previous up next ','..','$myPermit') ?> SYLLABUS  Previous: 1.2.3 Boundary / initial conditions  Up: 1.2 Differential Equations  Next: 1.2.5 Moments and conservation


1.2.4 Characteristics and dispersion relations

Slide : [ characteristics elliptic - parabolic - hyperbolic || local dispersion relation || VIDEO 99) echo " modem - ISDN - LAN "; else echo "login";?> ]

The characteristics of a PDE can loosely be defined as the trajectories $ x(t)$ along which discontinuities and the initial conditions propagate: think of the path a heat pulse takes in an inhomogeneous material. The chain rule can be used more formally to classify second order equations (1.2.2#eq.1) with D=0 using an ansatz $ f(x,t)=f(x(t),t)$

$\displaystyle \frac{\partial^2 f}{\partial x^2}\left[ A \left(\frac{\partial x}... ...grightarrow \frac{\partial x}{\partial t}=A^{-1}\left(-B\pm\sqrt{B^2-AC}\right)$ (1.2.4#eq.1)

Three categories of equations depend on the sign of the discriminant: The characteristics play an important role and will be exploited in the Lagrangian methods in chapter 6.

The local properties of a linear equation are conveniently investigated by introducing the harmonic ansatz $ f(t,x)=f_0\exp(-i\omega t + ikx)$ , which transforms differential operators in $ (t,x)$ into algebraic expression in $ (\omega,k)$ . In effect, you substitute $ \partial_t \rightarrow -i\omega$ and $ \partial_x \rightarrow +ik$

$\displaystyle \frac{\partial f}{\partial t} + u \frac{\partial f}{\partial x}=0 \quad\quad \Longrightarrow \quad\quad -\omega+uk=0$ (1.2.4#eq.2)

to produce a dispersion relation (or Von Neuman stability relation), relating the PHYSICAL phase velocity $ \Re e(\omega)/k$ or the growth rate $ \Im m(\omega)$ to the typical scale of the solution $ k$ . Assuming a homogeneous grid in space $ x_j=j\Delta x$ and time $ t_n=n\Delta t$ , it is moreover possible to assess the quality of the numerical approximation, by relating the effective NUMERICAL phase velocity or growth rate (i.e. obtained after discretization) to the the spatial $ k\Delta x$ or temporal resolutions $ \omega \Delta t$ .

For example, take the effective wavenumber $ k_\mathrm{eff}=-i f^\prime/f$ that results when the mid-point rule (1.2.1#eq.3) is used to approximate the first derivative of the spatial function $ f(x)=\exp(ikx)$ :

$\displaystyle k_\mathrm{eff}=-i\frac{\exp(ik\Delta x)-\exp(-ik\Delta x)}{2\Delta x} = k \frac{\sin(k\Delta x)}{k\Delta x}$ (1.2.4#eq.3)

The wave number is under-estimated for poor resolution $ k\Delta x\rightarrow\pi$ and even changes sign with less than two mesh points per wavelength. The forward difference (1.2.1#eq.2)

$\displaystyle k_\mathrm{eff}=-i\frac{\exp(ik\Delta x) -1}{\Delta x} = k \frac{\... ...s\left(\frac{k\Delta x}{2}\right) +i\sin\left(\frac{k\Delta x}{2}\right)\right]$ (1.2.4#eq.4)

has an imaginary part, showing that small wavelengths $ k\Delta x\rightarrow\pi$ will be strongly damped $ f\propto \exp[-\Im m(k_\mathrm{eff})\Delta x]$ .

SYLLABUS  Previous: 1.2.3 Boundary / initial conditions  Up: 1.2 Differential Equations  Next: 1.2.5 Moments and conservation