MATHEMATICS

B. M. BUDAK

ON HOMOGENEOUS DIFFERENTIAL-DIFFERENCE SCHEMES OF SECOND-ORDER ACCURACY FOR PARABOLIC AND HYPERBOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician A. A. Dorodnitsyn on 14 X 1961)

1. In this note we shall consider the solution, by the method of lines, of boundary value problems for the equations

\[ L'(u)\equiv \rho(x,t)u_t-(k(x,t)u_x)_x-\chi(x,t)u_x+q(x,t)u-f(x,t)=0, \]
\[ 0<x<l,\qquad 0<t<T; \tag{1,1'} \]

\[ L''(u)\equiv \rho(x,t)u_{tt}-(k(x,t)u_x)_x-\chi(x,t)u_x+q(x,t)u-f(x,t)=0. \]
\[ 0<x<l,\qquad 0<t<T, \tag{1,1''} \]

with boundary conditions of the form

\[ u(0,t)=\varphi_1(t),\qquad u(l,t)=\varphi_2(t),\qquad 0\leq t\leq T, \tag{1,2a} \]

or of the form

\[ k(0,t)u_x(0,t)=\psi_1(t),\qquad u(l,t)=\psi_2(t),\qquad 0\leq t\leq T, \tag{1,2b} \]

and initial conditions

\[ u(x,0)=\omega(x),\qquad 0\leq x\leq l; \tag{1,3'} \]

\[ u(x,0)=\omega_1(x),\qquad u_t(x,0)=\omega_2(x),\qquad 0\leq x\leq l, \tag{1,3''} \]

respectively. Here \(\rho(x,t)\geq \rho_{\min}>0,\quad k(x,t)\geq k_{\min}>0,\) \(q(x,t)\geq 0\). It is assumed that \(\rho,k,q,\chi,f\), together with their derivatives, undergo jump discontinuities on a finite number of straight lines \(x=X_k\), \(0<X_1<X_2<\cdots<X_N<l\). We shall also write the equations of the boundary lines in the form \(x\equiv X_0=0,\ x\equiv X_{N+1}=0\). The functions \(\rho,k,\chi,q,f,\varphi_1,\varphi_2,\psi_1,\psi_2,\omega,\omega_1,\omega_2\) will be assumed such that each of the boundary value problems \(((1,1'),(1,2a,b),(1,3'));\ ((1,1''),(1,2a,b),(1,3''))\) has in the rectangle \(\Pi:\ 0\leq x\leq l,\ 0\leq t\leq T\) a continuous unique solution satisfying on the straight lines \(x\equiv X_k,\ k=1,2,\ldots,N\), the conjugacy conditions

\[ u\big|_{X_k-0}=u\big|_{X_k+0},\qquad (ku_x)\big|_{X_k-0}=(ku_x)\big|_{X_k+0} \tag{1,4} \]

and possessing uniformly bounded derivatives \(u_x,u_{xx},u_{xxx},u_{xxxx},u_t,u_{tx}\) between the straight lines \(x\equiv X_k,\ k=0,1,\ldots,N+1\)*. In the case of boundary conditions (1,2b) we shall regard the prescribed functions and the solutions of the boundary value problems as smoothly continued through the straight line \(x\equiv X_0=0\) to the left, with preservation of the differential properties. In the estimates we shall also use the uniform boundedness of certain derivatives of the functions \(\rho,k,\chi,q,f,\omega,\omega_1,\omega_2\) between the straight lines \(x\equiv X_k,\ k=0,1,\ldots,N+1\), and the uniform boundedness of certain derivatives of the functions \(\varphi_1,\varphi_2,\psi_1,\psi_2\).

2. In the case of boundary conditions (1,2a) we take the mesh of nodes \(S_1\{x_i=ih,\ i=0,1,\ldots,(n+1);\ (n+1)h=l\}\). In the case of boundary conditions (1,2b) we take the mesh of nodes \(S_2\{x_i=(i-\tfrac12)h,\ i=0,1,\ldots,(n+1);\ (n+1)h=l\}\). We shall assume that each of the straight lines \(x\equiv X_k,\ k=1,\ldots,N+1\), passes through one of the mesh nodes**.

* The existence and uniqueness of the solution and the investigation of differential properties for an equation with discontinuous coefficients were considered in \((^{2,5})\).

** In order that this can be achieved with a step \(h\) common to the entire interval \(0\leq x\leq l\), it may be necessary to make a change of the variable \(x\).

