View source for Transport equations, numerical methods

Methods for solving integro-differential equations describing particle or radiation transport. The equations take the following form for stationary problems:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939003.png" /> is a unit vector, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939004.png" /> is the particle flux at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939005.png" /> for particles with velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939006.png" />, and the positive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939008.png" /> describe the interaction of the particles with matter, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t0939009.png" /> is the source. Two basic aspects are considered: 1) finding the solution to (1) in a (convex) domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390010.png" /> such that at its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390011.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390013.png" /> is the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390014.png" />; and 2) finding the largest eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390015.png" />, and the corresponding eigenfunction of (1)–(2), in which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Equation (1) contains six independent variables; as it is essentially higher dimensional, it must be reduced to simpler equations. In (1) and (3) one replaces the integral with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390017.png" /> by a quadrature formula containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390018.png" /> terms and assumes that the scattering is isotropic, which gives a system of so-called multi-velocity equations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390020.png" /></td> </tr></table>

are the zero moments, while the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390023.png" /> are obtained by applying averaging methods and using the solutions to conjugate problems. For the problem (2) one similarly gets

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390025.png" />, one gets the single-velocity equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390027.png" />. Equation (6) takes the following form for a planar layer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390028.png" />, where the solution depends only on one coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390029.png" /> and one angular variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390031.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390035.png" />. The characteristics of the left-hand side in (6) are all the straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390037.png" />, and along each of them, (6) takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

If one makes the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390039.png" /> in (6), it becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

The solution to (9) minimizes the quadratic Vladimirov functional

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390042.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390043.png" /></td> </tr></table>

Let the boundary value problems be written in operator form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

A characteristic feature of the transport problem, which is used in numerical algorithms, is that the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390045.png" /> is found from a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390046.png" /> by a direct method involving the integration of (8) along the characteristics. On this basis, from (11) one obtains the Peierls integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

for the zero moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390048.png" />.

The method of spherical harmonics (a form of Galerkin's method) has been substantially developed for solving transport problems. An approximate solution (approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390049.png" />) is found in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390051.png" /> are unknown functions while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390052.png" /> are spherical harmonics of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390053.png" />. Substituting (13) into (6), multiplying the result by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390054.png" /> and integrating with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390055.png" /> leads to a system of partial differential equations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390056.png" /> (cf. [[Spherical harmonics, method of|Spherical harmonics, method of]]). In approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390057.png" />, the system takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390061.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390062.png" /> (14) implies the diffusion approximation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390064.png" />, which is an elliptic problem whose solution can be found by variational or grid methods.

To solve one-dimensional cases, analytic methods have been developed based on expanding the solution in terms of generalized eigenfunctions. The [[Monte-Carlo method|Monte-Carlo method]] is used to find functionals in the solutions to complex multi-dimensional problems.

Finite-difference approximation methods are widely used for transport equations. For example, one can use a quadrature formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390065.png" /> and replace (12) by a system of linear equations. One can approximate the integral operator in (4), (5), (6), or (8) by means of quadrature formulas for a sphere. The [[Gauss quadrature formula|Gauss quadrature formula]] for a sphere is known up to the 35th algebraic order of accuracy. In the [[Method of characteristics|method of characteristics]], a family of characteristics is drawn through each point in the spatial grid along directions corresponding to the nodes of the quadrature for a sphere, and the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390066.png" /> in (8) is replaced by a difference one. The difference equations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390068.png" /> method are obtained by integrating (6) over a grid cell in the phase space on the assumption that the solution is linear in the independent variables within the cell. In Galerkin's method, the solution is sought in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>

If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390070.png" /> are given, one obtains a system of degenerate integral equations for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390071.png" />; if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390072.png" /> are functions of compact support, one obtains the finite-element method; and if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390073.png" /> are given functions of compact support and (16) minimizes (10), one gets the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390075.png" /> equations.

Iterative methods for solving difference transport problems have the specific feature that the convergence usually becomes slower as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390076.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390077.png" />, and that to derive the next approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390078.png" /> one uses only part of the information on a preceding approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390079.png" /> of substantially-fewer dimensions — one stores and uses only the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390080.png" />. In iterative methods an intermediate operation (an operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390081.png" />) is often that of solving the following problem:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>

Then the error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390083.png" /> satisfies (11) with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390084.png" /> which is, as well as the discrepancy, independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390085.png" />. This feature enables one to accelerate the convergence. Consider a periodic problem for (7) with constant coefficients, with a source that is even in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390086.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390087.png" />. In this application, below the following iterative methods are considered. For (7) one constructs a grid with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390088.png" /> nodes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390090.png" /> angular directions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390091.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390092.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390093.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390094.png" /></td> </tr></table>

For convergent iterative methods, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390096.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390097.png" /> be the price (number of operations) in the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390098.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t09390099.png" /> is the price of a complete iteration and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900100.png" />. The following relationships apply for the various methods.

1) Simple iteration: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900103.png" />.

