The Tricomi Problem
In this chapter we use the method of difference potentials for computing solutions of some difference analogs of the Tricomi problem. The classical Tricomi problem is a typical example of numerous problems arising in the study of plane transonic flows of
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In this chapter we use the method of difference potentials for computing solutions of some difference analogs of the Tricomi problem. The classical Tricomi problem is a typical example of numerous problems arising in the study of plane transonic flows of compressible gas, e.g., of flows in the Laval nozzle or flows around contours. The Tricomi equation has the form
cPu
8 2u y 8x 2 + 8 y 2 =
o.
(I)
It is an equation of mixed type, since it is elliptic for y > 0 and hyperbolic for y < o.
y
o
x
B'
Fig. 1.1.
The Tricomi problem is stated as follows. We define a domain jj bounded by some curve Fr = ABG and by two characteristics r2 = AB' and r3 = GB' of (I) (Fig. 1.1). On r 1 and r2 we define functions epr 1 and ip r2 .
V. S. Ryaben'kii, Method of Difference Potentials and Its Applications © Springer-Verlag Berlin Heidelberg 2002
330
1. The Tricomi Problem
It is required to find a solution of (I) that is continuous in the closed domain D U T 1 U T 2 U T 3 , belongs to some function class, and satisfies the boundary conditions
(II) Already known are the smoothness conditions on T 1 and the functions CP TIl CP T2' the conjugation conditions at the point A, and the conditions on the function class to which the solution belongs, which provide the existence and uniqueness of the desired solution [131]. We describe the scheme of the method of difference potentials used to obtain solutions of the difference analog of the Tricomi problem constructed by A. F. Filippov [132] and of some other difference analog of the Tricomi problem, which employs some elements of the Filippov constructions and was proposed (without any justification of convergence) by M. Yu. Lokhanov. To compute the solutions of these two difference analogs of the Tricomi problem, we present different realizations of the general MDP scheme. The main results of this chapter were obtained by M. Yu. Lokhanov in his diploma research (Moscow Institute of Physics and Technology, 1982) and in [41].
1.1 Difference Analogs of the Tricomi Problem We describe the difference analogs of the Tricomi problem whose solutions will be obtained by the method of difference potentials. 1.1.1 Filippov's Scheme
On the plane xOy we draw the following two families of straight lines:
+ ih,
x
=
Xi,
Xi
=
y
= Yj,
Yj
= ( '2 lj 1h
XA
3
)2/3
signj,
i=O,±I, ... ,
(1.1)
j = 0, ±1, ±2, ... ,
(1.2)
where XA is the abscissa of the point A. We choose the step h so that for some n the abscissa XB of the point B satisfies the relation Xc = XA + 2nh. For Y ~ 0, we consider the grid formed by intersection of the lines (1.1) and (1.2). For Y < 0, we construct the grid as follows. We draw the characteristics X
2 1 . + 3(-y)3 2 = XA + 2th,
X-
~(_y)3/2
= xA
+ 2ih,
(1.3) i = 0,1,2, ... , k,
1.1 Difference Analogs of the Tricomi Problem
331
B
Y1 H+--+--+-+-lI-f-+--+--+--1-l
B'
Fig. 1.2.
of the Tricomi equation (I) , which pass through the points XA + 2ih on the x-axis. The points of intersection of these characteristics form the grid used for Y < (Fig. 1.2).
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