We shall approximate equations \((1,1')\) and \((1,1'')\) by the equations

\[ L'_h(u_i)=\widetilde{\rho}_i\frac{du_i}{dt} -\frac{1}{h^2}\,[k_{i+1/2}(u_{i+1}-u_i)-k_{i-1/2}(u_i-u_{i-1})]- \]
\[ -\varkappa_{i+1/2}\frac{u_{i+1}-u_i}{2h} -\varkappa_{i-1/2}\frac{u_i-u_{i-1}}{2h} +\widetilde{q}_i u_i-\widetilde{f}_i=0; \tag{2,1'} \]

\[ L''_h(u_i)=\widetilde{\rho}_i\frac{d^2u_i}{dt^2} -\frac{1}{h^2}\,[k_{i+1/2}(u_{i+1}-u_i)-k_{i-1/2}(u_i-u_{i-1})]- \]
\[ -\varkappa_{i+1/2}\frac{u_{i+1}-u_i}{2h} -\varkappa_{i-1/2}\frac{u_i-u_{i-1}}{2h} +\widetilde{q}_i u_i-\widetilde{f}_i=0, \tag{2,1''} \]

\[ \widetilde{\rho}_i=\frac12[\rho(x_i-0,t)+\rho(x_i+0,t)],\quad \widetilde{q}_i=\frac12[q(x_i-0,t)+q(x_i+0,t)], \]

\[ \widetilde{f}_i=\frac12[f(x_i-0,t)+f(x_i+0,t)],\quad k_{i\pm1/2}=k(x_i\pm h/2,t),\quad \varkappa(x_i\pm h/2,t)=\varkappa_{i\pm1/2},\quad 1\leq i\leq n. \]

The boundary conditions \((1,2a)\) and \((1,2)\) shall be replaced, respectively, by the conditions

\[ u_0(t)=\varphi_1(t),\qquad u_{n+1}(t)=\varphi_2(t),\qquad 0\leq t\leq T; \tag{2,2a} \]

\[ l_h(u_i)\equiv k_{1/2}\frac{u_1-u_0}{h}-\varphi_1(t)=0,\qquad u_{n+1}=\varphi_2(t),\qquad 0\leq t\leq T. \tag{2,2б} \]

The initial conditions \((1,3')\) and \((1,3'')\) shall be replaced, respectively, by the conditions

\[ u_i(0)=\omega(x_i),\qquad i=0,1,\ldots,n+1; \tag{2,3'} \]

\[ u_i(0)=\omega_1(x_i),\qquad u'_i(0)=\omega_2(x_i),\qquad i=0,1,\ldots,n+1. \tag{2,3''} \]

1. Let \(u^{(a)}(x,t)\) and \(u^{(б)}(x,t)\) denote the solutions of the boundary-value problems with boundary conditions \((1,2a)\) and \((1,2б)\), and let \(u_i^{(a)}(t)\) and \(u_i^{(б)}(t)\) be the solutions of the corresponding differential-difference boundary-value problems with boundary conditions \((2,2a)\) and \((2,2б)\)*.

Theorem. Under the formulated conditions, the errors
\(\delta_i^{(a)}=u^{(a)}(x_i,t)-u_i^{(a)}(t)\),
\(\delta_i^{(б)}=u^{(б)}(x_i,t)-u_i^{(б)}(t)\)
are of order \(O(h^2)\) as \(h\to0\).

Proof. We give the outline of the proof. First we determine the order of approximation for the equations and boundary conditions, i.e., the order with respect to \(h\) of the residual terms obtained when the exact solutions of the differential boundary-value problems are substituted into the differential-difference equations and boundary conditions. Then we obtain differential-difference boundary-value problems for the errors \(\delta_i^{(a)}\) and \(\delta_i^{(б)}\), after which we estimate \(\delta_i^{(a)}\) and \(\delta_i^{(б)}\), using the “energy integral” that we previously applied in the case of equations with continuous coefficients.

1. Let us find the approximation error for equations \((2,1')\), \((2,1'')\). If \(x=x_i\) is not a point of discontinuity, then, applying Taylor expansions for \(k(x,t)\), \(\varkappa(x,t)\), and \(u(x,t)\), we easily obtain

\[ L'_h(u(x_i,t))=L'(u(x,t))_{x=x_i}+O(h^2)=O(h^2); \tag{4,1'_1} \]

\[ L''_h(u(x_i,t))=L''(u(x,t))_{x=x_i}+O(h^2)=O(h^2). \tag{4,1''_1} \]

