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MATHEMATICS
V. Ya. GOLDIN
A CHARACTERISTIC DIFFERENCE SCHEME FOR A NONSTATIONARY KINETIC EQUATION
(Presented by Academician M. V. Keldysh, 1 IV 1960)
The paper gives a method for constructing characteristic difference schemes for a nonstationary kinetic equation. The solution of the resulting system of equations is simplified by reduction to a difference scheme for the corresponding integral equation. The stability of the schemes obtained is determined only by the time step.
- Neutron multiplication in the nonstationary regime of a reactor is characterized by the density of the number of neutrons \(\psi(P,\mathbf e,v,t)\), where \(P\) is a point of the region \(G\) of three-dimensional Euclidean space \(R_3\) (\(G\) is the region occupied by the reactor), \(\mathbf e\) is the unit vector in the direction of the velocity, \(v\) is the neutron velocity, and \(t\) is time. The function \(\psi\) satisfies the linear integro-differential (kinetic) Boltzmann equation \((^{1-3})\), which is a neutron balance equation. Numerous works are devoted to the approximate determination of \(\psi\) in stationary problems (a detailed bibliography is given in \((^{3,4})\)).
We shall consider a nonstationary problem in the one-group approximation. For the case of isotropic scattering, the equation for \(\psi(P,\mathbf e,t)\) takes the form
\[ \frac{\partial \psi}{\partial t}+\frac{\partial \psi}{\partial \mathbf e}+\alpha\psi-\beta n = q(P,\mathbf e,t). \tag{1} \]
Here \(v=1\); \(\partial\psi/\partial\mathbf e\) is the derivative in the direction \(\mathbf e\); \(\alpha\) is the absorption coefficient; \(\beta\) is the multiplication coefficient; \(n(P,t)=\frac{1}{4\pi}\int \psi(P,\mathbf e,t)\,d\mathbf e\) is the mean density of the number of neutrons; \(q(P,\mathbf e,t)\) is the density of neutron sources. Methods for calculating (1) are treated in \((^{5,6})\). The solution is determined in a convex region \(G\), bounded by a piecewise smooth surface \(S\), under the initial and boundary conditions
\[ \psi(P,\mathbf e,t)\big|_{t=t^0}=\psi_0(P,\mathbf e); \tag{2} \]
\[ \psi(P,\mathbf e,t)\big|_{P\in S}=0 \quad \text{for } (\mathbf e\mathbf n)<0, \tag{3} \]
where \(\mathbf n\) is the outward normal to \(S\) at the point \(P\). Questions of existence and uniqueness of the solution of equation (1) are considered in \((^2)\).
- We shall consider the solution of equation (1) in the six-dimensional phase space \((^4)\) \(R_3\times\Omega\times\Theta\), where \(R_3\) is three-dimensional Euclidean space; \(\Omega\) is the space formed by the unit vectors \(\mathbf e\); \(\Theta\) is a one-dimensional space \((t^0\le t<\infty)\).
An arbitrary point in phase space \(R_3\times\Omega\times\Theta\) will be denoted by \(\widetilde P=(P,\mathbf e,t)\), \(P\in R_3\), \(\mathbf e\in\Omega\), \(t\in\Theta\).
We shall consider the solution of equation (1) in the phase region \(\widetilde G=G\times\Omega\times[t^0,T]\) of the phase space \(R_3\times\Omega\times\Theta\). In the phase space \(R_3\times\Omega\), we shall call the corresponding region \(\overline G=G\times\Omega\), and the corresponding point \(\overline P=(P,\mathbf e)\).
The trajectories of neutrons between collisions coincide with the characteristics of the differential operator \(L=\partial/\partial t+\partial/\partial\mathbf e\). Let two points in \(\widetilde G\), \(\widetilde P_2\) and \(\widetilde P_1\), lie on one characteristic, i.e.
\[ P_1=P_2-\mathbf e_2\tau,\qquad \tau=t^2-t^1,\qquad \mathbf e_1=\mathbf e_2 . \tag{4} \]
Integrating (1) from \(\widetilde P_1\) to \(\widetilde P_2\) along the characteristic, we obtain
\[ \mathscr L_{\widetilde P_2,\tau}\psi=Q_{\widetilde P_2,\tau}, \tag{5} \]
where
\[ \mathscr L_{\widetilde P_2,\tau}\psi = \psi(\widetilde P_2) - \psi(\widetilde P_1)\exp\left(-\int_{\widetilde P_1}^{\widetilde P_2}\alpha\,d\xi\right) - \int_{\widetilde P_1}^{\widetilde P_2} \beta n(P')\exp\left(-\int_{P'}^{\widetilde P_2}\alpha\,d\xi\right)\,ds, \tag{6} \]
\[ Q_{\widetilde P_2,\tau} = \int_{\widetilde P_1}^{\widetilde P_2} q(p')\exp\left(-\int_{P'}^{\widetilde P_2}\alpha\,d\xi\right)\,ds . \tag{7} \]
Integrating (5) with respect to \(\mathbf e_2\) leads to the nonstationary Peierls equation \(({}^1)\).
