MATHEMATICS

V. I. LEBEDEV

THE METHOD OF ORTHOGONAL PROJECTIONS FOR A FINITE-DIFFERENCE ANALOGUE OF A CERTAIN SYSTEM OF EQUATIONS

(Presented by Academician S. L. Sobolev on 16 XI 1956)

In the present note we investigate properties of the solutions of a finite-difference analogue of the system of equations:

\[ \frac{\partial \mathbf U}{\partial t}=A\mathbf U-\operatorname{grad}p+\mathbf F,\qquad \operatorname{div}\mathbf U=0, \tag{1} \]

where \(\mathbf U=(u_1(x_1,x_2,x_3,t),u_2(x_1,x_2,x_3,t),u_3(x_1,x_2,x_3,t))\); \(\mathbf F=(f_1,f_2,f_3)\); \(A\) is a matrix with bounded elements.

For system (1) one poses either the Cauchy problem, in which case one is given

\[ \mathbf U\big|_{t=0}=\mathbf U_0(x_1,x_2,x_3), \tag{2} \]

or a mixed problem: one seeks a solution of (1) in a simply connected domain \(\Omega\), satisfying condition (2) and one more of the two conditions: either

\[ p\big|_{S}=0, \tag{3} \]

or

\[ \sum_{i=1}^{3} u_i\cos(n,x_i)\big|_{S}=U_n\big|_{S}=0, \tag{4} \]

where \(S\) is the boundary of the domain \(\Omega\), and \(n\) is the normal to the boundary. A system of type (1) was investigated by S. L. Sobolev \((^1)\); the proofs of existence of solutions of the listed problems for system (1) do not differ in principle from the proofs in \((^1)\).

Let the space \(R_3(x_1,x_2,x_3)\) be given. The set of points \(x\in R_3\) with coordinates \(x_i=k_i h,\ i=1,2,3\), where \(h>0\) and \(k_i\) are integers, will be denoted by \(M_h\). The set of points \(x\in M_h\) for which \(\sum_{i=1}^{3} k_i=2j,\ j=0,\pm1,\pm2,\ldots\), will be denoted by \(M_{1h}\), and the set of points \(x\in M_h\) for which \(\sum_{i=1}^{3} k_i=2j+1\) will be denoted by \(M_{2h}\). If the domain \(\Omega\) is finite, then we shall consider that \(x(k_1h,k_2h,k_3h)\in\Omega_{2h}\) if \(x\in M_{2h}\) and the octahedron with center at the point \(x\) and with diagonals parallel to the coordinate axes and of length \(4h\) belongs to \(\overline{\Omega}\). Define the boundary points for \(\Omega_{2h}\): we shall say that \(x\in S_{2h}\) if at distance \(2h\) from the point \(x\) there are points of \(M_{2h}\) both belonging and not belonging to \(\Omega_{2h}\); denote \(\overline{\Omega}_{2h}=\Omega_{2h}+S_{2h}\). We shall say that \(x\in\Omega_{1h}\) if \(x\in M_{1h}\) and at distance \(h\) from the point \(x\) there are 6 points of the set \(\overline{\Omega}_{2h}\). Let a function \(\varphi(x_1,x_2,x_3)\) be given on \(M_{2h}\); then denote:

\[ \varphi_{x_1}(x_1,x_2,x_3)=\frac{1}{2h}\bigl(\varphi(x_1+h,x_2,x_3)-\varphi(x_1-h,x_2,x_3)\bigr), \]

where \(x(x_1,x_2,x_3)\in M_{1h}\); \(\varphi_{x_2}, \varphi_{x_3}\) are defined analogously, and \(\varphi_{x_i}\), \(i=1,2,3\), are regarded as defined at the points \(x\in M_{1h}\); \(\varphi_{x_i}\) are defined analogously at the points \(x\in M_{2h}\) through the values of the function \(\varphi\) prescribed on the set \(M_{1h}\).