If \(x=x_i\) lies on one of the internal straight lines \(x=X_k\), i.e., is a point of discontinuity, then, integrating \((1,1')\) with respect to \(x\) from \(x_i-h/2\) to \(x_i\) and from \(x_i\) to \(x_i+h/2\), we obtain:

\[ \int_{x_i-h/2}^{x_i}\rho(x,t)\frac{\partial u}{\partial t}\,dx = \left[k(x,t)\frac{\partial u}{\partial x}\right]_{x_i-0} - \left[k(x,t)\frac{\partial u}{\partial x}\right]_{x_i-h/2} + \]

* The actual solution of these problems for the systems of ordinary differential equations \((2,1')\) and \((2,1'')\) can be carried out by numerical integration, as in the case of the more general method of integral relations (1), using any standard program.

\[ + \int_{x_i-h/2}^{x_i} \left[\varkappa(x,t)\frac{\partial u}{\partial x}-q(x,t)u+f(x,t)\right]\,dx, \]

\[ \int_{x_i}^{x_i+h/2}\rho(x,t)\frac{\partial u}{\partial t}\,dx = \left[k(x,t)\frac{\partial u}{\partial x}\right]_{x_i+h/2} - \left[k(x,t)\frac{\partial u}{\partial x}\right]_{x_i+0} + \int_{x_i}^{x_i+h/2} \left[\varkappa(x,t)\frac{\partial u}{\partial x}-q(x,t)u+f(x,t)\right]\,dx . \]

We transform these equations, using rectangular quadrature formulas of the form

\[ \int_{x_i}^{x_i+h/2} F(x)\,dx = \frac{h}{2}F(x_i+0)+\frac{h^2}{2}F'(\xi_1), \]

\[ \int_{x_i-h/2}^{x_i} F(x)\,dx = \frac{h}{2}F(x_i-0)+\frac{h^2}{2}F'(\xi_2)* \]

and the formulas

\[ \left.\frac{\partial u}{\partial x}\right|_{x_i-h/2} = \frac{u(x_i,t)-u(x_i-h,t)}{h}+O(h^2), \]

\[ \left.\frac{\partial u}{\partial x}\right|_{x_i+h/2} = \frac{u(x_i+h,t)-u(x_i,t)}{h}+O(h^2), \]

and then subtract one from the other and, using conditions (1,4), divide the result by \(h\). This gives

\[ L'_h(u(x_l,t))=O(h). \tag{4,1\(_2'\)} \]

Similarly we obtain:

\[ L''_h(u(x_l,t))=O(h). \tag{4,1\(_2''\)} \]

For the first boundary condition (2,26) we obtain

\[ l_h(u(x_l,t))\equiv k_{1/2}\frac{u(x_1,t)-u(x_0,t)}{h}-\psi_1(t)=O(h^2). \tag{4,26} \]

The remaining boundary and initial conditions are approximated exactly.

1. Now for \(\delta_i^{(a)}\) and \(\delta_i^{(b)}\) we obtain differential-difference boundary-value problems and estimate \(\delta_i^{(a)}\) and \(\delta_i^{(b)}\) by means of the “energy integral.” Let us consider in detail, for example, the case of the first boundary-value problem for equation \((1,1'')\). We have

\[ \widetilde{\rho}_i\frac{d^2\delta_i}{dt^2} = \frac{1}{h^2}\left[k_{i+1/2}(\delta_{i+1}-\delta_i)-k_{i-1/2}(\delta_i-\delta_{i-1})\right] + \]

\[ +\varkappa_{i+1/2}\frac{\delta_{i+1}-\delta_i}{2h} + \varkappa_{i-1/2}\frac{\delta_i-\delta_{i-1}}{2h} - \widetilde{q}_i\delta_i + R_i h^{\sigma_i}, \qquad 0<i<n+1; \tag{5,1''} \]

\[ \delta_0(t)=0,\qquad \delta_{n+2}(t)=0; \tag{5,2a} \]

\[ \delta_i(0)=0,\qquad \delta_i'(0)=0,\qquad 0\le i\le n+1, \tag{5,3} \]

where \(\sigma_i=2\) if \(x_i\) is a point of continuity, and \(\sigma_i=1\) if \(x_i\) is a point of discontinuity, while \(R_i=O(1)\). Consider the “energy integral”

\[ I(t)= \sum_{i=1}^{n}\widetilde{\rho}_i\dot{\delta}_i^{\,2} + \sum_{i=0}^{n}k_{i+1/2} \left(\frac{\delta_{i+1}-\delta_i}{h}\right)^2 + \sum_{i=1}^{n}\widetilde{q}_i\delta_i^2. \tag{5,4} \]

* In the right-hand sides of these equalities, when integrating \(\varkappa(x,t)\partial u/\partial x\), instead of \(x_i\pm0\) one should take \(x_i\pm h/2\), respectively.

