Transport Equation Characteristics

Worked Example

Consider the advection equation u_t + c u_x = 0 on the real line, with the initial profile u(x,0) = f(x). The solution is u(x,t) = f(x - ct). The curves x(t) = x₀ + ct are the characteristics, along which the solution is constant.

Let f(x) = exp(-x²), speed c = 1, at time t = 1, and probe x = 1.

1. Characteristic Foot: The characteristic passing through (x,t)=(1,1) intersects the t=0 axis at x₀ = x - ct = 1 - 1(1) = 0.
2. Solution: u(1,1) = f(x₀) = exp(-0²) = 1.0.
3. Spatial Derivative: u_x(x,t) = f'(x - ct). Here f'(x) = -2x exp(-x²), so u_x(1,1) = f'(0) = 0.0.
4. Time Derivative: u_t(x,t) = -c f'(x - ct) = -1 * 0 = -0.0.
5. Residual: u_t + c u_x = 0 + 1(0) = 0.

Experiment

Explore this concept in the Transport Equation Characteristics module.

1. Select the Profile as "Gaussian" and Speed c as 1.
2. Change the Time t to observe the profile translating to the right.
3. Adjust the Probe x and verify that the calculated residual is always zero, confirming exact advection without diffusion.

Limitations

This module demonstrates pure advection with a constant coefficient c. It does not cover variable speed equations (where c depends on x or t), nonlinear transport equations like Burgers' equation (where characteristics can intersect to form shocks), or transport with diffusion (which requires parabolic PDE solvers). The rendering is also restricted to a finite spatial window and a small set of simple analytical profiles.

References

For a comprehensive introduction to the method of characteristics for first-order PDEs, see the lecture notes by Gautam Iyer.

• Method of Characteristics - Math 2026-632