2.2 Partial Differential Equations | 2.2 Partial Differential Equations | Unit 2: Numerical Methods for Partial Differential Equations | Computational Methods in Aerospace Engineering | Aeronautics and Astronautics

<Partial Differential Equations
2.2.1Conservation Laws in Integral and Differential Form
2.2.2One-Dimensional Burgers Equation
2.2.3Convection
2.2.4Characteristics for One-Dimensional Burgers Equation
2.2.5Diffusion
2.2.6Convection-Diffusion
2.2.7Linear Elasticity
>Convection

2.2.2 One-Dimensional Burgers' Equation

Burgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas, such as modeling of gas dynamics and traffic flow. It is named for Johannes Martinus Burgers (1895-1981). The one-dimensional Burgers' equation is given in differential form as:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0 \label{equ:burgers}\]

(2.17)

Which of the following is the correct mapping between the terms of the Burgers' equation and the terms of the canonical conservation equation?

\(U=u, \vec{F}=u\)

\(U=u, \vec{F}=u^2\)

\(U=\frac{1}{2}u, \vec{F}=u^2\)

\(U=u,\vec{F}=\frac{1}{2}u^2\)

Answer: This solution can be verified by plugging in \(\vec{F} =\frac{1}{2} u^2\) and noting that \(\frac{\partial \vec{f}}{\partial x} = \frac{\partial \vec{F}}{\partial u}\frac{\partial u}{\partial x}\) and we arrive that the above differential form.