2) Lyusternik's method: For certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900104.png" />, one uses the simple iteration

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900105.png" /></td> </tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900106.png" /> is the largest eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900107.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900109.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900110.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900112.png" />).

3) The method of estimating iterative deviations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900113.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900114.png" /> is the solution of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900115.png" /></td> </tr></table>

then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900116.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900117.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900119.png" />).

4) The balancing-multiplier method: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900120.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900121.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900123.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900124.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900126.png" />).

5) The mean-flux method (rebalance method):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900127.png" /></td> </tr></table>

where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900128.png" /> is selected to minimize the functional (10) or else to minimize it in some finite-dimensional subspace: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900129.png" />, and then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900130.png" /> satisfy a certain system of equations.

6) The quasi-diffusion method:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900131.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900132.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900133.png" />.

7) Splitting methods:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900134.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900135.png" /></td> </tr></table>

The methods 4)–6) are non-linear, and their convergence may slow down as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900136.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900137.png" />; method 7) requires the storage of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900138.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900139.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900141.png" />).

8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900143.png" />-methods: The correction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900144.png" /> is determined as the solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900145.png" /> of the boundary value problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900146.png" /></td> <td valign="top" style="width:5%;text-align:right;">(18)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900147.png" /> are second-order linear differential operators, and one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900148.png" />. In one of the forms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900149.png" />-method, (18) takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900150.png" /></td> <td valign="top" style="width:5%;text-align:right;">(19)</td></tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900151.png" /> for (19); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900152.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900153.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900154.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900155.png" /> are the roots of the Jacobi polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900157.png" />, the geometric mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900158.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900159.png" /> iterations is close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900160.png" />. In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900161.png" />-method, the convergence of the iterations does not slow down as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900163.png" />.

Seidel's iterative method is used to solve the multi-velocity problem (4):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900164.png" /></td> <td valign="top" style="width:5%;text-align:right;">(20)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900165.png" /></td> </tr></table>

and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900166.png" /> in each equation in (20) is found by an iterative method for the one-velocity equation.

To solve multi-velocity problems for the eigenvalues of (4) and (5), these two iterative loops are supplemented with a further outer one to find the maximal value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900167.png" /> and the corresponding eigenfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900168.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900170.png" />, then problem (4), (5) becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900171.png" /></td> <td valign="top" style="width:5%;text-align:right;">(21)</td></tr></table>

To find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900173.png" />, iterative methods are used with the Chebyshev parameters:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900174.png" /></td> <td valign="top" style="width:5%;text-align:right;">(22)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900175.png" /></td> <td valign="top" style="width:5%;text-align:right;">(23)</td></tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900176.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900178.png" /> are parameters. One assumes that the spectrum is non-negative and first finds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900179.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900180.png" />, which are the largest eigenvalues of (21), on the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900182.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900183.png" /> is a lower bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900184.png" />, and takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900185.png" /> as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900186.png" />-sequence (see below). The values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900187.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900188.png" /> are determined by the generalized Aitken method, which incorporates the shifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900189.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900190.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900191.png" /> have been found, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900192.png" /> is found from (22) and (23) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900193.png" />. The infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900195.png" />-sequence is formed, correspondingly, from the specially ordered roots of the Chebyshev polynomials of the first kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900196.png" />, while the initial segment of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900197.png" />-segment of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900198.png" /> consists of all numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900200.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900201.png" /></td> </tr></table>

Any segment of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900202.png" />-sequence of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900203.png" /> ensures an optimal suppression in a certain sense of the error and stability in the iterative method (22), (23).

The following are used to solve non-stationary problems

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900204.png" /></td> </tr></table>

the method of characteristics in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900205.png" />-space, Galerkin's method, and finite-difference methods amounting to explicit and implicit difference schemes or to operator splitting methods. In the case of implicit schemes, the solution on the upper layer may be found by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093900/t093900206.png" />-method.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Mathematical methods of uniform-velocity transport theory"  ''Trudy Mat. Inst. Steklov.'' , '''61'''  (1961)  pp. 1–158  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.I. Marchuk,   V.I. Lebedev,   "Numerical methods in the theory of neutron transport" , Harwood  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Lebedev,   S.A. Finogenov,   "Utilization of ordered Chebyshev parameters in iterative methods"  ''USSR Comp. Math. Math. Phys.'' , '''16''' :  4  (1976)  pp. 70–83  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''16''' :  4  (1976)  pp. 895–907</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Lebedev,   "An iterative method with Chebyshev parameters for finding the maximum eigenvalue and corresponding eigenfunction"  ''USSR Comp. Math. Math. Phys.'' , '''17''' :  1  (1977)  pp. 92–101  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''17''' :  1  (1977)  pp. 100–108</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Davison,   "Neutron transport theory" , Oxford Univ. Press  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.J. Bell,   S. Glasstone,   "Nuclear reactor theory" , v. Nostrand  (1971)</TD></TR></table>