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MATHEMATICS
E. G. DYAKONOV
ON THE APPLICATION OF DIFFERENCE SCHEMES WITH A SPLITTING OPERATOR TO HYPERBOLIC EQUATIONS WITH VARIABLE COEFFICIENTS
(Presented by Academician S. L. Sobolev on 23 II 1963)
The present work is devoted to the study of economical difference methods for equations of hyperbolic type with variable coefficients and is a direct continuation and strengthening of the results of works \((^{1-5})\); the methodology used in it essentially coincides with the methodology of work \((^{4})\). We note that works \((^{6,7})\) are also devoted to a closely related circle of questions.
1. The initial problem is the following.
Problem 1. In the cylinder \(Q_T=\overline{\Omega}\times[0\le x_0\le T]\), where \(\overline{\Omega}\) is a closed domain in the space \(x'=(x_1,x_2,\ldots,x_p)\), composed of a finite number of parallelepipeds with faces parallel to the coordinate planes, one seeks a solution of the equation
\[ D_0^2u=\sum_{k,s=1}^{p}D_k\bigl(a_{ks}(x)D_su\bigr)+\sum_{s=1}^{p}\bigl(b_s(x)D_su+c_s(x)u\bigr)+f(x), \tag{1} \]
satisfying the initial and boundary conditions
\[ u\bigm|_{x_0=0}=\varphi(x');\qquad D_0u\bigm|_{x_0=0}=\mu(x');\qquad u\bigm|_{S}=\psi(x),\quad x\in S. \tag{2} \]
Here \(D_s=\partial/\partial x_s\) \((s=0,1,\ldots,p)\), \(x=(x_0,x')\), \(S\) is the lateral surface of \(Q_T\),
\[ a_{ks}(x)=a_{sk}(x),\qquad a_s=a_{ss}(x)\ge \gamma=\mathrm{const}>0 \]
\[ \left|\sum_{k\ne s} a_{ks}\xi_k\xi_s\right|\le (1-\sigma)\sum_{s=1}^{p}a_s\xi_s^2, \tag{3} \]
\(\sigma>0\), \(\xi=(\xi_1,\xi_2,\ldots,\xi_p)\) is an arbitrary real vector.
2. To construct a difference analogue of problem 1, introduce the notation:
\(i=(i_1,i_2,\ldots,i_p)\) (\(i_s\) is an integer), \(h>0\) is the step of the spatial grid, \(\{ih\}\) is the set of points of the spatial grid; accordingly the set of interior points \(\Omega_h\), the set of boundary points \(\Gamma_h\), and \(\overline{\Omega}_h=\Omega_h\cup\Gamma_h\) are defined; the value of the grid function \(v\) at the point \((n\tau,ih)\), where \(\tau>0\) is the step in \(x_0\), is denoted by \(v_i^n\),
\[ v_i=v(ih),\qquad {}^{\pm s}\!i=(i_1,i_2,\ldots,i_s\pm1,\ldots,i_p),\qquad {}^{\pm s}\!v_i=v({}^{\pm s}\!i\,h), \]
\[ \Delta_s v_i=\frac{1}{h}\bigl({}^{+s}\!v_i-v_i\bigr),\qquad \overline{\Delta}_s v_i=\frac{1}{h}\bigl(v_i-{}^{-s}\!v_i\bigr),\qquad \widetilde{\Delta}_s v_i=\frac12(\Delta_s+\overline{\Delta}_s)v_i, \]
\[ (s=1,2,\ldots,p), \]
\[ \Delta_0 v_i^n=\frac{1}{\tau}\bigl(v_i^{n+1}-v_i^n\bigr),\qquad {}^{+\hat{s}}\!i=\left(i_1,\ldots,i_s+\frac12,\ldots,i_p\right), \]
\[ {}^{+\hat{s}}\!a_i=a({}^{+\hat{s}}\!i\,h),\qquad t_n=n\tau. \]
\[ L_s^n v_i = L_s v_i = \overline{\Delta}_s\!\left(a_s(t_n,ih)\,\Delta_s v_i\right) + b_s(t_n,ih)\,\widetilde{\Delta}_s v_i + c_s(t_n,ih)\,v_i,\qquad A^n v_i = A_s v_i = \left(E-\frac{\tau^2}{2}L_s\right)v_i, \]
\(E\) is the identity operator,
\[ A v_i=\prod_{s=1}^{p} A_s v_i,\qquad Bv_i^n=\left(2E+\tau^2\sum_{k\ne s}\left(D_k a_{ks}^n\widetilde{\Delta}_s+a_{ks}^n\widetilde{\Delta}_k\widetilde{\Delta}_s\right)\right)v_i^n . \]
