Direct method

A method for numerically solving partial differential equations (cf. [1], [2], [3]). It is applicable to non-linear equations and systems of equations of elliptic [4], hyperbolic [5] and parabolic [6] type of any order. The direct method makes it possible to perform numerical computations in regions with curvilinear boundaries [7]. The direct method is used in solving various problems in mechanics [8].

In the direct method one obtains approximations for differentiation operators in various directions; this makes it possible to lower the dimension of the problem and to replace solving the original system of partial differential equations by computations of a system of lower order approximating it.

Several problems in gas dynamics have been solved by the direct method [9]. Here, the problem of interpolating the original system of partial differential equations leads to the computation of a system of ordinary differential equations approximating it. In the direct method the region of integration is partioned, diametrically to the shock wave, by straight rays into strips. The enclosing ray lies in the region of supersonicity. The rays correspond, respectively, to the nodes of a Chebyshev polynomial or a uniform polynomial. The gas-dynamical functions are piecewise linearly approximated along each strip or are approximated by polynomials with nodes at each ray. The resulting system of ordinary differential equations can then be integrated along each ray from the shock wave to the body. As distinct from the integral-relation method there are no integral relations formed and one does not distinguish a minimal attraction region of blunts, which leads to a lesser degree of exactness but simplifies the form of the approximating system (cf. [10]).

By the direct method one has performed computations on two-dimensional flow around bodies of revolution by a perfect [11], a homogeneous [12] and an inhomogeneous [13] gas. One has also solved problems of supersonic flow around spheres using models of detonation waves [14], shock waves and water fronts [15], etc. By using trigonometric approximation along the meridian angle, the direct method has been extended to the three-dimensional spatial case (cf. [16], [17]), including inhomogeneous flow [18].

References

[1]	E.H. Rothe, "Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben" Math. Ann. , 102 (1930) pp. 650–670
[2]	A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, Byull. Moskov. Gos. Univ. Sekts. A , 1 : 6 (1937) pp. 1–26
[3]	A.A. Dorodnitsyn, , Conf. "Ways of development of Soviet mathematical machine-constructions and instrument engineering" , Moscow (1956) (In Russian)
[4]	E.Kh. Kostyukovich, Dokl. Akad. Nauk SSSR , 118 : 3 (1958) pp. 433–435
[5]	V.I. Lebedev, "The equations and convergence of a differential-difference method (the method of lines)" Vestn. Moskov. Gos. Univ. , 10 (1955) pp. 47–57 (In Russian)
[6]	O.A. Oleinik, A.S. Kalashnikov, Yu Lin Chzhou, "The Cauchy problem and boundary problems for equations of non-stationary filtration type" Izv. Akad. Nauk SSSR Ser. Mat. , 22 : 5 (1958) pp. 667–704 (In Russian)
[7]	B.M. Budak, A.D. Gorbunov, "The method of lines for solving a nonlinear boundary problem in the region bounded by a curve" Dokl. Akad. Nauk SSSR , 118 : 5 (1958) pp. 858–861 (In Russian)
[8]	Ya.I. Alikhashkin, "Solution of the problem of incomplete device by the method of lines" Vychisl. Mat. , 1 (1957) pp. 136–152 (In Russian)
[9]	S.M. Gilinskii, G.F. Telenin, G.P. Tinyakov, Izv. Akad. Nauk SSSR. Mekh. i Mashinostr. , 4 (1964) pp. 9–28
[10]	O.M. Belotserkovskii, P.I. Chyshkin, "A numerical method of integral relations" USSR Comp. Math. Math. Phys. , 2 : 5 (1963) pp. 823–858 Zh. Vychisl. Mat. i Mat. Fiz. , 2 : 5 (1962) pp. 731–759
[11]	G.S. Roslyakov, G.F. Telenin, "Survey of work on the numerical integration of exterior and interior problems of aerodynamics carried out at Moscow University" , Sb. Rabot. V.Ts. Moskov. Gosudarst. Univ. , 11 (1968) pp. 93–112 (In Russian)
[12]	G.F. Telenin, G.P. Tinyakov, Dokl. Akad. Nauk SSSR , 159 : 1 (1964) pp. 39–42
[13]	V.P. Stulov, , Some applications of grid methods in gas dynamics , 5 , Moscow (1974) pp. 140–227 (In Russian)
[14]	S.M. Gilinskii, Z.D. Zapryanov, G.G. Chernyi, Izv. Akad. Nauk SSSR. Mekh. Zhidkost. i Gaza , 5 (1966) pp. 8–13
[15]	S.M. Gilinskii, G.G. Chernyi, Izv. Akad. Nauk SSSR. Mekh. Zhidkost. i Gaza , 1 (1968) pp. 20–32
[16]	V.B. Minotsev, G.F. Telenin, G.P. Tinyakov, Dokl. Akad. Nauk SSSR , 179 : 2 (1968) pp. 304–307
[17]	A.P. Bazzhin, I.F. Chelisheva, Izv. Akad. Nauk SSSR. Mekh. Zhidkost. i Gaza , 3 (1967) pp. 119–123
[18]	O.N. Semenikhina, V.P. Shkalova, Izv. Akad. Nauk SSSR. Mekh. Zhidkost. i Gaza , 2 (1973) pp. 99–103