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THE METHOD OF INTEGRAL RELATIONS IN SOLVING THE GOURSAT PROBLEM FOR A SYSTEM OF TWO FIRST-ORDER EQUATIONS
V. V. BOBKOV
A little more than ten years ago A. A. Dorodnitsyn proposed an approximate method for solving partial differential equations, called the method of integral relations [1]. A fairly complete review of the method is given in the note [2]. The method was developed for systems of first-order differential equations of divergent form (see [1]) and approximately reduces the latter to systems of ordinary differential equations.
Here we shall consider a certain modification of the method as applied to linear (or quasilinear) hyperbolic systems of two first-order equations of the general form
\[ \begin{aligned} a_1 u_x + a_2 u_y + a_3 v_x + a_4 v_y + a_5 u + a_6 v + a_7 &= 0,\\ b_1 u_x + b_2 u_y + b_3 v_x + b_4 v_y + b_5 u + b_6 v + b_7 &= 0 \end{aligned} \Bigg\}, \tag{1} \]
where \(a_i\) and \(b_i\) \((i=1,2,\ldots,7)\) are known, while \(u\) and \(v\) are the unknown functions of two independent variables \(x\) and \(y\).
Applying the idea of integration with subsequent interpolation of the integrand functions not to the original system written in divergent form, but directly to the conditions on the characteristics of system (1), we obtain, as the approximating system, a system of linear algebraic equations.
The main attention in the note will be devoted to the Goursat problem for system (1), which consists in finding a solution \(u, v\) of the original system, if on two of its characteristics issuing from one point the values of \(u\) and \(v\) are prescribed (of course, the values of the functions prescribed on the characteristics must be consistent). Here computational formulas will be obtained for finding an approximate solution of the problem, convergence with rate of order \(h^2\) will be proved, and a priori estimates of the error of the method will be given; attainable estimates for the exact solution of the original problem will also be presented. The approach used in the conclusion carries over to the Cauchy problem and to both mixed problems.
As already noted, we shall consider only hyperbolic systems of the form (1) (we shall not write out here the well-known hyperbolicity conditions for such systems). The characteristics of our system do not depend on the solution of the posed problem and can be found in advance. We take them as coordinate lines in a new coordinate system \(\xi, \eta\).
Let \(\varphi_1(\xi)\) and \(\varphi_2(\xi)\) be the values of \(u\) and \(v\) prescribed, in the case of the Goursat problem, on the boundary characteristic \(\eta=0\), and let \(\psi_1(\eta)\) and \(\psi_2(\eta)\) be the corresponding values on the characteristic \(\xi=0\).
Using the known [3] conditions on the characteristics, we obtain the following relations between \(u\) and \(v\):
\[ \text{along the curve } \xi=\mathrm{const}\quad \overline A_1 u_\eta+\overline B_1 v_\eta+\overline C_1u+\overline D_1v+\overline F_1=0, \tag{2} \]
\[ \text{along the curve } \eta=\mathrm{const}\quad \overline A_2 u_\xi+\overline B_2 v_\xi+\overline C_2u+\overline D_2v+\overline F_2=0 \tag{3} \]
(here and everywhere below the notation from [4] is used).
For a given \(h>0\), construct a grid of characteristics
\(\xi=\xi^i=ih,\ \eta=\eta^j=jh,\ i,j=0,1,2,\ldots\). We shall seek the approximate solution \(u_{mn}, v_{mn}\) of the problem posed, and the error of the method \(\gamma_{mn}=u(\xi^m,\eta^n)-u_{mn}\), \(\delta_{mn}=v(\xi^m,\eta^n)-v_{mn}\) at an arbitrary point \((\xi^m,\eta^n)\) of our grid.