By the difference analogue of problem I, or problem \(I_h\), we shall mean the problem of finding a grid function \(v\) satisfying the relation with the splitting operator \(A\)
\[ \frac{1}{\tau^2}A\left(v_i^{n+1}+v_i^{n-1}\right) = \frac{1}{\tau^2}Bv_i^n+f_i^n,\qquad ih\in\Omega_h \tag{4} \]
and the initial and boundary conditions
\[ v_i^0=\varphi_i,\qquad v_i^1=g_i\qquad (ih\in\Omega_h), \]
\[ v_i^k=\psi_i^k\qquad \left(ih\in\Gamma_h,\ k=0,1,\ldots,\frac{T}{\tau}\right), \tag{5} \]
where the function \(g_i\), expressed explicitly in terms of \(\varphi,\mu,f\), is such that \(g_i=u_i^1+O(\tau^q)\), where \(u\) is the exact solution of problem I, \(q\ge 2\).
It can be verified that, for sufficiently smooth coefficients of equation (1), problem \(I_h\), on the class of sufficiently smooth solutions of problem I, will approximate the latter in the metric of the space \(C\) with order of approximation \(O(\tau^2+h^2)\). Moreover, as follows from (5), the transition in problem \(I_h\) from one time layer to another requires \(\sim 1/h^p\) arithmetic operations.
- To formulate requirements on the coefficients of equation (1), let us introduce the following function spaces: \(H^0\) is the space of functions bounded in \(Q_T\):
\[ \|u\|_{H^0}=\sup_{x\in Q_T}|u(x)|; \]
\(H_l(m)\) \((l=0,1,\ldots,p)\) is the space of functions having in \(Q_T\) bounded derivatives with respect to \(x_l\) of order not higher than \(m\), satisfying the Lipschitz condition with respect to \(x_l\); \(D_s^m\) \((s=1,2,\ldots,p)\) is the space of functions having bounded derivatives in \(Q_T\), containing no more than \(m\) differentiations with respect to each \(x_l\) \((l\ne 0,\ l\ne s)\). The norms in the enumerated spaces are introduced as sums of the \(H^0\)-norms of all derivatives entering into the definition of the corresponding space.
Theorem 1. If conditions (3) are satisfied and
\[ a_s\in H_s(2),\qquad a_s\in D_s^1,\qquad b_s\in D_s^1,\qquad c_s\in D_s^1\quad (s=1,2,\ldots,p); \tag{6} \]
\[ a_s\in H_0(0),\qquad D_k a_{ks}\in H^0,\qquad a_{ks}\in H_0(0)\quad (k\ne s),\qquad f\in H^0, \]
then there exist such \(\tau_0>0\) and \(h_0>0\) that, for all \(\tau\le\tau_0,\ h\le h_0\), for the grid function \(y\), which is a solution of problem \(I_h\) with zero boundary conditions \((\psi=0)\), the difference a priori estimate
\[ \begin{aligned} V(y^k,y^{k-1}) \equiv {}& [(\Delta_0 y^{k-1})^2,1] +\sum_{|\nabla|\le 1}\left[((\nabla y^k)^2+(\nabla y^{k-1})^2),1\right] \\ &+\tau^2\sum_{|\nabla|\le 2}\left[((\nabla y^k)^2+(\nabla y^{k-1})^2),1\right] +\cdots +\tau^{2(p-1)}\sum_{|\nabla|\le p}\left[((\nabla y^k)^2+(\nabla y^{k-1})^2),1\right] \\ &\le M\left\{\tau\sum_{n=1}^{k-1}[(f^n)^2,1]+V(y^1,y^0)\right\}, \end{aligned} \tag{7} \]
where \([z,w]=h^p \sum_{ih\in\bar\omega_h} z_i w_i,\ M>0\) is a certain constant, and \(\nabla y\) is a mixed difference with respect to the spatial variables of order \(|\nabla|\), containing no more than one “differentiation” with respect to each spatial variable, \(k\le T/\tau\).
