Burgers' equation

Summary

Burgers' equation $u_t + u\, u_x = \nu\, u_{xx}$ is the simplest equation that combines nonlinear transport with diffusion. For $\nu = 0$ it is the inviscid Burgers' equation, the model scalar conservation law, whose smooth solutions break down and form shock waves. For $\nu > 0$ the Cole–Hopf transformation turns it into the heat equation, so it can be solved exactly. It is a standard test case for theory and for numerical methods in fluid dynamics. See the definition.