Interpolating the functions \(u(\xi^i,\eta)\) and \(v(\xi^i,\eta)\), \(u(\xi,\eta^j)\) and \(v(\xi,\eta^j)\) linearly by their values, respectively, on the lines \(\eta=\eta^j\) and \(\eta=\eta^{j+1}\), \(\xi=\xi^i\) and \(\xi=\xi^{i+1}\), and integrating relations (2), (3), written on the chosen grid of characteristics, respectively in \(\eta\) from \(\eta=\eta^j\) to \(\eta=\eta^{j+1}\) and in \(\xi\) from \(\xi=\xi^i\) to \(\xi=\xi^{i+1}\), we obtain, for finding \(u_{mn}, v_{mn}\) and \(\gamma_{mn}, \delta_{mn}\), systems of linear algebraic equations:
\[ \begin{gathered} \alpha_{ij+1}u_{ij+1}+\beta_{ij+1}v_{ij+1}-\xi_{ij}u_{ij}-\eta_{ij}v_{ij}=f_{ij+1},\quad i=1,2,\ldots,m;\\ j=0,1,\ldots,n-1,\\ \alpha^*_{i+1j}u_{i+1j}+\beta^*_{i+1j}v_{i+1j}-\xi^*_{ij}u_{ij}-\eta^*_{ij}v_{ij}=f^*_{i+1j},\quad i=0,1,\ldots,m-1;\\ j=1,2,\ldots,n,\\ u_{i0}=\varphi_1(\xi^i),\quad v_{i0}=\varphi_2(\xi^i),\quad i=1,2,\ldots,m;\quad u_{0j}=\psi_1(\eta^j),\\ v_{0j}=\psi_2(\eta^j),\quad j=1,2,\ldots,n \end{gathered} \tag{4} \]
\[ \begin{gathered} \alpha_{ij+1}\gamma_{ij+1}+\beta_{ij+1}\delta_{ij+1}-\xi_{ij}\gamma_{ij}-\eta_{ij}\delta_{ij}=r_{ij+1},\quad i=1,2,\ldots,m;\\ j=0,1,\ldots,n-1,\\ \alpha^*_{i+1j}\gamma_{i+1j}+\beta^*_{i+1j}\delta_{i+1j}-\xi^*_{ij}\gamma_{ij}-\eta^*_{ij}\delta_{ij}=r^*_{i+1j},\quad i=0,1,\ldots,m-1;\\ j=1,2,\ldots,n,\\ \gamma_{i0}=\delta_{i0}=0,\quad i=1,2,\ldots,m;\quad \gamma_{0j}=\delta_{0j}=0,\quad j=1,2,\ldots,n. \end{gathered} \tag{5} \]
The coefficient matrix for the unknowns of system (5) can be written in the following form:
\[ \overline P= \begin{bmatrix} A & B\\ C & D \end{bmatrix}, \]
where
\[ A= \begin{bmatrix} \alpha_1&0&0&\cdot&0\\ 0&\alpha_2&0&\cdot&0\\ 0&0&\alpha_3&\cdot&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdot&\alpha_m \end{bmatrix}, \quad \alpha_i= \begin{bmatrix} \alpha_{i1}&0&0&\cdot&0&0\\ -\xi_{i1}&\alpha_{i2}&0&\cdot&0&0\\ 0&-\xi_{i2}&\alpha_{i3}&\cdot&0&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdot&-\xi_{in-1}&\alpha_{in} \end{bmatrix}, \quad i=1,2,\ldots,m; \]
\[ B= \begin{bmatrix} \beta_1&0&0&\cdot&0\\ 0&\beta_2&0&\cdot&0\\ 0&0&\beta_3&\cdot&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdot&\beta_m \end{bmatrix}, \quad \beta_i= \begin{bmatrix} \beta_{i1}&0&0&\cdot&0&0\\ -\eta_{i1}&\beta_{i2}&0&\cdot&0&0\\ 0&-\eta_{i2}&\beta_{i3}&\cdot&0&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdot&-\eta_{in-1}&\beta_{in} \end{bmatrix}, \quad i=1,2,\ldots,m. \]