Let \(H_{ih}\), \(i=1,2\), be the spaces of vectors \(\mathbf v\) defined in \(\Omega_{ih}\), such that \(h^3\sum_{\Omega_{ih}}\|\mathbf v\|^2<C\). We introduce scalar products in \(H_{ih}\):

\[ (\mathbf v^{(1)},\mathbf v^{(2)})_{ih} = h^3\sum_{\Omega_{ih}} \left( v_1^{(1)}v_1^{(2)} + v_2^{(1)}v_2^{(2)} + v_3^{(1)}v_3^{(2)} \right); \tag{5} \]

the notions of difference gradient, difference curl, and difference divergence:

\[ \operatorname{grad}_h\varphi=(\varphi_{x_1},\varphi_{x_2},\varphi_{x_3}), \tag{6} \]

\[ \operatorname{rot}_h\vec\psi= \bigl((\psi_{3x_2}-\psi_{2x_3}),(\psi_{1x_3}-\psi_{3x_1}),(\psi_{2x_1}-\psi_{1x_2})\bigr), \tag{7} \]

\[ \operatorname{div}_h\mathbf v=v_{1x_1}+v_{2x_2}+v_{3x_3}. \tag{8} \]

Theorem 1. For a vector \(\mathbf v\), defined on \(M_{1h}\), to be representable in the form \(\mathbf v=\operatorname{grad}_h\varphi\), where \(\varphi\) is defined on \(M_{2h}\), it is necessary and sufficient that

\[ \operatorname{rot}_h\mathbf v=0. \]

Theorem 2. For a vector \(\mathbf v\), defined on \(M_{1h}\), to be representable in the form \(\mathbf v=\operatorname{rot}_h\vec\psi\), where the vector \(\vec\psi\) is defined on \(M_{2h}\), it is necessary and sufficient that

\[ \operatorname{div}_h\mathbf v=0. \]

The method of proof of Theorems 1 and 2 is the same as in the continuous case \((^2)\).

In \(H_{1h}\) lie the linear manifold \(G_{1h}\) of vectors of the form \(\mathbf v_1=\operatorname{grad}_h\varphi\) and the linear manifold \(J_{1h}\) of vectors of the form \(\mathbf v_2=\operatorname{rot}_h\vec\psi\). We also introduce the linear manifolds of vectors \(H^0_{1h}, G_{0h}, J_{0h}\). We assume that: \(\mathbf v\in H^0_{1h}\) if \(\mathbf v\in H_{1h}\) and \(\mathbf v\equiv0\) outside \(\Omega_{vh}\subseteq\Omega_{1h}\); \(\mathbf v_1\in G_{0h}\) if \(\mathbf v_1=\operatorname{grad}_h\varphi\in G_{1h}\) and \(\varphi\equiv0\) outside \(\Omega_{v_1h}\subseteq\Omega_{2h}\); \(\mathbf v_2\in J_{0h}\), if \(\mathbf v_2=\operatorname{rot}_h\vec\psi\in J_{1h}\) and \(\vec\psi\equiv0\) outside \(\Omega_{v_2h}\subseteq\Omega_{2h}\).

Lemma 1. In the case when \(\Omega\) is the whole space, an element \(\mathbf v\) of \(H_{1h}\), orthogonal to all elements of \(G_{0h}\) and \(J_{0h}\), can only be identically zero.

Indeed, since \(\mathbf v\perp J_{0h}\) and \(G_{0h}\), then \(\mathbf v\perp \Delta_{2h}\mathbf w\), if \(\mathbf w\in H^0_{1h}\), because

\[ \Delta_{2h}\mathbf w=\operatorname{grad}_h\operatorname{div}_h\mathbf w-\operatorname{rot}_h\operatorname{rot}_h\mathbf w, \]

and then \(\Delta_{2h}v_i=0\), i.e. \(\mathbf v\equiv0\), since \(h^3\sum_{M_{1h}}\|v\|^2<C\).

Lemma 2. The manifold \(G_{0h}\) is orthogonal to the manifold \(J_{1h}\).

Lemma 3. The manifold \(G_{1h}\) is orthogonal to the manifold \(J_{0h}\).

Corollary. The manifold \(J_{0h}\) is orthogonal to the manifold \(G_{0h}\).

Theorem 3. In the case when \(\Omega\) is the whole space, the space \(H_{1h}\) can be represented in the form

\[ H_{1h}=J_h\oplus G_h, \]

where \(J_h=\overline{J_{0h}}=\overline{J_{1h}}\), \(G_h=\overline{G_{0h}}=\overline{G_{1h}}\) (the bar over a letter denotes the closure of the corresponding space).

For finite domains the following results hold.

Lemma 4. Every vector \(\mathbf v\) from \(H_{1h}\), orthogonal to \(J_{0h}\) and \(G_{0h}\) simultaneously, is a harmonic vector, i.e. its curl, its divergence, and \(\Delta_{2h}\mathbf v\) are equal to zero.

