$$ \newcommand{\D}{\displaystyle} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})} $$




5 Mathematical properties of partial differential equations

5.1 Model equations

In Appendix 2 in the Numeriske Beregninger Navier-Stokes equations for an incompressible fluid: $$ \begin{align} \frac{Du}{Dt}\equiv\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y} & =-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\right) \tag{5.1}\\ \frac{Dv}{Dt}\equiv\frac{\partial v}{\partial t}+u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y} & =-\frac{1}{\rho}\frac{\partial p}{\partial y}+\nu\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2v}{\partial y^2}\right) \tag{5.2}\\ \frac{DT}{Dt}\equiv\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y} &=\alpha\left(\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2}\right) \end{align} \tag{5.3} $$

The left-hand side terms represent transport (advection/convection), while the right-hand side stands for diffusive processes. It follows that transport is expressed by first order terms and are, in this case, non-linear, while diffusive terms are given by second order derivatives, which in this case are linear. Many important engineering problems can be described by special cases of these equations, such as boundary layer problems. Since such special cases are, of course, easier to solve and analyze, these will be often used when we are going to derive a numerical scheme. These are usually one-dimensional, non-stationary cases.

5.1.1 List of some model equations

Poisson equation. $$ \begin{equation} \frac{\partial^2u}{\partial x^2}+\frac{\partial^2}{\partial y^2}=f(x,y) \tag{5.4} \end{equation} $$ Laplace for \( f(x,y)=0 \) Potential flow theory, stationary heat conduction, etc.

One-dimensional diffusion equation. $$ \begin{equation} \frac{\partial u}{\partial t}=\alpha \frac{\partial^2u}{\partial x^2} \tag{5.5} \end{equation} $$

Wave equation. Fundamental equation for acoustics and other applications. $$ \begin{equation} \frac{\partial^2u}{\partial t^2}=\alpha_0^2\frac{\partial^2u}{\partial x^2} \tag{5.6} \end{equation} $$ 1. order linear advection equation. $$ \begin{equation} \frac{\partial u}{\partial t}+\alpha_0\frac{\partial u}{\partial x}=0 \tag{5.7} \end{equation} $$

Inviscid Burger's equation. Model for Euler's equation of gas dynamics. $$ \begin{equation} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0 \tag{5.8} \end{equation} $$ Burger's equation. Model for incompressible Navier-Stokes equations. $$ \begin{equation} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2u}{\partial x^2} \tag{5.9} \end{equation} $$

Tricomi equation. Model for transonic flow. $$ \begin{equation} y \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0 \tag{5.10} \end{equation} $$

Convection-diffusion equation. Linear advection and diffusion. $$ \begin{equation} \frac{\partial u}{\partial t}+u_0\frac{\partial u}{\partial x}=\alpha \frac{\partial^2u}{\partial x^2} \tag{5.11} \end{equation} $$

The expressions transport, convection and advection are equivalent. (5.7) above is known as the advection equation.