\[ C= \begin{bmatrix} \alpha_1^* & 0 & 0 & \cdot & 0 & 0\\ \xi_1 & \alpha_2^* & 0 & \cdot & 0 & 0\\ 0 & \xi_2 & \alpha_3^* & \cdot & 0 & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \cdot & \xi_{m-1} & \alpha_m^* \end{bmatrix}, \qquad D= \begin{bmatrix} \beta_1^* & 0 & 0 & \cdot & 0 & 0\\ \eta_1 & \beta_2^* & 0 & \cdot & 0 & 0\\ 0 & \eta_2 & \beta_3^* & \cdot & 0 & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \cdot & \eta_{m-1} & \beta_m^* \end{bmatrix}; \]
\[ \alpha_i^*= \begin{bmatrix} \alpha_{i1}^* & 0 & \cdot & 0\\ 0 & \alpha_{i2}^* & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \alpha_{in}^* \end{bmatrix}, \quad i=1,2,\ldots,m; \qquad \xi_j=- \begin{bmatrix} \xi_{j1}^* & 0 & \cdot & 0\\ 0 & \xi_{j2}^* & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \xi_{jn}^* \end{bmatrix}, \quad j=1,2,\ldots,m-1; \]
\[ \beta_i^*= \begin{bmatrix} \beta_{i1}^* & 0 & 0\\ 0 & \beta_{i2}^* & \cdot 0\\ \cdot & \cdot & \cdot\\ 0 & 0 & \cdot \beta_{in}^* \end{bmatrix}, \quad i=1,2,\ldots,m; \qquad \eta_j=- \begin{bmatrix} \eta_{j1}^* & 0 & \cdot 0\\ 0 & \eta_{j2}^* & \cdot 0\\ \cdot & \cdot & \cdot\\ 0 & 0 & \cdot \eta_{jn}^* \end{bmatrix}, \quad j=1,2,\ldots,m-1. \]
That is, system (5) can be rewritten in the form
\[ \left. \begin{aligned} A\Gamma+B\Delta&=R,\\ C\Gamma+D\Delta&=F \end{aligned} \right\}, \tag{6} \]
where
\[ \Gamma= \begin{bmatrix} \gamma_1\\ \gamma_2\\ \vdots\\ \gamma_m \end{bmatrix}, \qquad \gamma_i= \begin{bmatrix} \gamma_{i1}\\ \gamma_{i2}\\ \vdots\\ \gamma_{in} \end{bmatrix}, \quad i=1,2,\ldots,m; \qquad \Delta= \begin{bmatrix} \delta_1\\ \delta_2\\ \vdots\\ \delta_m \end{bmatrix}, \]
\[ \delta_i= \begin{bmatrix} \delta_{i1}\\ \delta_{i2}\\ \vdots\\ \delta_{in} \end{bmatrix}, \quad i=1,2,\ldots,m; \]
\[ R= \begin{bmatrix} r_1\\ r_2\\ \vdots\\ r_m \end{bmatrix}, \qquad r_i= \begin{bmatrix} r_{i1}\\ r_{i2}\\ \vdots\\ r_{in} \end{bmatrix}, \quad i=1,2,\ldots,m; \qquad F= \begin{bmatrix} r_1^*\\ r_2^*\\ \vdots\\ r_m^* \end{bmatrix}, \]
\[ r_i^*= \begin{bmatrix} r_{i1}^*\\ r_{i2}^*\\ \vdots\\ r_{in}^* \end{bmatrix}, \quad i=1,2,\ldots,m. \]
From (6) we obtain
\[ (B^{-1}A-D^{-1}C)\Gamma=B^{-1}R-D^{-1}F, \tag{7} \]
\[ (A^{-1}B-C^{-1}D)\Delta=A^{-1}R-C^{-1}F. \tag{8} \]
Let us find the elements of the square matrices \(B^{-1}A-D^{-1}C\), \(A^{-1}B-C^{-1}D\) of order \(mn\), and of the column vectors \(B^{-1}R-D^{-1}F\), \(A^{-1}R-C^{-1}F\) of the same dimension:
\[ B^{-1}A-D^{-1}C= \begin{bmatrix} c_{11} & 0 & \cdot & 0\\ c_{21} & c_{22} & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot\\ c_{m1} & c_{m2} & \cdot & c_{mm} \end{bmatrix}, \qquad B^{-1}R-D^{-1}F= \begin{bmatrix} \rho_1\\ \rho_2\\ \vdots\\ \rho_m \end{bmatrix}, \]
where
\[ c_{ii}= \begin{bmatrix} \dfrac{q_{i1}}{\beta_{i1}\beta^{*}_{i1}} & 0 & \cdot & 0 & 0\\[1.1em] \dfrac{t_{i1}}{\beta_{i1}\beta_{i2}} & \dfrac{q_{i2}}{\beta_{i2}\beta^{*}_{i2}} & \cdot & 0 & 0\\[1.1em] \dfrac{t_{i1}\eta_{i2}}{\beta_{i1}\beta_{i2}\beta_{i3}} & \dfrac{t_{i2}}{\beta_{i2}\beta_{i3}} & \cdot & 0 & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\[0.8em] \dfrac{t_{i1}\eta_{i2}\ldots\eta_{in-1}}{\beta_{i1}\beta_{i2}\ldots\beta_{in}} & \dfrac{t_{i2}\eta_{i3}\ldots\eta_{in-1}}{\beta_{i2}\beta_{i3}\ldots\beta_{in}} & \cdot & \dfrac{t_{in-1}}{\beta_{in-1}\beta_{in}} & \dfrac{q_{in}}{\beta_{in}\beta^{*}_{in}} \end{bmatrix}, \qquad i=1,2,\ldots,m; \]
\[ c_{i+1\,i}=-\, \begin{bmatrix} \dfrac{t^{*}_{i1}}{\beta^{*}_{i1}\beta^{*}_{i+1\,1}} & 0 & \cdot & 0\\[1.1em] 0 & \dfrac{t^{*}_{i2}}{\beta^{*}_{i2}\beta^{*}_{i+1\,2}} & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \dfrac{t^{*}_{in}}{\beta^{*}_{in}\beta^{*}_{i+1\,n}} \end{bmatrix}, \qquad i=1,2,\ldots,m-1; \]
\[ c_{i+j\,i}=-\, \begin{bmatrix} \dfrac{t^{*}_{i1}\eta^{*}_{i+1\,1}\ldots\eta^{*}_{i+j-1\,1}} {\beta^{*}_{i1}\beta^{*}_{i+1\,1}\ldots\beta^{*}_{i+j\,1}} & 0 & \cdot & 0\\[1.1em] 0 & \dfrac{t^{*}_{i2}\eta^{*}_{i+1\,2}\ldots\eta^{*}_{i+j-1\,2}} {\beta^{*}_{i2}\beta^{*}_{i+1\,2}\ldots\beta^{*}_{i+j\,2}} & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot\\[0.8em] 0 & 0 & \cdot & \dfrac{t^{*}_{in}\eta^{*}_{i+1\,n}\ldots\eta^{*}_{i+j-1\,n}} {\beta^{*}_{in}\beta^{*}_{i+1\,n}\ldots\beta^{*}_{i+j\,n}} \end{bmatrix}, \]
\[ j=2,3,\ldots,m-1;\qquad i=1,2,\ldots,m-j; \]
\[ \rho_i= \begin{bmatrix} \dfrac{R_{i1}}{\beta_{i1}\beta^{*}_{i1}} -\dfrac{r^{*}_{i-1\,1}\eta^{*}_{i-1\,1}}{\beta^{*}_{i-1\,1}\beta^{*}_{i1}} -\ldots -\dfrac{r^{*}_{11}\eta^{*}_{11}\ldots\eta^{*}_{i-1\,1}} {\beta^{*}_{11}\beta^{*}_{21}\ldots\beta^{*}_{i1}} \\[1.3em] \dfrac{r_{i1}\eta_{i1}}{\beta_{i1}\beta_{i2}} +\dfrac{R_{i2}}{\beta_{i2}\beta^{*}_{i2}} -\dfrac{r^{*}_{i-1\,2}\eta^{*}_{i-1\,2}}{\beta^{*}_{i-1\,2}\beta^{*}_{i2}} -\ldots -\dfrac{r^{*}_{12}\eta^{*}_{12}\ldots\eta^{*}_{i-1\,2}} {\beta^{*}_{12}\beta^{*}_{22}\ldots\beta^{*}_{i2}} \\ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \\[0.8em] \dfrac{r_{i1}\eta_{i1}\ldots\eta_{in-1}}{\beta_{i1}\beta_{i2}\ldots\beta_{in}} +\ldots+ \dfrac{r_{in-1}\eta_{in-1}}{\beta_{in-1}\beta_{in}} +\dfrac{R_{in}}{\beta_{in}\beta^{*}_{in}} -\dfrac{r^{*}_{i-1\,n}\eta^{*}_{i-1\,n}}{\beta^{*}_{i-1\,n}\beta^{*}_{in}} -\ldots \\[1.3em] \ldots- \dfrac{r^{*}_{1n}\eta^{*}_{1n}\ldots\eta^{*}_{i-1\,n}} {\beta^{*}_{1n}\beta^{*}_{2n}\ldots\beta^{*}_{in}}, \end{bmatrix}. \]
\[ i=1,\,2,\ldots,m. \]
The elements of the matrix \(A^{-1}B-C^{-1}D\) and of the column vector \(A^{-1}R-C^{-1}F\) are obtained, respectively, from the elements of \(B^{-1}A-D^{-1}C\) and \(B^{-1}R-D^{-1}F\), if one interchanges \(\alpha_{ij}\) and \(\beta_{ij}\), \(\xi_{ij}\) and \(\eta_{ij}\), \(\alpha_{ij}^{*}\) and \(\beta_{ij}^{*}\), \(\xi_{ij}^{*}\) and \(\eta_{ij}^{*}\).
Let us note that for sufficiently small \(h\) all \(q_{ij}\ne 0\), which is an immediate consequence of the hyperbolicity of the original system (for specific restrictions on the step \(h\), see [4]).
We solve equations (7), (8). The fact that the matrices \(B^{-1}A-D^{-1}C\) and \(A^{-1}B-C^{-1}D\) are triangular makes it possible, from the indicated equations, to find successively
\(\gamma_{11}, \delta_{11}, \gamma_{12}, \delta_{12}, \ldots, \gamma_{1n}, \delta_{1n}, \gamma_{21}, \delta_{21}, \ldots, \gamma_{mn}, \delta_{mn}\).
By induction we obtain
\[ \gamma_{1n} = \frac{1}{q_{1n}} \left( R_{1n} + \frac{\beta_{1n}^{*}}{q_{1n-1}} \sum_{i=1}^{n-1} Q_{1i} \prod_{j=1}^{\,n-i-1} \frac{p_{1n-j}}{q_{1n-j-1}} \right), \]
\[ \delta_{1n} = - \frac{1}{q_{1n}} \left( F_{1n} + \frac{\alpha_{1n}^{*}}{q_{1n-1}} \sum_{i=1}^{n-1} Q_{1i} \prod_{j=1}^{\,n-i-1} \frac{p_{1n-j}}{q_{1n-j-1}} \right). \]
From the formulas obtained, taking (5) into account, the following recurrence relations follow directly:
\[ \gamma_{in} = \frac{1}{q_{in}} \left\{ R_{in} - \beta_{in} \bigl( \xi_{i-1\,n}^{*}\gamma_{i-1\,n} + \eta_{i-1\,n}^{*}\delta_{i-1\,n} \bigr) + \right. \]
\[ \left. + \frac{\beta_{in}^{*}}{q_{in-1}} \sum_{j=1}^{n-1} \left[ Q_{ij} + t_{ij} \bigl( \xi_{i-1\,j}^{*}\gamma_{i-1\,j} + \eta_{i-1\,j}^{*}\delta_{i-1\,j} \bigr) \right] \prod_{k=1}^{\,n-j-1} \frac{p_{in-k}}{q_{in-k-1}} \right\}, \]