The constants \(\tau_0\) and \(h_0\) in Theorem 1 depend, first, on the dimensions of the domain \(Q_T\) and, second, on the norms of the coefficients in the corresponding spaces (conditions (6)). In a number of special cases the restrictions on the smallness of \(\tau\) and \(h\) due to the second reason become superfluous. For example, if all \(a_{ks}\equiv 0\) for \(k\ne s\), or \(a_s=\mathrm{const}\), then the restriction on the smallness of \(h\) disappears; if the coefficient \(a_s\) does not depend on \(x_0\), then the restriction on the smallness of \(\tau\) disappears.
It follows obviously from Theorem 1 that, for \(\tau\le \tau_0,\ h\le h_0\), problem \(I_h\) is well posed; here we note the fact that no restrictions on the smallness of \(\tau/h\) were imposed.
Moreover, if the approximation conditions are satisfied, then for the difference of the solutions of problems \(I\) and \(I_h\), \(z=u-v\), using the a priori estimate (7), one can obtain
\[ V\left(z^k,z^{k-1}\right) = O\left(\tau^2+h^2\right)^2 + \sum_{r=0}^{p-1} O\left(\frac{\tau^{q+r}}{h^{r+1}}\right)^2 . \tag{8}^* \]
In particular, if \(q\ge 3\) and \(\tau/h\) is bounded above, then
\[ V\left(z^k,z^{k-1}\right)=O\left(\tau^2+h^2\right)^2 . \tag{9} \]
4. Convergence of the solutions of problem \(I_h\) to the solution of the problem with the rate of convergence (9) can be obtained under weaker conditions than the approximation conditions, if one refines estimate (7) for the case of a function \(f\) representable in “divergence” form. Namely, let
\[ f = F - \left(\frac{\tau^2}{2}\right)^3 \sum_{s_1\,s_4} L_{s_1}L_{s_2}L_{s_3}L_{s_4}\Phi + \cdots + \left(-\frac{\tau^2}{2}\right)^{p-1} L_1L_2\cdots L_p\Phi, \tag{10} \]
where \(\sum_{s_l\,s_k}\) denotes summation over all \(s_l\) \((l=1,2,\ldots,k)\), \(s_1<s_2<\cdots<s_k,\ s_l=1,2,\ldots,p\), and the function \(\Phi\) has in \(Q_T\) bounded derivatives with respect to all spatial variables, containing no more than one differentiation with respect to each \(x_s\) \((s=1,2,\ldots,p)\). Then the following theorem is valid:
Theorem 2. Under the conditions of Theorem 1, for the mesh function \(y\) that is the solution of problem \(I_h\) with zero boundary conditions and right-hand side \(f\) representable in the form (10), there exist \(\tau_0>0,\ h_0>0\) such that for all \(\tau\le\tau_0,\ h\le h_0\) the a priori estimate holds
\[ \begin{aligned} V\left(y^k,y^{k-1}\right) \le M\Bigg\{ &V\left(y^1,y^0\right) + \tau\sum_{n=1}^{k-1} \Bigg\{ \left[(F^n)^2,1\right] + \tau^4 \sum_{|\nabla|\le 4} \left[(\nabla\Phi^n)^2,1\right] \\ &\quad + \tau^6 \sum_{|\nabla|\le 5} \left[(\nabla\Phi^n)^2,1\right] + \cdots + \tau^{2p-4} \sum_{|\nabla|\le p} \left[(\nabla\Phi^n)^2,1\right] \Bigg\} \Bigg\} \\ \le M\left\{ &V\left(y^1,y^0\right) + \tau\sum_{n=1}^{k-1} \left[(F^n)^2,1\right] \right\} + O\left(\tau^4\right). \end{aligned} \tag{11} \]
If it is additionally assumed that \(\dfrac{\tau^2}{h}\le l<\infty\) (\(l\) arbitrary), then the a priori estimate (11) can also be obtained in the case when
\[ \text{* Estimate (9) will also be valid for } q=2,\ \text{if one assumes } D_sD_0^3u\in H^0\ (s=1,2,\ldots,p). \]
the function \(\Phi\) from (10) has in \(Q_T\) bounded derivatives only of the form
\[ D_{s_1}D_{s_2}D_{s_3}D_{s_4}\Phi(s_1>s_2>s_3>s_4;\ s_k=1,2,\ldots,p;\ k=1,2,3,4)^*. \]
- A priori estimates of the type (7), (11) can be obtained for an entire class of difference schemes with a splitting operator, but their description is rather cumbersome. Therefore we indicate only a few cases.
