View source for Integral-relation method

''strip method''

A method for solving a system of partial differential equations, based on approximately reducing a partial differential equation to a system of ordinary differential equations. It is applicable to differential equations of various types. It was introduced by A.A. Dorodnitsyn [[#References|[1]]] as a generalization of the method of straight lines, a generalization of it with the introduction of smoothened functions was given in [[#References|[2]]], while in [[#References|[3]]] the method was evaluated and further developed.

Suppose one has a system of partial differential equations in divergence 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/i/i051/i051610/i0516101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</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/i/i051/i051610/i0516102.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516105.png" /> are given functions in the independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516107.png" /> and in the unknown variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516108.png" />. Suppose further that a solution to (1) is sought for in the curvilinear rectangle with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i0516109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161012.png" />, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161013.png" /> conditions are posed, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161014.png" /> of them are posed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161016.png" />. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161017.png" /> is unknown in advance, one only needs one additional condition. If the boundary contains singular points, then instead of corresponding boundary conditions one uses regularity conditions.

In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161018.png" />-th approximation the domain of integration is decomposed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161019.png" /> strips by the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161021.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161022.png" /> one chooses a closed system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161023.png" /> linearly independent functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161024.png" />. By multiplying each initial equation in (1) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161025.png" /> and after integration, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161026.png" />, across all strips, one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161027.png" /> integral relations of 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/i/i051/i051610/i05161028.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/i/i051/i051610/i05161029.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/i/i051/i051610/i05161030.png" /></td> </tr></table>

The integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161033.png" /> are approximated by using interpolation formulas with respect to their values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161036.png" /> on the boundaries of the strips. The integrals participating in the integral relations have 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/i/i051/i051610/i05161037.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161038.png" /> are numerical coefficients depending on the choice of the interpolation formulas and the form of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161039.png" />. The result is a system of ordinary differential equations, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161040.png" />, relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161041.png" /> values of the unknown functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161042.png" /> on the entire boundaries of the strips. This system is closed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161043.png" /> boundary conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161045.png" />.

The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161046.png" /> can be chosen rather arbitrarily. Choosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161048.png" />, i.e. delta-functions, leads to the method of straight lines. In it the derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161049.png" /> are replaced by difference expressions, corresponding to the interpolation formulas chosen. When using the step-functions

<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/i/i051/i051610/i05161050.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161051.png" />, one simply speaks of the integral-relation method, and the initial equations are integrated across each strip. Moreover, the conservation laws chosen for (1) are written as the following integral relations for the strips:

<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/i/i051/i051610/i05161052.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/i/i051/i051610/i05161053.png" /></td> </tr></table>

For systems of quasi-linear hyperbolic-type equations one has investigated convergence and error of the integral-relation method [[#References|[4]]]. In [[#References|[4]]] one has also established results that indicate the superiority of the integral-relation method over that of straight lines. Later papers have investigated analogous results for elliptic-type equations.

The merits of the integral-relation method are, among others: the possibility of choosing the interpolation formulas and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051610/i05161054.png" /> in dependence on the behaviour of the solution; the exact integration with respect to one of the variables (a result of the divergence notation of the original equations); the simplicity of the computational algorithm; and the relatively-small amount of computer memory when calculating on a computer. In the integral-relation method one approximates by an integral: this increases the accuracy of the approximation on account of a reduction of the coefficient of the remainder term; the integral is a smoother function than the integrand, thus one may reduce the number of interpolation nodes. If the integrand has discontinuities of the first kind, then the integral is a continuous function. The integral-relation method is most efficient when the solution can be obtained with good accuracy for a small number of strips.

The approximation in two variables of the integral-relation method can be generalized to the case of equations in three variables. Approximation in two variables can be conducted also in the two-dimensional case (the domain of integration is not partitioned into strips but into subdomains then); the system of approximate equations is then a system of non-linear algebraic or transcendental equations. Elements of the integral-relation method are used in other numerical methods, e.g. in the [[Large-particle method|large-particle method]].