- On the time interval \([t^0,T]\) introduce the difference grid
\(H_t(t^0,t^1,\ldots,t^j,t^{j+1},\ldots,t^N=T)\). In the domain \(\overline{\widetilde G}\) define the grid \(\widetilde H_\rho\) as the product of the grid \(H_\rho\) in the domain \(G\) and \(H_\Omega\) in the space \(\Omega\). The grids \(H_\rho\) and \(H_\Omega\) divide, respectively, \(G\) and \(\Omega\) into cells. We include boundary surfaces among the coordinate surfaces forming the grid \(H_\rho\). The nonstandard form of boundary cells must be taken into account by special interpolation.
The general grid \(\widetilde H_\rho\) in \(\widetilde G\) is equal to \(\overline H_\rho\times H_t\). Denote the nodes of the grid \(H_\rho\) by \(P_1,P_2,\ldots,P_i,\ldots,P_I\), the nodes of the grid \(\overline H_\rho\) by \(\overline P_{ik}(P_i,\mathbf e_k)\), and those of the grid \(\widetilde H_\rho\) by \(\widetilde P^{\,j}_{ik}(P_i,\mathbf e_k,t^j)\). In a number of cases in \(\Omega\) one uses a local coordinate system associated with the position of the point \(P\). In this case the vector \(\mathbf e_k\) depends on \(P_i\), \(\mathbf e_k=\mathbf e_k(P_i)\) (for example, in the case of spherical symmetry). For the grid \(\overline H_\rho\) define the diameter \(\rho\) so that, as \(\rho\to0\), all dimensions of all cells tend to zero. To each point \(\widetilde P^{\,j+1}_{ik}\) there corresponds a point \(\widetilde P^{\,j}_{(ik)}\), determined according to (4). Obviously, the point \(\widetilde P^{\,j}_{(ik)}\) is not (in general) a nodal point of the grid \(\widetilde H_\rho\).
We shall call a function defined only at the nodes of \(\widetilde H_\rho\) a grid function \(\psi^j_{ik}\). Associate with it, by interpolation, the functions \(\psi_H\) and \(n_H\):
\[ \psi_H^j(\overline P)=\sum \zeta_{ik}(\overline P)\psi^j_{ik}, \tag{8} \]
where \(\overline P_{ik}\) are the vertices of the cell \(\overline G_H\) in which \(\overline P\) lies; \(\zeta_{ik}(\overline P)\) are nonnegative interpolation coefficients, chosen so that
\[ \sum \zeta_{ik}(\overline P)=1. \]
At the nodes of the grid \(H_\rho\) define \(n_H^j\) by integrating \(\psi_H^j\):
\[ n_H^j(P_i)=\frac{1}{4\pi}\int \psi_H(P_i,\mathbf e)\,d\mathbf e = \sum \gamma_k\psi^j(\overline P_{ik}). \tag{9} \]
Extend \(n_H\) outside the nodes on \(G\) and on \(S\) by interpolation of the type (8).
To determine the function \(n_H\) on characteristics for \(t^j\leq t\leq t^{j+1}\), define \(n_H\) at the points where the characteristic intersects the faces \(G_H\), interpolating in \(t\) between \(n_H^j\) and \(n_H^{j+1}\). On the segment of a characteristic lying in a cell \(G_H\), \(n_H\) is determined by interpolation along the characteristic from the values at the ends of the segment.
The adopted interpolation process assigns to any discontinuous function \(\psi(\widetilde P)\) the function \(\psi_H\) for the chosen grid \(\widetilde H_\rho\). In this case the grid function is \(\psi_{ik}^j=\psi(\widetilde P_{ik}^j)\). From the function \(\psi_H\) at the nodes of the grid \(\widetilde H\) one can compute \(\mathcal L_{\widetilde P_{ik}^{j+1},\tau^j}\psi_H\) by formula (6). It is obvious that in this case \(\mathcal L\psi_H\) is determined through \(\psi_{ik}^j\), i.e., the operator \(\mathcal L_{\widetilde P_{ik}^{j+1},\tau^j}\psi_H\) is a difference operator.
The accuracy of the adopted interpolation on the chosen grid for the function \(\psi\) determines the accuracy of the local approximation of the difference operator
\[ \mathcal E_\psi(\tau,\rho)=\sup\left|\frac{1}{\tau^j}\mathcal L_{\widetilde P_{ik}^{j+1},\tau^j}(\psi-\psi_H)\right|. \tag{10} \]
To determine the approximate grid function and the corresponding function \(\psi_H\), we require that equation (5) be satisfied for the function \(\psi_H\) at the nodal points of the grid.
