Abstract
Full Text
UDC 533.601.172
MATHEMATICS
N. B. MASLOVA
ON A METHOD FOR SOLVING RELAXATION EQUATIONS
(Presented by Academician V. I. Smirnov on 30 January 1967)
This note proposes a method for solving a system of nonlinear ordinary differential equations describing relaxation in a nonequilibrium gas with internal degrees of freedom. A sequence of functions is constructed that converges to the solution of the Cauchy problem uniformly on any finite interval of variation of the argument.
1°. Nonstationary spatially homogeneous motion is described by the system of equations (see (¹))
\[ dn_i/dt=\Phi_i(n)-n_iQ_i(n),\qquad i=1,2,\ldots,r; \tag{1} \]
\[ \Phi_i(n)=\sum_{k,l,m} n_kn_lP_{kl}^{im},\qquad Q_i=\sum_{k,l,m} n_mP_{im}^{kl}; \tag{2} \]
\[ {}^{3}\!/_{2}\,kT(t)\sum_{i=1}^{r} n_i(t)+\sum_{i=1}^{r}\varepsilon_i n_i(t)=E_0; \tag{3} \]
\[ n_i\big|_{t=0}=n_i(0)>0,\qquad T\big|_{t=0}=T(0)>0, \]
where \(n_i\) is the number density of particles of species \(i\) having internal energy \(\varepsilon_i\); \(P_{kl}^{im}\) are given nonnegative functions of the temperature \(T\), proportional to transition probabilities. From (1) and (2) it follows, obviously, that
\[ \sum_{i=1}^{r} n_i(t)=\sum_{i=1}^{r} n_i(0). \tag{4} \]
Let us first consider the special case of the system (1)—(3), when
\[ \sum_{i=1}^{r}\varepsilon_i n_i(t) \]
and \(T\) are conserved separately in time (there is no exchange of energy between the internal and translational degrees of freedom). Then the problem consists in constructing a solution of the system (1), with \(P_{kl}^{im}\) being constant quantities.
Without loss of generality, we shall assume that \(\sum_{i=1}^{r} n_i(0)=1\). We transform (1) into the equivalent system of integral equations:
\[ n_i(t)=n_i(0)\exp\left\{-\int_{0}^{t} Q_i(s)\,ds\right\} +\int_{0}^{t}\Phi_i(\tau)\exp\left\{-\int_{\tau}^{t} Q_i(s)\,ds\right\}\,d\tau \equiv V_i(n,T). \tag{5} \]
The system (5) can be solved by the usual iteration method
\[ n_i^{(n)}=V_i\bigl(n^{(n-1)},T\bigr). \]
The sequence \(n_i^{(n)}\) converges to the solution on some sufficiently small time interval. It is not difficult to see that the iterations \(V_i\bigl(n^{(n-1)},T\bigr)\) do not preserve the property of the solution (4). It is proposed to correct each iteration in such a way that the conservation law (4) is satisfied in every approximation. More precisely, the following method is proposed—
method of successive approximations:
\[ n_i^{(0)}=\bar n_i^{(0)}= \begin{cases} n_i(0), & t=0,\\ \bar n_{ip}, & t>0, \end{cases} \tag{6} \]
\[ n_i^{(n)}=V_i\bigl(\bar n^{(n-1)},T\bigr),\qquad \bar n_i^{(n-1)}=n_i^{(n-1)}\bigg/ \sum_{i=1}^{r} n_i^{(n-1)}, \tag{7} \]
where \(\bar n_p=\{\bar n_{1p},\ldots,\bar n_{rp}\}\) is the normalized \(\left(\sum_{i=1}^{r}\bar n_{ip}=1\right)\) solution of the system \(\Phi_i(n)-n_iQ_i(n)=0\).
