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''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

$$ \tag{1 }

\frac \partial {\partial  x }
 P _ {i} ( x , y , u _ {1} \dots u _ {k} ) +

\frac \partial {\partial  y }
 Q _ {i} ( x , y , u _ {1} \dots u _ {k} ) =
$$

$$ 
= \ 
F _ {i} ( x , y , u _ {1} \dots u _ {k} ) ,\  i = 1 \dots k,
$$

where  $  P _ {i} $, 
$  Q _ {i} $, 
$  F _ {i} $
are given functions in the independent variables  $  x $, 
$  y $
and in the unknown variables  $  u _ {1} \dots u _ {k} $. 
Suppose further that a solution to (1) is sought for in the curvilinear rectangle with boundary  $  x = a $, 
$  x = b $, 
$  y = 0 $, 
$  y = \Delta ( x) $, 
on which  $  2k $
conditions are posed, and where  $  k $
of them are posed on  $  x = a $
and  $  x = b $. 
If the function  $  \Delta ( x) $
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  $  N $-
th approximation the domain of integration is decomposed into  $  N $
strips by the lines  $  y = y _ {n} ( x) = n \Delta ( x) / N $, 
$  n = 1 \dots N $. 
For each  $  N $
one chooses a closed system of  $  N $
linearly independent functions  $  f _ {n} ( y) $. 
By multiplying each initial equation in (1) by  $  f _ {n} ( y) $
and after integration, with respect to  $  y $, 
across all strips, one obtains  $  kN $
integral relations of the form

$$ 
\left . 
\frac{d}{dx}
 \int\limits _ { 0 } ^ {  \Delta   ( x) } f _ {n} ( y) P  dy -
\Delta  ^  \prime  ( x) f _ {n} ( \Delta ( x) ) P \right | _ {y = \Delta ( x) }  +
$$

$$ 
+ 
\left . f _ {n} ( \Delta ( x)) Q \right | _ {y = \Delta ( x) }  - f _ {n} ( 0) Q ( 0) +
$$

$$ 
- \int\limits _ { 0 } ^ {  \Delta   ( x) } f _ {n} ^ { \prime } ( y) Q  dy
 =  \int\limits _ { 0 } ^ {  \Delta   ( x) } f _ {n} ( y) F  dy .
$$

The integrals  $  P $, 
$  Q $, 
$  F $
are approximated by using interpolation formulas with respect to their values  $  P _ {n} $, 
$  Q _ {n} $, 
$  F _ {n} $
on the boundaries of the strips. The integrals participating in the integral relations have the form

$$ 
\int\limits _ { 0 } ^ {  \Delta   ( x) } f _ {n} ( y) P  dy  \approx  \Delta ( x)
\sum _ { n= } 0 ^ { N }  C _ {n} P _ {n} y _ {n} ( x) ,
$$

where  $  C _ {n} $
are numerical coefficients depending on the choice of the interpolation formulas and the form of the  $  f _ {n} ( y) $. 
The result is a system of ordinary differential equations, with respect to  $  x $, 
relative to  $  K ( N+ 1) $
values of the unknown functions  $  u _ {n} $
on the entire boundaries of the strips. This system is closed by  $  k $
boundary conditions for  $  y = 0 $
and  $  y = \Delta ( x) $.

The functions  $  f _ {n} ( y) $
can be chosen rather arbitrarily. Choosing  $  f _ {n} ( y) = \delta ( y - y _ {n} ) $, 
$  n = 1 \dots N $, 
i.e. delta-functions, leads to the method of straight lines. In it the derivatives with respect to  $  y $
are replaced by difference expressions, corresponding to the interpolation formulas chosen. When using the step-functions

$$ 
f _ {n} ( y)  = \ 
\left \{

\begin{array}{l}
0 \  \textrm{ for }  y < y _ {n-} 1 ,  \\
1 \  \textrm{ for }  y _ {n-} 1 \leq  y \leq  y _ {n} ,  \\
0 \  \textrm{ for }  y > y _ {n} ,  \\
\end{array}
 \right .
$$

$  n = 1 \dots N $, 
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:

$$ 

\frac{d}{dx}
 \int\limits _ { y _ {n-} 1 } ^ { {y _ n } } P  dy - y _ {n}  ^  \prime 
P ( y _ {n} )+ y _ {n-} 1  ^  \prime  P ( y _ {n-} 1 ) + Q ( y _ {n} ) - Q ( y _ {n-} 1 ) =
$$

$$ 
= \ 
\int\limits _ { y _ {n-} 1 } ^ { {y _ n } } F  dy .
$$

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  $  f( y) $
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>