To determine \(\psi_H^{j+1}\) and \(n_H^{j+1}\) at the nodal points, given \(\psi_H^j\) and \(n_H^j\), we obtain the system of equations
\[ \mathcal L_{\widetilde P_{ik}^{j+1},\tau^j}\psi_H=Q_{\widetilde P_{ik}^{j+1},\tau^j}, \tag{11} \]
\[ n_H^{j+1}(P_i)=\sum\nolimits' \gamma_k\psi^{j+1}(\overline P_{ik}). \tag{12} \]
The resulting system of linear equations can be solved by the method of iterations; the number of unknowns is then equal to \(\sim I\times K\). However, the resulting system of equations can be reduced to a system of linear equations only with respect to \(n_i^{j+1}\), i.e., one obtains a difference scheme for the nonstationary Peierls integral equation. In this case the number of unknowns is \(\sim I\).
Multiplying (11) by \(\gamma_k\) and summing, we obtain a system of linear equations for \(n_H^{j+1}(P_i)\). If \(n_H\) on the characteristic is determined by interpolation between \(n_H^{j+1}(P_i)\) and \(n_H^j(P_i-e_k\tau^j)\), then the system of equations for determining \(n_H^{j+1}\) decouples, and each \(n_H^{j+1}(P_i)\) is determined independently.
Convergence of the resulting difference scheme in the class of continuous solutions is ensured by the fulfillment of the maximum principle (stability) and by the local approximation (7). More precisely, the following estimates hold:
1) Maximum principle
\[ \|\psi\|^{j+1}\le \frac{\|\psi\|^j+\|Q\|^{j+1}\tau^j}{1-\omega\tau^j} \quad \text{for } \tau^j<\frac{1}{\omega}, \]
where
\[ \|\psi\|^j=\sup|\psi_H^j|,\qquad \omega=\sup\left(\frac{1}{\tau^j}\int_{\widetilde P_{(ik)}^j}^{\widetilde P_{ik}^{j+1}} \beta\exp\left(-\int_{P'} \alpha\,d\xi\right)\,ds\right), \]
\[ \|Q\|^{j+1}=\sup\left|\frac{1}{\tau^j}Q_{\widetilde P_{ik}^{j+1},\tau^j}\right|. \]
2) The accuracy of the difference scheme is determined by the difference \(v_H=\widetilde\psi_H-\psi_H\), where \(\widetilde\psi_H\) is the interpolating function of the exact solution \(\widetilde\psi\) of equation (5), while \(\psi_H\) is determined by the solution of the difference scheme. For the error \(v_H\) po-
we obtain the estimate \(\sup |v_H|^j \leqslant \varepsilon^j\), where \(\varepsilon^j\) is defined by the recurrence formula
\[ \varepsilon^j=\frac{\varepsilon^{j-1}+\varepsilon(\tau,\rho)\tau^j}{1-\omega\tau^j}. \]
For \(\tau^j=\tau \to 0\), for a given \(t\) we obtain
\[ \varepsilon(t)\sim \varepsilon(t^0)\exp\left[ \omega(t-t^0)\left(1-\frac{\omega\tau}{2}+\frac{\omega^2\tau^2}{3}\right)\right]+ \]
\[ +\frac{\varepsilon(\tau,\rho)}{\omega} \left\{\exp\left[ \omega(t-t^0)\left(1-\frac{\omega\tau}{2}+\frac{\omega^2\tau^2}{3}\right)\right]-1\right\}. \]
Here \(\varepsilon(\tau,\rho)=\varepsilon_{\psi}(\tau,\rho)\) on the exact solution \(\widetilde{\psi}\).
Proceeding from the definition of \(\varepsilon(\tau,\rho)\), we obtain sufficient conditions for approximation and convergence
\[ \text{a)}\quad \left| \frac{ \widetilde{\psi}_{H}\left(\widetilde{P}_{ik}^{\,j+1}-e_k\tau\right) - \widetilde{\psi}\left(\widetilde{P}_{ik}^{\,j+1}-e_k\tau\right) }{\tau} \right| \leqslant \varepsilon_1(\tau,\rho), \quad \text{where }\varepsilon_1(\tau,\rho)\xrightarrow[\tau\to0,\ \rho\to0]{}0, \]
\[ \text{b)}\quad \left|\widetilde{n}_{H}(\widetilde{P})-\widetilde{n}(\widetilde{P})\right| \leqslant \varepsilon_2(\tau,\rho), \quad \text{where }\varepsilon_2(\tau,\rho)\xrightarrow[\tau\to0,\ \rho\to0]{}0. \]
For the difference scheme obtained, as for the original equation, there is continuous dependence on the parameters \(\alpha,\beta\) and on the sources \(q\), by virtue of the fulfillment of the maximum principle. Therefore, in the difference scheme the exact values \(\alpha,\beta\), and \(q\) may be replaced by mean values over the mesh. A generalization of the method to the case of stationary discontinuities of the solution and weak anisotropy of scattering is possible.
In conclusion I express my gratitude to A. N. Tikhonov, A. A. Samarskii, and M. I. Volchinskaya for discussion.
Received
31 III 1960
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