Such a construction is possible if, for any pair \(k,l\), \(\sum_{i,m}P_{kl}^{im}>0\), since from (7) there follows the inequality
\[ \rho^{(n)}=\sum_{i=1}^{r} n_i^{(n)} \geq \min\left(1,\min_{k,l}\sum_{i,m}P_{kl}^{im}\bigg/ \max_{i,m}\sum_{k,l}P_{im}^{kl}\right) =\rho_{\min}>0. \tag{8} \]
It can be proved that the sequences \(n_i^{(n)}\) and \(\bar n_i^{(n)}\) converge to the solution of system (5) uniformly on any finite time interval. From equations (7) and the inequality
\[ \min\left[1,\inf_t\left(\sum_i\Phi_i(\bar n^{(n-1)})\bigg/ \sum_i\bar n_i^{(n)}Q_i(\bar n^{(n-1)})\right)\right]\leq \rho^{(n)}\leq \]
\[ \leq \max\left[1,\sup_t\left(\sum_i\Phi_i(\bar n^{(n-1)})\bigg/ \sum_i\bar n_i^{(n)}Q_i(\bar n^{(n-1)})\right)\right] \]
it follows that \(\lim_{n\to\infty}n_i^{(n)}(t)=\lim_{n\to\infty}\bar n_i^{(n)}(t)=n_i(t)\) is a solution of the original system of equations. The following estimates of the closeness of the approximate solution to the exact one can be obtained:
\[ \bigl|n_i(t)-n_i^{(n)}(t)\bigr| \leq \max_i \bigl|n_i(0)-\bar n_{ip}\bigr|\exp\{-tQ_{\min}\} \sum_{k=n}^{\infty}\frac{(Lt)^k}{k!}, \]
\[ \bigl|n_i(t)-\bar n_i^{(n)}(t)\bigr| \leq \frac{r+1}{\rho_{\min}}\max_i \bigl|n_i(0)-\bar n_{ip}\bigr|\exp\{-tQ_{\min}\} \sum_{k=n}^{\infty}\frac{(Lt)^k}{k!}, \tag{9} \]
where \(L\) is a constant.
For some specific transition probabilities, from system (1) one can obtain a closed equation for \(\sum_{i=1}^{r}\varepsilon_i n_i(t)\), whose solution, together with (3), makes it possible to find \(T(t)\) (1). Then problem (1)—(3) reduces to finding the solution of system (1), in which \(P_{kl}^{im}\) are known functions of time. In this case the method described above requires no changes except replacing \(\min_{i,m}\sum P_{kl}^{im}\) and \(\max_{i,m}\sum P_{kl}^{im}\) by the lower and upper bounds of the corresponding functions of time. In the general case, for solving problem (1)—(8), the following method of successive approximations is proposed:
\[ n_i^{(0)}=\bar n_i^{(0)}= \begin{cases} n_i(0), & t=0,\\ n_{ip}\bigl(T^{(0)}\bigr), & t>0; \end{cases} \]
\[ n_i^{(n)}=V_i\bigl(\bar n^{(n-1)},T^{(n-1)}\bigr),\qquad \bar n_i^{(n-1)}=\frac{n_i^{(n-1)}}{\rho^{(n-1)}}\sum_{i=1}^{r}n_i(0); \]
\(T^{(n)}\) is determined from the equation
\[ {}^{3}\!/_{2}kT^{(n)}\sum_{i=1}^{r} n_i(0) + \sum_{i=1}^{r}\varepsilon_i \bar n_i^{(n)} = E_0 . \tag{10} \]
The sequences \(n_i^{(n)}, T^{(n)}\) converge to the solution of problem (1)—(3) uniformly on any finite interval of time, if
\[ E_0 > \max \varepsilon_i \sum_{i=1}^{r} n_i(0); \qquad \min_{k,l}\sum_{i,m} P_{kl}^{im}(T) > 0, \quad 0<T<\infty . \tag{11} \]
The solution of problem (1)—(3) is unique.
If the principle of detailed balance is satisfied, then the solution of system (1)—(3) tends, as \(t \to \infty\), to the solution of the system
\[ \Phi_i - n_i Q_i(n) = 0; \qquad {}^{3}\!/_{2}kT\sum_{i=1}^{r}n_i + \sum_{i=1}^{r}\varepsilon_i n_i = E_0 . \tag{12} \]
This assertion is proved by Carleman’s method \((^2)\), using the inequality
\[ \frac{dH}{dt} = \frac{1}{4}\sum_{i,k,l,m} \left[ n_k n_l \left(\frac{\mu_k\mu_l}{\mu_i\mu_m}\right)^{3/2} \exp\frac{\varepsilon_k+\varepsilon_l-\varepsilon_i-\varepsilon_m}{kT} - n_i n_m \right] P_{im}^{kl} \ln \left\{ \frac{n_i n_m}{n_k n_l} \left(\frac{\mu_i\mu_m}{\mu_k\mu_l}\right)^{3/2} \exp\frac{\varepsilon_i+\varepsilon_m-\varepsilon_k-\varepsilon_l}{kT} \right\} \le 0 \tag{13} \]
and the equality
\[ P_{kl}^{im} = \left(\frac{\mu_k\mu_l}{\mu_i\mu_m}\right)^{3/2} \exp\left\{\frac{\varepsilon_k+\varepsilon_l-\varepsilon_i-\varepsilon_m}{kT}\right\} P_{im}^{kl}, \tag{14} \]
where \(\mu_i\) is the mass of a particle of species \(i\);
\[ H=\sum_{i=1}^{r} n_i \left\{ \ln n_i\left(\frac{\mu_i}{2\pi kT}\right)^{3/2} - {}^{3}\!/_{2} \right\}. \]
Equality (14) is a consequence of the principle of detailed balance.