Let \(\mathbf v_1=\operatorname{grad}_h\varphi_1\in G_{0h}\); then
\[ (\mathbf v,\mathbf v_1)_{1h}=0=(\mathbf v,\operatorname{grad}_h\varphi_1)_{1h} =-(\varphi_1,\operatorname{div}_h\mathbf v)_{2h}, \]
i.e. \(\operatorname{div}_h\mathbf v=0\) in \(\Omega_{2h}\).

Let \(\mathbf v_2=\operatorname{rot}_h\vec\psi\in J_{0h}\); then
\[ (\mathbf v,\mathbf v_2)_{1h}=0=(\mathbf v,\operatorname{rot}_h\vec\psi)_{1h} =(\vec\psi,\operatorname{rot}_h\mathbf v)_{2h}, \]
i.e. \(\operatorname{rot}_h\mathbf v=0\) in \(\Omega_{2h}\).

The fact that \(\Delta_{2h}\mathbf v=0\) follows from the formula for \(\Delta_{2h}\mathbf v\).

Lemma 5. A vector \(\mathbf v\) orthogonal to \(G_{0h}\) and \(J_{1h}\) simultaneously is identically zero.

Indeed, since \(\mathbf v\perp G_{0h}\), \(\operatorname{div}_h\mathbf v=0\), i.e. \(\mathbf v\in J_{1h}\), and since \(\mathbf v\perp J_{1h}\), it follows that \(\mathbf v=0\).

Lemma 6. A vector \(\mathbf v\) orthogonal to \(G_{1h}\) and \(J_{0h}\) simultaneously is identically zero.

Indeed, since \(\mathbf v\perp J_{0h}\), \(\operatorname{rot}_h\mathbf v=0\), i.e. \(\mathbf v\in G_{1h}\), and since \(\mathbf v\perp G_{1h}\), it follows that \(\mathbf v=0\).

Theorem 4. The space \(H_{1h}\) admits the representation
\[ H_{1h}=G_{0h}\oplus I_h\oplus J_{0h}, \]
where \(I_h=G_{1h}J_{1h}\).

We shall now construct a difference analogue of system (1). In the space \(R_4(x_1,x_2,x_3,t)\) consider the set of points \((x,t)\) with coordinates \(x_i=k_i h,\ t=k_0\Delta t,\ i=1,2,3,\ \Delta t>0\); denote the set of points \((x,t)\) such that \(x\in\Omega_{ih}\) by \(D_{ih}\). Put
\[ \mathbf U_t(x_1,x_2,x_3,t)=\frac{1}{\Delta t}\bigl(\mathbf U(x_1,x_2,x_3,t)-\mathbf U(x_1,x_2,x_3,t-\Delta t)\bigr), \]
\[ \mathbf U_{\mathrm{cp}}(x_1,x_2,x_3,t)=\alpha\mathbf U(x_1,x_2,x_3,t)+\beta\mathbf U(x_1,x_2,x_3,t-\Delta t), \]
where \(\alpha\ge 0,\ \beta\ge 0\), and \(\alpha+\beta=1\).

At the points \((x,t)\in D_{1h}\) replace system (1) by the equations
\[ \mathbf U_t=A\mathbf U_{\mathrm{cp}}-\operatorname{grad}p+\mathbf F \quad\text{and}\quad \operatorname{div}_h\mathbf U=0 \tag{9} \]
at the points \((x,t)\in D_{2h}\). We shall prove the existence of a solution of (9).

I. Condition (4) for smooth functions and a smooth boundary of the domain \(\Omega\) is equivalent to condition \((1)\)
\[ \int_\Omega (\mathbf U,\operatorname{grad}\varphi)\,d\Omega=0 \quad\text{for any }\varphi. \tag{10} \]

For system (9), replace condition (10) by the condition that
\[ (\mathbf U,\operatorname{grad}_h\varphi)_{1h}=0 \quad\text{for any }\varphi. \tag{11} \]

Consider the solution of (9), replacing (4) by the requirement that \(\mathbf U\) be an arbitrary element of \(J_{0h}\). Then \(\mathbf v_1=\operatorname{grad}_h p\) is determined by the formula
\[ \mathbf v_1=P_{0h}^{*}\{A\mathbf U_{\mathrm{cp}}+\mathbf F\}, \tag{12} \]
and
\[ \mathbf U_t=P_{0h}\{A\mathbf U_{\mathrm{cp}}+\mathbf F\}, \tag{13} \]
where \(P_{0h}^{*},P_{0h}\) are the projection operators of \(H_{1h}\), respectively onto \(G_{1h}\) and \(J_{0h}\).