\[ \delta_{in} = - \frac{1}{q_{in}} \left\{ F_{in} - \alpha_{in} \bigl( \eta_{i-1\,n}^{*}\delta_{i-1\,n} + \xi_{i-1\,n}^{*}\gamma_{i-1\,n} \bigr) + \right. \]
\[ \left. + \frac{\alpha_{in}^{*}}{q_{in-1}} \sum_{j=1}^{n-1} \left[ Q_{ij} + t_{ij} \bigl( \eta_{i-1\,j}^{*}\delta_{i-1\,j} + \xi_{i-1\,j}^{*}\gamma_{i-1\,j} \bigr) \right] \prod_{k=1}^{\,n-j-1} \frac{p_{in-k}}{q_{in-k-1}} \right\}, \]
\[ \gamma_{0j}=\delta_{0j}=0,\qquad i=1,\,2,\ldots,m;\quad j=1,\,2,\ldots,n. \]
The latter, in turn, by induction give the following computational formulas:
\[ \left. \begin{aligned} \gamma_{mn} &= \frac{1}{q_{mn}} \left[ R_{mn} + \frac{\beta_{mn}}{q_{m-1\,n}}\sigma_{m-1\,n} + \frac{\beta_{mn}^{*}}{q_{mn-1}} \sum_{j=1}^{n-1} \left( Q_{mj} - \frac{t_{mj}}{q_{m-1\,j}}\sigma_{m-1\,j} \right) \prod_{i=1}^{\,n-j-1} \frac{p_{mn-i}}{q_{mn-i-1}} \right], \\[1.2em] \delta_{mn} &= - \frac{1}{q_{mn}} \left[ F_{mn} + \frac{\alpha_{mn}}{q_{m-1\,n}}\sigma_{m-1\,n} + \frac{\alpha_{mn}^{*}}{q_{mn-1}} \sum_{j=1}^{n-1} \left( Q_{mj} - \frac{t_{mj}}{q_{m-1\,j}}\sigma_{m-1\,j} \right) \prod_{i=1}^{\,n-j-1} \frac{p_{mn-i}}{q_{mn-i-1}} \right] \end{aligned} \right\} \tag{9} \]
where
\[ \sigma_{il}=\frac{t_{il}^{*}}{q_{il-1}}\sum_{j=1}^{l-1} \left(Q_{ij}-\frac{t_{ij}}{q_{i-1j}}\sigma_{i-1j}\right) \prod_{k=1}^{l-j-1}\frac{p_{il-k}}{q_{il-k-1}} -\frac{p_{il}^{*}}{q_{i-1l}}\sigma_{i-1l}-Q_{il}^{*}, \]
\[ \sigma_{0l}=0,\qquad l=1,\,2,\ldots,n;\quad i=1,\,2,\ldots,m-1. \]
The solution of system (4) is equivalent to the solution of the following system:
\[ \left. \begin{aligned} &\alpha_{ij+1}u_{ij+1}+\beta_{ij+1}v_{ij+1}-\xi_{ij}u_{ij}-\eta_{ij}v_{ij} =\rho_{ij+1},\quad i=1,2,\ldots,m;\\ &\hspace{43mm} j=0,1,\ldots,n-1,\\ &\alpha_{i+1j}^{*}u_{i+1j}+\beta_{i+1j}^{*}v_{i+1j} -\xi_{ij}^{*}u_{ij}-\eta_{ij}^{*}v_{ij} =\rho_{i+1j}^{*},\quad i=0,1,\ldots,m-1;\\ &\hspace{43mm} j=1,2,\ldots,n,\\ &u_{i0}=v_{i0}=0,\quad i=1,\ldots,m;\qquad u_{0j}=v_{0j}=0,\quad j=1,\ldots,n. \end{aligned} \right\} \tag{10} \]
Comparing systems (5) and (10), it is not difficult to observe that the corresponding computational formulas for \(u_{mn}\) and \(v_{mn}\) are obtained from (9) by replacing \(\breve{R}_{mn}\), \(F_{mn}\), \(\sigma_{ij}\), \(Q_{mj}\), \(Q_{il}^{*}\) by \(\overline{R}_{mn}\), \(\overline{F}_{mn}\), \(\overline{\sigma}_{ij}\), \(\overline{Q}_{mj}\), \(\overline{Q}_{il}^{*}\), respectively.
We shall prove convergence and give an error estimate for the method presented.
