Method of Characteristics

The Method of Characteristics with Applications to Conservation Laws

2. Special Case: b(x, t) = 1 and c(x, t) = 0

We will examine the method of characteristics for three different PDEs:

For all three examples, the initial conditions are specified as . The examples take a slightly simpler form than the general equation (1). All three examples have b(x,t) = 1 and c(x,t) = 0. In this case characteristic equation (2b) becomes ,.and the solution is t = s+ k. Using the initial condition t(0) = 0, we determine that the constant is k = 0, so s = t. In this special case with b(x,t) = 1, we only have one characteristic equation to solve. Before proceeding to the examples, we restate the general strategy in terms of this special case.

Step1. Solve the characteristic equation (2a),

with the initial condition

Step 2. Solve the ODE (3), which in this case simplifies to

, with initial condition

Step 3. We now have a solution

. Solve for

in terms of x and t, using the results of Step 1, and substitute for

to get the solution to the original PDE as

2.1 Constant Coefficient Advection Equation

The advection equation is the PDE

, (4)

where a is a real constant, the wave speed or velocity of propagation. This is the rate at which the solution will propagate along the characteristics. The velocity is constant, so all points on the solution profile will move at the same speed a.

Now we apply the method of characteristics outlined in the 3 steps above.

Step 1. The characteristic equation (2a) to solve is

with the initial condition

. The solution

gives the characteristic curves, where

is the point at which each curve intersects the x-axis in the x-t plane.

Step 2. Solve equation (3),

, with initial condition

. The solution is

Step 3. The characteristic curves are determined by

, so

, and the solution of the PDE is

To verify that does indeed remain constant along the characteristics, we differentiate along one of these curves to find the rate of change of u along the characteristic:

So we find that the rate of change is zero, verifying that u is constant along the curve.

By writing the characteristic curves as , we see that, in the x-t plane, the characteristics are parallel lines with slope 1/a, so the slope of the characteristics depends only on the constant a.

Experiment in the applet with different values of the wave speed a, which can be changed through the dialog box displayed when the parameters button is pressed. Set to observe the solution profile moving in the opposite direction, and observe the negative slope of the characteristics.

2.2 Variable Coefficient Advection Equation

The variable coefficient advection equation is

. (5)

For this example, we take , so the wave speed depends on the spatial coordinate x. That is, the speed of a point on the solution profile will depend on the horizontal coordinate x of the point.

Step 1. The characteristic equation (2a) is

with the initial condition

. Thus the characteristic curves are

Step 2. The solution of equation (3),

with initial condition

, is

Step 3. The characteristic curves are determined by

, so

, and the solution of the PDE in the original variables is

View the variable coefficient advection equation simulation in the applet. Notice that the characteristics are not straight lines. Also observe that the characteristics do not intersect.