3.1.1 Viscous and inviscid Burgers' equation

Viscous Burgers' equation:

$$\frac{\partial v_{}}{\partial t} + \frac{\partial \frac{1}{... ...rtial x} = \epsilon \frac{\partial^2 v_{}}{{\partial x}^2}$$ (158)

inviscid Burgers' equation in conservation form (with flux $\frac{1}{2} v_{}^2$)

$$\frac{\partial v_{}}{\partial t} + \frac{\partial \frac{1}{2} v_{}^2}{\partial x} = 0 \enspace ,$$ (159)

quasi-linear form

$$\frac{\partial v_{}}{\partial t} + v_{} \frac{\partial v_{}}{\partial x} = 0 \enspace ,$$ (160)

and integral form

$$\int_{x_0}^{x_1} \left[ v_{} \! \left( x , t_1 \right) - ... ...frac{1}{2} v_{}^2 \! \left( x_0 , t \right) \right] dt = 0$$ (161)

• It resembles the linear advection equation (63).
• However, it is non-linear with $v_{}\left( x, t \right)$ instead of $v_{} = \mbox{const}$.
• It describes the transport of $v_{}$ with velocity $v_{}$.