From (9), having obtained by induction an estimate for \(|\sigma_{m-1n}|\), we find
\[ |\gamma_{mn}|\le h^{2}Q\{(l_n-h)PQR_{1}\lambda_n\beta^{*} +(l_m-h)Q\sigma_n\Lambda_{mn}[\beta+(l_n-h)QT_{1}\lambda_n\beta^{*}]+ \]
\[ +h[R_{1}\beta^{*}+R_{2}\beta+(l_n-h)QR_{2}T_{1}\lambda_n\beta^{*}]\}, \]
\[ |\delta_{mn}|\le h^{2}Q\{(l_n-h)PQR_{1}\lambda_n\alpha^{*} +(l_m-h)Q\sigma_n\Lambda_{mn}[\alpha+(l_n-h)QT_{1}\lambda_n\alpha^{*}]+ \]
\[ +h[R_{1}\alpha^{*}+R_{2}\alpha+(l_n-h)QR_{2}T_{1}\lambda_n\alpha^{*}]\}. \]
Now the convergence of the chosen method with rate of order \(h^{2}\) in any bounded domain is completely obvious.
In addition to the notation adopted in [4], let us explain here that
\[ \beta=\max_{1\le i\le m}\max_{1\le j\le n}|\beta_{ij}|,\qquad \beta^{*}=\max_{1\le i\le m}\max_{1\le j\le n}|\beta_{ij}^{*}|, \]
\[ \alpha=\max_{1\le i\le m}\max_{1\le j\le n}|\alpha_{ij}|,\qquad \alpha^{*}=\max_{1\le i\le m}\max_{1\le j\le n}|\alpha_{ij}^{*}|, \]
\[ \lambda\ge \lambda_n=\lambda_0^{\,n-2},\qquad \lambda_0=\left[1+\frac{h}{b_{12}}\left(B_{12}^{\prime}+H\right)\right]\Big/\left(1-\frac{h}{b_{12}}H\right), \]
\[ \Lambda\ge \Lambda_{mn}=\Lambda_n^{\,m-2},\qquad \Lambda_n=\lambda_0\left[1+h(l_n-h)T_1T_2\lambda_0^{\,n-3}Q^2\right], \]
\[ R_1=h^{-3}\max_{1\le i\le m}\max_{1\le j\le n}|r_{ij}|,\qquad R_2=h^{-3}\max_{1\le i\le m}\max_{1\le j\le n}|r_{ij}^{*}|. \]
Let us note that the problem posed is solved analogously also in the case of quasilinear systems.
In a similar way, passing here to the limit as \(h\to 0\), we obtain an estimate for the exact solution of the problem (for simplicity we shall take the Goursat boundary conditions to be zero):
\[ \begin{gathered} |u(\xi,\eta)| \leq \frac{1}{2}\,\tilde B_{12}\tilde b_{12}^{-2} \bigl[B_1\tilde F_2(\Lambda+\lambda_{(1)})\xi+ \tilde B_2\tilde F_1(\Lambda_{(1)}+\lambda)\eta+{}\\ +\tilde b_{12}^{-1}(\tilde B_1\tilde F_1\tilde T_2+ \tilde B_2\tilde F_2\tilde T_1)(\Lambda\lambda+\Lambda_{(1)}\lambda_{(1)})\xi\eta +\tilde b_{12}^{-2}\tilde T_1\tilde T_2 (\tilde B_1\tilde F_2\Lambda_{(1)}\lambda_{(1)}^2\xi+{}\\ +\tilde B_2\tilde F_1\Lambda\lambda^2\eta)\xi\eta\bigr], \\[1ex] |v(\xi,\eta)| \leq \frac{1}{2}\,\tilde B_{12}\tilde b_{12}^{-2} \bigl[\tilde A_1\tilde F_2(\Lambda+\lambda_{(1)})\xi+ \tilde A_2\tilde F_1(\Lambda_{(1)}+\lambda)\eta+{}\\ +\tilde b_{12}^{-1}(\tilde A_1\tilde F_1\tilde T_2+ \tilde A_2\tilde F_2\tilde T_1)(\Lambda\lambda+\Lambda_{(1)}\lambda_{(1)})\xi\eta +\tilde b_{12}^{-2}\tilde T_1\tilde T_2 (\tilde A_1\tilde F_2\Lambda_{(1)}\lambda_{(1)}^2\xi+{}\\ +\tilde A_2\tilde F_1\Lambda\lambda^2\eta)\xi\eta\bigr]. \end{gathered} \tag{11} \]
where the tilde above denotes the limiting values (as \(h\to 0\)) of the corresponding quantities (see [4]).
