What Is a Characteristic Line?

Method of characteristics: A similar method for solving hyperbolic partial differential equations based on feature theory. It originated earlier and was effectively used by people at the end of the 19th century. After the advent of the electronic computer, it has been further developed, and has been widely used in problems such as one-dimensional unsteady flow and two-dimensional steady flow.

Method of characteristics: A similar method for solving hyperbolic partial differential equations based on feature theory. It originated earlier and was effectively used by people at the end of the 19th century. After the advent of the electronic computer, it has been further developed, and has been widely used in problems such as one-dimensional unsteady flow and two-dimensional steady flow.

Chinese name

Feature line method

Foreign name

method of characteristics

Produce

Late 19th century

Foundation

Feature theory of partial differential equations

Application

widely

Overview of feature line method

Based on the characteristic theory of partial differential equations, an approximate calculation method for solving hyperbolic partial differential equations. If the problem is relatively simple, this method can be used to obtain an analytical solution or an approximate analytical solution; if the problem is complex, a highly accurate numerical solution can also be obtained. In addition, the characteristic line method can also be used to make a qualitative analysis of hyperbolic problems, especially to study how to give initial conditions and boundary conditions to make the problem well-posed. This has guiding significance for designing other types of numerical methods for solving hyperbolic differential equations. The characteristic line method has appeared as early as the end of the 19th century, and many problems have been solved by hand counting in the 1930s and 1940s. With the advent of electronic computers, this method has become more sophisticated and widely used.

Physical meaning

Figure 1 Characteristic lines of shallow water waves

Figure 2 Shock waves and characteristic lines when a bullet passes through air at supersonic speed

Although the characteristic line is an abstract mathematical concept, its physical meaning is clear in some problems. The steady two-dimensional shallow water wave shown in Figure 1 can be seen by the naked eye. Fig. 1a shows that the water flow is dropping from the inclined plane at a flow velocity v. There is a small angular pebble in the current, and each small angular disturbance to the current appears as a ripple (Figure 1b). When the average velocity v exceeds

(G is the acceleration of gravity and h is the average thickness of the water), the water waves will not propagate upwards but will be carried downstream by the water, that is, the stones will not affect the water on the left of a, b, and c in Figure 1b. The boundary between the area affected by a small disturbance at a certain point and the area not affected is actually a characteristic line, and this characteristic line is visible to the naked eye. In general, the characteristic line is invisible to the naked eye. For example, a bullet with a streaked surface passes through the air at supersonic speed, and the characteristic line caused by the streak (Figure 2) can only be observed with the aid of an instrument. The one-dimensional unsteady motion of a gas can be described by the following basic equations:

Where u is the particle velocity; is the density; p is the pressure; S is the entropy; x is the coordinate; and t is the time. In order to solve (1), the speed of sound c and the equation of state c = c (p, S) and = (, S) are also introduced. Equation (1) is a system of hyperbolic equations with two independent variables and three unknown functions. It is non-linear. The main points of the characteristic line method are explained by solving this system of equations. (1) can be transformed into an equivalent system of equations (2) through transformation, and each equation of (2) contains only a derivative in a certain direction. This direction is the "feature direction". The third formula of (1) is a differential quotient along the direction dx = udt, so dx = udt is a characteristic direction.

Is the corresponding characteristic relationship along this direction. (1) The first, second, and two equations can be obtained by simple transformation:

The two formulas in (2) only have the differential quotients along the directions dx = (u + c) dt and dx = (u-c) dt, respectively. Therefore, dx = (u ± c) dt is the other two characteristic directions of (1).

It is a "characteristic relationship" along these two directions. In the (x, t) plane, the corresponding curve determined by the feature direction is the feature line of (1). Summarizing the three characteristic directions of (1) and the corresponding characteristic relations, we get the ordinary differential equations equivalent to (1):

The characteristic line method transforms the problem of solving the system of partial differential equations (1) into the problem of solving the system of ordinary differential equations (3) which is much simpler through the above transformation.

Considering two sufficiently close points Q ₁ and Q ₂ on the (x, t) plane (Fig. 3), suppose that u, p, , S, and c at these two points are known. The intersection of dx = (u + c) dt and the characteristic line dx = (uc) dt passing Q ₂ is denoted as Q ₃ . Then, from Q ₃ to the direction with a small time, the characteristic line Q ₃ Q ₄ is dx / dt-u, and the value of u is temporarily the average value of u at two points Q ₁ and Q ₂ . The intersection of Q ₃ Q ₄ and Q ₁ Q ₂ is Q ₄ . At the points Q ₁ Q ₄ , all the quantities (u, p, , c, S, x, t) are represented by corresponding symbols. In order to find x ₃ , t ₃ , u ₃ , p ₃ , S ₃ at the Q ₃ point, and then use the equation of state to find ₃ , c ₃ , the equation (3) can be approximated as:

Figure 3 Feature lines on the x-t plane-Find Q3 from Q1, Q2

(4) Medium

It can be obtained by interpolation based on the entropy values of Q ₁ and Q ₂ points. From (4), we can find the approximate position of Q ₃

And above

,

,

. The above method is only equivalent to replacing the curve near the tangent point with a tangent point on the curve, so the numerical calculation is called first-order approximation (also called initial calculation). Based on the results of the first-order approximation (also called recalculation), a much more accurate second-order approximation is obtained. The method is to use the first-order approximation of the Q3 point

,

,

It is averaged with the known values u ₁ , c ₁ , and ₁ at the point Q ₁ to replace (u ₁ + c ₁ ) and ₁ c ₁ in formula (4). This is equivalent to replacing the curve with a secant. Of course, both theoretically and practically. Similarly, the average value of the Q ₃ point initial value and the Q ₂ point known value is used to replace u ₂ -c ₂ and ₂ c ₂ in the formula (4). Of course, the position of Q ₄ and S ₄ must be recalculated. The x ₃ , t ₃ , u ₃ , p ₃ , ₃ , c ₃ , and S _{3 obtained} in this way are quite good numerical results of using the characteristic line method to find the quantities of Q ₃ points.

If an initial value is given on a curve segment MN that is not tangent to the characteristic line and is close to the t-axis direction (Figure 4), then the above method can be used to find the characteristic line at the point M.

And feature lines that cross N points

The approximate positions of the intersections of the characteristic lines and the corresponding unknown function values in the enclosed area MNP.

Figure 4 Feature line grid on the x-t plane

The simplest and essential case of solving (1) is described above. For the general narrow hyperbolic equations where the n independent feature directions of two independent variables and n unknown functions are not the same, you need to find n characteristic directions and the corresponding n characteristic relations, and use a method similar to the above To solve. As for solving the system of three independent variables, the feature line method can be generalized, but they are all complicated, and there are still some problems that need to be solved. It is not widely used. The more common method is the difference method.

Feature Line Method Bibliography

R. Courant and KO Friedrichs, Supersonic Flow and Waves, Interscience Pub., London, 1956.