\(2^\circ\). The solution of the equations describing a one-dimensional stationary flow in the relaxation zone of a shock wave can be obtained by the same method. The corresponding system of equations, after introducing the new functions
\[ N_i(x)=n_i(x)\bigg/\sum_{i=1}^{r} n_i(x), \]
has the form
\[ N_i(x) = N_i(0)\exp\left\{-\int_{0}^{x}\overline{Q}_i(s)\,ds\right\} + \int_{0}^{x}\overline{\Phi}_i(\tau) \exp\left\{-\int_{\tau}^{x}\overline{Q}_i(s)\,ds\right\}d\tau = \overline{V}_i(N,T,\rho), \tag{15} \]
\[ \overline{\Phi}_i=\sum_{k,l,m} N_kN_l\Pi_{kl}^{im}, \qquad \overline{Q}_i=\sum_{k,l,m} N_m\Pi_{im}^{kl}, \qquad \Pi_{kl}^{im}=\frac{\rho^2}{C_1\mu}P_{kl}^{im}, \]
\[ \rho U=C_1; \qquad \frac{k}{\mu}\rho T+\rho U^2=C_2, \]
\[ \frac{\gamma}{\gamma-1}\frac{k}{\mu}T+\frac{U^2}{2} +\frac{1}{\mu}\sum_{i=1}^{r}\varepsilon_i N_i=C_3, \tag{16} \]
where \(\mu,\gamma,C_i\) are constants. If the functions \(\rho,T\) at infinity are prescribed, then equation (16) in some domain \(\Omega\) of the values of \(\sum_{i=1}^{r}\varepsilon_i N_i,\ C=\{C_1,C_2,C_3\}\) deter-
define single-valued positive functions: \(\rho\left(\sum_{i=1}^{r}\varepsilon_i N_i, C\right)\), \(T\left(\sum_{i=1}^{r}\varepsilon_i N_i, C\right)\). If from equation (15) one can obtain a closed equation for \(\sum_{i=1}^{r}\varepsilon_i N_i\), then the problem reduces to that described above. In the general case the following method of successive approximations is proposed:
\[ N_i^{(0)}=\overline N_i^{(0)}= \begin{cases} N_i(0), & x=0,\\ N_{ip}(T_\infty,\rho_\infty), & x>0; \end{cases} \]
\[ N_i^{(n)}=\overline V_i\left(\overline N^{(n-1)}, T^{(n-1)}, \rho^{(n-1)}\right), \qquad \overline N_i^{(n-1)}=N_i^{(n-1)}\bigg/ \sum_{i=1}^{r}N_i^{(n-1)}, \]
\[ T^{(n)}=T\left(\sum_{i=1}^{r}\varepsilon_i\overline N^{(n-1)}, C\right), \qquad \rho^{(n)}=\rho\left(\sum_{i=1}^{r}\varepsilon_i\overline N^{(n-1)}, C\right). \]
Obviously,
\[ \min_i \varepsilon_i \leq \sum_{i=1}^{r}\varepsilon_i\overline N^{(n)} \leq \max_i \varepsilon_i. \]
If \(\min_i \varepsilon_i, C\) and \(\max_i \varepsilon_i, C\) belong to the domain \(\Omega\), and if
\[ \min_{k,l}\sum_{i,m}P_{kl}^{im}(T)>0 \]
for \(0<T<\infty\), then the functions
\[ \sum_{i,m}^{k,l}\Pi_{kl}^{im}\left(T^{(n)}(x),\rho^{(n)}(x)\right) \]
have a nonzero lower bound and a finite upper bound with respect to \(n\) and \(x\). Under these conditions the sequences \(N_i^{(n)}\), \(\rho^{(n)}\), \(T^{(n)}\) converge to the solution of the system (15), (16) uniformly on any finite interval of variation of \(x\).
As \(x\to\infty\), the functions \(N_i(x)\), \(\rho(x)\), \(T(x)\) have the limits \(N_{ip}(T_\infty,\rho_\infty)\), \(\rho_\infty\), \(T_\infty\).
The author expresses gratitude to Prof. S. V. Vallander for his attention to the work.
Leningrad State University
named after A. A. Zhdanov
Received
20 I 1967
References
- S. V. Vallander, E. A. Nagnibeda, Vestn. Leningrad Univ., No. 13, Ser. Math., Mech. and Astr., 3 (1963).
- T. Carleman, Mathematical Problems of the Kinetic Theory of Gases, IL, 1960.