The estimate obtained is attainable. Thus, for example, if system (2), (3) has the form
\(A_1^0u_\eta-B_1^0v_\eta-F_1^0=0,\quad A_2^0u_\xi-B_2^0v_\xi+F_2^0=0\), where
\(A_i^0,\ B_i^0,\ F_i^0\ (i=1,2)\) are nonnegative constant numbers, and moreover
\(A_1^0B_2^0-A_2^0B_1^0>0\), then the result given by estimate (11) completely coincides with the exact solution of the corresponding Goursat problem.
Let us note that one may proceed in a similar way when solving other problems as well.
Consider, for example, the Cauchy problem, which consists [5] in finding a solution of the original system, if the functions \(u\) and \(v\) are prescribed on some arc of a smooth curve \(C\) having no characteristic directions at any point.
As in the case of the Goursat problem, after first transferring the initial data from the arc of the curve \(C\) to the nearest nodes of the mesh, here too we obtain systems of linear algebraic equations for finding the approximate solution and the error of the method, similar respectively to systems (4) and (5). The number of equations in the systems obtained will depend essentially also on the chosen arc of the curve \(C\), and the free terms will also reflect the new boundary values. Let us note that the approximate solution of the Cauchy problem (and, analogously, for the error of the method) at the point \((i,j)\) of the mesh is found, according to the proposed method, directly if its values at the points \((i-1,j)\) and \((i,j-1)\) are known; moreover, the circle of operations necessary for this is determined by the interpolation indicated earlier, followed by integration, and coincides exactly with the operations necessary for this in the case of the Goursat problem. This makes it possible to carry out the computation successively along the rows or columns of the mesh in exactly the same way as was done in the preceding problem, introducing each time corrections for the new boundary values and for the number of steps (depending on the arc of the curve \(C\)).
Carefully tracing how the estimates of interest to us were obtained in the case of the Goursat problem, it is not difficult to observe that, if the initial data in the Cauchy problem are also transferred by means of linear interpolation of \(u\) and \(v\), followed by integration of the relations on the characteristics with variable step, then in the case of the Cauchy problem as well one can obtain estimates entirely analogous to those given earlier, with the natural change of the closed domains over which the extremal values of the quantities entering into the constants of the final estimates are taken.
The problem can be solved analogously in the quasilinear case as well.
The approach considered above is carried over in a similar way to both mixed problems [5], for which the corresponding boundary conditions are prescribed on a characteristic issuing from a single point and on some line that nowhere assumes a characteristic direction.
Without dwelling in greater detail on the latter problems, let us note in conclusion that the characteristic mesh need not necessarily be constructed with one and the same constant step along both characteristic directions. Similar results can be achieved when a mesh with different steps is chosen, for example, so that the boundary of the mesh domain is completely superposed on the boundary of the original domain. Such a mesh can, for instance, be obtained by dividing the line on which the boundary conditions are prescribed into segments of length \(h\) and drawing through each point of division one characteristic of each family. In this case the need to transfer the Cauchy conditions to the boundary mesh points disappears, and the question of the number of steps on each layer is solved automatically. The changes connected with the transition to a variable step do not, in this case, introduce any fundamental difficulties, and, following the construction scheme considered by us earlier, one can obtain results analogous to those presented.
References
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A. A. Dorodnitsyn, Proceedings of the Third All-Union Mathematical Congress, 1956, 3, Academy of Sciences of the USSR, Moscow, 1958, pp. 447–453.
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V. M. Belotserkovskii, P. I. Chushkin, Journal of Computational Mathematics and Mathematical Physics, 2, No. 5, 1962, pp. 731–759.
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L. Collatz, Numerical Methods for the Solution of Differential Equations, IL, Moscow, 1953.
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V. V. Bobkov, V. I. Krylov, Differential Equations, 1, No. 2, 1965.
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I. S. Berezin, N. P. Zhidkov, Methods of Computation, 2, Moscow, 1959.
Received by the editors
December 1, 1964
Belorussian State University
named after V. I. Lenin