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|=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}+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); 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}
\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),},\varphi_{x_2},\varphi_{x_3
\tag{6}
]

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

[
\operatorname{div}h\mathbf v=v.}+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), 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), 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}) of vectors of the form (\mathbf v_2=\operatorname{roth\vec\psi). We also introduce the linear manifolds of vectors (H^0}, 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{gradh\varphi\in G}) and (\varphi\equiv0) outside (\Omega_{v_1h}\subseteq\Omega_{2h}); (\mathbf v_2\in J_{0h}), if (\mathbf v_2=\operatorname{roth\vec\psi\in J).}) 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); then
[
(\mathbf v,\mathbf v_1){1h}=0=(\mathbf v,\operatorname{grad}_h\varphi_1)
=-(\varphi_1,\operatorname{div}h\mathbf v),
]
i.e. (\operatorname{div}h\mathbf v=0) in (\Omega).

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

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), it follows that (\mathbf v=0).}), and since (\mathbf v\perp J_{1h

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), it follows that (\mathbf v=0).}), and since (\mathbf v\perp G_{1h

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). 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)=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^{}{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)=0
\quad\text{for any }\mathbf U\in J_{1h}.
\tag{15}
]

Then

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

[
\mathbf{U}t=P},}{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}},}}+\mathbf{F
\tag{18}
]

[
\mathbf{U}t=P_h{A\mathbf{U}},}}+\mathbf{F
\tag{19}
]

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

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},\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_0.}^{\infty}\frac{t_i^n}{n!}(\Gamma_{h\Delta t})^n\mathbf{v
\tag{20}
]

For the nonhomogeneous equation

[
\mathbf{U}t=B_h\mathbf{U}}}+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}^{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.