For example, relation (4) may be replaced by any of the following:
\[ \frac{1}{\tau^2}Av_i^{n+1} = \frac{1}{\tau^2}Bv_i^n - \frac{1}{\tau^2} \left(E-\frac{\tau^2}{2}\sum_{s=1}^{p}L_s\right)v_i^{n-1} + f_i^n, \tag{12} \]
\[ \frac{1}{\tau^2} \left\{\bar A\left(v_i^{n+1}+v_i^{n-1}\right)-2\bar Bv_i^n\right\} = f_i^n; \tag{13} \]
where
\[ \bar A=\prod_{s=1}^{p}\left(E-\frac{\tau^2}{2}L_s\right), \qquad \bar B=\bar A+\frac{\tau^2}{2}\sum_{s=1}^{p}L_s+\frac{1}{2}(B-2E). \]
Moreover, in all the indicated schemes the operator \(L_s\) may be understood as the operator
\[ L_s=\bar\Delta_s\left(\hat a_s\Delta_s\right)+b_s\widetilde\Delta_s+c_sE. \]
Then, naturally, in the operators \(B,\bar B\) additional terms of the form
\[ \tau^2\sum_{s=1}^{p} \left\{ \bar\Delta_s\left((a_s-\hat a_s)\Delta_s\right) + (b_s-\widetilde b_s)\hat\Delta_s + (c_s-\widetilde c_s)E \right\} \]
must appear. In this case it is required that the coefficients \(a_s,b_s,c_s\) satisfy the same conditions (6) as the original coefficients, and that the coefficients \(a_s-\hat a_s\), \(b_s-\widetilde b_s\), \(c_s-\widetilde c_s\) satisfy weaker conditions. It is also required that the coefficients \(a_s-\hat a_s\) \((s=1,2,\ldots,p)\) be sufficiently small.
Remark. Analogous difference schemes and a priori estimates have been obtained for a differential problem more general than Problem I. Namely, equation (1) may be replaced by
\[ \rho(x)D_0^2u = \sum_{k,s=1}^{p}D_k\left(a_{ks}(x)D_su\right) + f(x,u,D_1u,\ldots,D_pu), \]
where \(\rho(x)\geq \rho>0\), \(\rho(x)\in H_0(0)\), the coefficients \(a_{ks}\) satisfy conditions (3), and the function \(f\) has bounded partial derivatives with respect to \(u,D_su\) \((s=1,2,\ldots,p)\).
Moscow State University
named after M. V. Lomonosov
Received
13 II 1963
References
- E. G. D’yakonov, DAN, 144, No. 1 (1962).
- E. G. D’yakonov, Zhurn. vychislit. matem. i matem. fiz., 2, No. 4 (1962).
- E. G. D’yakonov, UMN, 17, issue 4 (1962).
- E. G. D’yakonov, Dissertation, V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 1962.
- E. G. D’yakonov, Zhurn. vychislit. matem. i matem. fiz., 3, No. 2 (1963).
- A. N. Konovalov, DAN, 147, No. 1 (1962).
- A. A. Samarskii, Zhurn. vychislit. matem. i matem. fiz., 2, No. 5 (1962).
\[ \text{—} \]
\(^*\) In those cases when, for a difference problem written in the form \(Lv=\mathscr L F\) (\(L\) and \(\mathscr L\) are certain difference operators, \(F\) is a known function, \(v\) is the solution of the difference problem), it is possible to obtain an estimate analogous to (11) of the form \(\|v\|_1\leq M\|F\|_2\), it is apparently appropriate to say that the difference problem approximates the differential problem if \(Lv-Lu=\mathscr L(f)\) (\(u\) is the solution of the differential problem) and \(\|f\|_2\to0\) as the mesh size decreases.