The integral-relation method is mostly used in gas dynamics, where a number of practically-important problems have been solved by using it. Three schemes of the integral-relation method have been considered here: 1) the domain of integration is partitioned by lines passing through the surface of the body and the shock waves (Fig.a); 2) the domain of integration is partitioned by lines passing through the axis of symmetry and the boundary characteristics (Fig.b); and 3) the partitioning is into subdomains by two families of intersecting lines (Fig.c).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i051610a.gif" />

Figure: i051610a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i051610b.gif" />

Figure: i051610b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i051610c.gif" />

Figure: i051610c

The integral-relation method for computing supersonic flow around the fore part of a blunted body with propagating shock waves was developed in [[#References|[5]]], in which a numerical solution to this problem in a direct formulation was obtained for the first time. The solutions for most general cases have been found (bodies of various forms; spatial flow; the flow of a real gas under equilibrium and non-equilibrium physical-chemical changes and radiations; non-stationary motion; and the flow of a viscous gas, [[#References|[6]]]). The method has also been used for computing the potential flow of a gas [[#References|[7]]], and for solving some mixed problems: flow in sub- and transonic parts of jets [[#References|[8]]]. Pre-critical flow around a profile under an angle of attack with a place of the supersonic region has been considered [[#References|[9]]]. Supersonic conic gas flow has been clarified by the integral-relation method.

The integral-relation method has found wide application in computing the flow of a viscous gas in the theory of boundary layers. In the laminar incompressible case a solution has been obtained up to the point of discontinuity [[#References|[2]]]. Further, laminar boundary layers in a gas taking into account blowing-up or suction, radiation or heath conduction have been calculated. By the integral-relation method one has obtained solutions to the non-stationary problem of a point one-dimensional discontinuity in a gas taking counter-pressure into account, as well as in a gas with infinite conductivity in the presence of a magnetic field and in a heated gas.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Dorodnitsyn,   "On a method for numerically solving certain nonlinear problems of aerohydrodynamics" , ''Proc. third All-Union Math. Congress'' , '''3''' , Moscow  (1956)  pp. 447–453  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Dorodnitsyn,   "On a method for solving the laminar boundary layer equation"  ''Zh. Priklad. Mekh. i Tekhn. Fiz.'' , '''1''' :  3  (1960)  pp. 111–118  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.M. Belotserkovskii,   P.I. Chushkin,   "A numerical method of integral relations"  ''USSR Comp. Math. Math. Phys.'' , '''2'''  (1963)  pp. 823–858  ''Zh. Vychisl. Mat. Mat. Fiz.'' , '''2''' :  5  (1962)  pp. 731–759</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Bobkov,   V.I. Krylov,   "The method of integral relations for equations and systems of hyperbolic type"  ''Differentsial'nye Uravneniya'' , '''1''' :  2  (1965)  pp. 230–243  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.M. Belotserkovskii,   "Flow around a circular cylinder with a detached shockwave"  ''Dokl. Akad. Nauk SSSR'' , '''113''' :  3  (1957)  pp. 509–512  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  O.M. Belotserkovskii,   et al.,   "Flow of a supersonic gas around blunted bodies; theoretical and experimental investigations" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.I. Chuskin,   "Subsonic gas flow around ellipses and ellipsoids"  ''Vychisl. Mat.'' , '''2'''  (1957)  pp. 20–44  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Ya.I. Alikhashkin,   A.P. Favorskii,   P.I. Chushkin,   "On the calculation of the flow in a plane laval nozzle"  ''USSR Comp. Math. Math. Phys.'' , '''3''' :  6  (1963)  pp. 1552–1558  ''Zh. Vychisl. Mat. Mat. Fiz.'' , '''3''' :  6  (1963)  pp. 1130–1134</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  T.C. Tai,   "Transonic inviscid flow over lifting airfoils by the method of integral relations"  ''J. AIAA'' , '''12'''  (1974)  pp. 798–804</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V.P. Korobeinikov,   P.I. Chushkin,   "Plane, cylindrical, and spherical explosions in a gas with counterpressure"  ''Proc. Steklov Inst. Math.'' , '''87'''  (1967)  pp. 1–40  ''Trudy Mat. Inst. Steklov.'' , '''87'''  (1966)  pp. 4–34</TD></TR></table>