II. Condition (3) for a smooth function \(p\) and a smooth boundary of the domain \(\Omega\) is equivalent to condition \((1)\)
\[ \int_\Omega (\mathbf U,\operatorname{grad}p)\,d\Omega=0 \quad\text{for any }\mathbf U\in J_1. \tag{14} \]

For system (9), replace (14) by the condition that
\[ (\mathbf U,\operatorname{grad}_h p)_{1h}=0 \quad\text{for any }\mathbf U\in J_{1h}. \tag{15} \]

Then

\[ \mathbf{v}_1=\operatorname{grad}_h p=P_{1h}^{*}\{A\mathbf{U}_{\mathrm{cp}}+\mathbf{F}\}, \tag{16} \]

\[ \mathbf{U}_t=P_{1h}\{A\mathbf{U}_{\mathrm{cp}}+\mathbf{F}\}, \tag{17} \]

where \(P_{1h}^{*}, P_{1h}\) are the projection operators of \(H_{1h}\), respectively, onto \(G_{0h}\) and \(J_{1h}\).

III. Similarly, solutions of (9) are found for the Cauchy problem:

\[ \mathbf{v}_1=\operatorname{grad}_h p=P_h^{*}\{A\mathbf{U}_{\mathrm{cp}}+\mathbf{F}\}, \tag{18} \]

\[ \mathbf{U}_t=P_h\{A\mathbf{U}_{\mathrm{cp}}+\mathbf{F}\}, \tag{19} \]

where \(P_h^{*}, P_h\) are the projection operators of \(H_{1h}\), respectively, onto \(\overline{G}_{0h}\) and \(\overline{J}_{0h}\).

Let us find the solution of (13) in the form of a power series; for (17) and (19) the solutions are found analogously. If we denote

\[ P_{0h}A\mathbf{v}=B_h\mathbf{v},\qquad \Gamma_{h\Delta t}=\frac{1}{\Delta t}\ln\bigl[(E-B_h\alpha\Delta t)^{-1}(E+B_h\beta\Delta t)\bigr], \]

then the solution of the problem

\[ \mathbf{v}_t=B_h\mathbf{v}_{\mathrm{cp}},\qquad \mathbf{v}\big|_{t=0}=\mathbf{v}_0 \quad \text{for } t_i=i\Delta t \]

is given by the formula

\[ \mathbf{v}(t_i)=\exp[\Gamma_{h\Delta t}t_i]\mathbf{v}_0 =\sum_{n=0}^{\infty}\frac{t_i^n}{n!}(\Gamma_{h\Delta t})^n\mathbf{v}_0. \tag{20} \]

For the nonhomogeneous equation

\[ \mathbf{U}_t=B_h\mathbf{U}_{\mathrm{cp}}+P_{0h}\mathbf{F} \]

the solution is given by the formula

\[ \mathbf{U}(t_i)=\exp[\Gamma_{h\Delta t}t_i]\mathbf{U}_0 +\Delta t\sum_{\tau_j=\Delta t}^{t_i} \exp[\Gamma_{h\Delta t}(t_i-\tau_j)] (E-B_h\alpha\Delta t)^{-1}P_{0h}\mathbf{F}(\tau_j). \tag{21} \]

From the form of the formulas giving the explicit solution there follows the correctness of the solutions of the difference schemes considered, as well as the convergence in \(W_2^{(1)}\) of the approximate solutions to the exact ones, provided \(\mathbf{F}\) and \(\mathbf{U}_0\) have the corresponding smoothness.

In conclusion I express my deep gratitude to my scientific adviser, Acad. S. L. Sobolev, for valuable comments.

Moscow State University
named after M. V. Lomonosov

Received
16 XI 1956

References Cited

\({}^{1}\) S. L. Sobolev, Izv. AN SSSR, ser. matem., 18, No. 1 (1954).
\({}^{2}\) N. E. Kochin, Vector Calculus and the Elements of Tensor Calculus, Publishing House of the Academy of Sciences of the USSR, 1951.