xx}^{2s})_{x,0}=\bar e_s, \]
\[ V_{N-1}^{3s}=\overrightarrow{0},\qquad V_{N-1}^{4s}=\overrightarrow{0}, \]
\[ V_{x,N}^{3s}+\frac{h}{2}V_{xx,N}^{3s}=\overrightarrow{0},\qquad V_{x,N}^{4s}+\frac{h}{2}V_{xx,N}^{4s}=\overrightarrow{0}, \]
\[ (a^{(-1)}V_{xx}^{3s})_N=\bar e_s,\qquad (a^{(-1)}V_{xx}^{4s})_N=\overrightarrow{0}, \]
\[ (a^{(-1)}V_{xx}^{3s})_{x,N}=\overrightarrow{0},\qquad (a^{(-1)}V_{xx}^{4s})_{x,N}=\bar e_s, \]
where the symbol \(\overrightarrow{0}\) denotes the zero vector. The elements of the matrices \(\alpha_s(\xi)\), \(\beta_s(\xi)\), \(\gamma_s(\xi)\), and \(\delta_s(\xi)\) are found from the conditions:
\[ \bar G^{hs}(x,\xi)=\bar{\bar G}^{hs}(x,\xi),\quad \bar G_x^{hs}(x,\xi)=\bar{\bar G}_x^{hs}(x,\xi),\quad a\bar G_{xx}^{hs}(x,\xi)=a\bar{\bar G}_{xx}^{hs}(x,\xi), \]
\[ [a\bar G_{xx}^{hs}(x,\xi)]_{\bar x}-[a\bar{\bar G}_{xx}^{hs}(x,\xi)]_{\bar x}=-\bar e_s \quad \text{for } x=\xi . \tag{48} \]
From conditions (48), for fixed \(s\) we obtain \(r\) systems of algebraic equations, each of which contains four unknowns. For example, to determine the coefficients \(\alpha_s^{11}(\xi)\), \(\beta_s^{11}(\xi)\), \(\gamma_s^{11}(\xi)\), and \(\delta_s^{11}(\xi)\), we obtain the system
\[ \alpha_s^{11}(\xi)V_1^{1s}(\xi)+\beta_s^{11}(\xi)V_1^{2s}(\xi) = \gamma_s^{11}(\xi)V_1^{3s}(\xi)+\delta_s^{11}(\xi)V_1^{4s}(\xi), \]
\[ \alpha_s^{11}(\xi)V_{1x}^{1s}(\xi)+\beta_s^{11}(\xi)V_{1x}^{2s}(\xi) = \gamma_s^{11}(\xi)V_{1x}^{3s}(\xi)+\delta_s^{11}(\xi)V_{1x}^{4s}(\xi), \tag{49} \]
\[ \alpha_s^{11}(\xi)|a(\xi)|V_{1xx}^{1s}(\xi) +\beta_s^{11}(\xi)|a(\xi)|V_{1xx}^{2s}(\xi) = \gamma_s^{11}(\xi)|a(\xi)|V_{1xx}^{3s}(\xi) + \delta_s^{11}(\xi)|a(\xi)|V_{1xx}^{4s}(\xi), \]
\[ \alpha_s^{11}(\xi)\bigl(|a(\xi)|V_{1xx}^{1s}(\xi)\bigr)_{\bar x} +\beta_s^{11}(\xi)\bigl(|a(\xi)|V_{1xx}^{2s}(\xi)\bigr)_{\bar x} - \gamma_s^{11}(\xi)\bigl(|a(\xi)|V_{xx}^{3s}(\xi)\bigr)_{\bar x} - \delta_s^{11}(\xi)\bigl(|a(\xi)|V_{xx}^{4s}(\xi)\bigr)_{\bar x} = \]
\[ = (-1)^{s+1} \left| \begin{array}{ccccc} a^{12} & a^{13} & \ldots & a^{1r}\\ \ldots & \ldots & \ldots & \ldots\\ a^{s-1,2} & a^{s-1,3} & \ldots & a^{s-1,r}\\ a^{s+1,2} & a^{s+1,3} & \ldots & a^{s+1,r}\\ \ldots & \ldots & \ldots & \ldots\\ a^{r2} & a^{r3} & \ldots & a^{rr} \end{array} \right|_{x=\xi-h}, \]
where \(|a|=\det a\). A system of the form (49) has been studied in detail in [2]. Considering the remaining \(r-1\) systems and using Lemmas 1–7 of [2], we conclude that for the difference vector Green’s functions and their difference quotients the following estimates hold:
\[ |G^{hs}(x,\xi)| \leq M_s,\quad |G_x^{hs}(x,\xi)| \leq M_s,\quad |G_{xx}^{hs}(x,\xi)| \leq M_s, \]
\[ |(aG_{xx}^{hs})_{\bar x}| \leq M_s, \tag{50} \]
where \(M_s>0\) are constants independent of \(h\). With the aid of the functions \(G^{hs}(x,\xi)\), the solution of difference boundary-value problem II is written in the form
\[ y^s = ((G^{hs},\Phi))\quad (s=1,2,\ldots,r), \tag{51} \]
where under the summation sign stands the scalar product of the vectors \(G^{hs}\) and \(\Phi\). Introducing into consideration the special norm of the matrix \(T\)
\[ T_2=\left(1,\sqrt{\sum_{l,m}|t_{lm}|^2}\right), \tag{52} \]
it is easy to prove the co-stability of conservative difference schemes also for the case of a system under consideration. The results of item \(5^\circ\) also carry over to the case of a system of equations.
All the results obtained can, without particular difficulty, be carried over to the case of nonuniform grids.
I express my sincere gratitude to A. A. Samarskii for valuable comments and advice.
References
- Tikhonov A. N., Samarskii A. A. Zhurnal vych. matem. i matem. fiz., 1, No. 1, 5–63, 1961.
- Hao Shou. Zhurnal vych. matem. i matem. fiz., 3, No. 5, 841–860, 1963.
Received by the editors
November 3, 1964
Taras Shevchenko Kiev State University