Differentiating, we obtain, using equations \((5,1')\) and the boundary conditions \((5,2a)\):

\[ \frac{dI}{dt} = 2\sum_{i=1}^{n} \left( \chi_{i+1/2}\frac{\delta_{i+1}-\delta_i}{2h} + \chi_{i-1/2}\frac{\delta_i-\delta_{i-1}}{2h} + R_i h\delta_i \right)\delta_i' + \]

\[ + \sum_{i=1}^{n}\widetilde{\rho}_i\delta_i'^2 + \sum_{i=0}^{n}k_{i+1/2}' \left(\frac{\delta_{i+1}-\delta_i}{h}\right)^2 + \sum_{i=1}^{n}\widetilde{q}_i'\delta_i^2 . \tag{5,5} \]

Without loss of generality, in what follows we shall assume that there is only one line of discontinuity, \(x=x_{i_0}\). Accordingly, from the first sum on the right-hand side of \((5,5)\), in making estimates we shall single out the term with the first power of \(h\), i.e. \(R_{i_0}\delta_{i_0}'h\). Using the relation (see \((3)\))

\[ |\delta_i| \leq \sqrt{\frac{x_i(l-x_i)}{l k_{\min}}}\sqrt{I(t)}\,h^{1/2} \tag{5,6} \]

and expression \((5,4)\), we obtain

\[ \sum_{i=1}^{n}\widetilde{\rho}_i\delta_i'^2 + \sum_{i=0}^{n}k_{i+1/2}' \left(\frac{\delta_{i+1}-\delta_i}{h}\right)^2 + \sum_{i=1}^{n}\widetilde{q}_i'\delta_i^2 \leq C_1 I(t), \qquad C_1=\mathrm{const}>0 . \]

Using the Cauchy inequality, we find

\[ \left| 2\sum_{\substack{i=1\\ i\ne i_0}}^{n} R_i\delta_i' h^2 \right| + 2R_{i_0}\delta_{i_0}'h \leq C_2 h^{3/2}\sqrt{I(t)} + 2R_{i_0}\delta_{i_0}'h, \]

\[ \left| 2\sum_{i=1}^{n} \left( \chi_{i+1/2}\frac{\delta_{i+1}-\delta_i}{2h} + \chi_{i-1/2}\frac{\delta_i-\delta_{i-1}}{2h} \right)\delta_i' \right| \leq C_3 I(t). \]

As a result we obtain

\[ \frac{dI}{dt} \leq C_4 I + C_4 h^{3/2}\sqrt{I} + 2hR_{i_0}\delta_{i_0}' . \]

Integrating this inequality, taking into account the initial conditions \(I(0)=0\), \(\delta_{i_0}(0)=0\), and applying inequality \((5,6)\), we obtain

\[ I - C_4\int_{0}^{t} I\,dt \leq C_6 h^{3/2} \left( \int_{0}^{t}\sqrt{I}\,dt + \sqrt{I} \right). \tag{5,7} \]

Multiplying the left-hand side of this inequality by \(e^{-C_4t}\), we obtain

\[ \frac{d}{dt} \left( e^{-C_4t}\int_{0}^{t} I\,dt \right) \leq h^{3/2}C_6 \left( \int_{0}^{t}\sqrt{I}\,dt + \sqrt{I} \right). \]

Integrating this inequality from \(0\) to \(t\), substituting the result in \((5,7)\), and then applying the Cauchy inequality, we find

\[ \sqrt{\max_{0\leq t\leq T} I(t)} \leq Ch^{3/2}. \]

Consequently, by virtue of \((5,6)\), it follows that

\[ |\delta_i| \leq h^2 C^* \sqrt{\frac{x_i(l-x_i)}{l}} . \]

Other boundary-value problems are considered analogously; moreover, the energy integral for equation \((1,1')\) is written in the form of expression (10) on p. 11 in \((^3)\).

Homogeneous difference schemes for equations of parabolic type with discontinuous coefficients were considered by another method in \((^4)\).

Moscow State University
named after M. V. Lomonosov

Received
12 X 1961

References

1. A. A. Dorodnitsyn, Proceedings of the Third All-Union Mathematical Congress, Moscow, 1956.
2. A. A. Samarskii, DAN, 121, No. 2 (1958).
3. B. M. Budak, Vestn. MGU, No. 1, 3 (1956).
4. A. A. Samarskii, Proceedings of the All-Union Conference on Differential Equations in Yerevan, 1959.
5. Yu. V. Egorov, DAN, 134, No. 2 (1960).