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1.2.4 Characteristics and dispersion relations

Slide : [ characteristics elliptic - parabolic - hyperbolic || local dispersion relation || VIDEO 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:
  • $ B^2-AC<0$ the equation has no characteristic and is called elliptic (Laplace eq.)
  • $ B^2-AC=0$ the equation has one characteristic and is called parabolic (heat eq.)
  • $ B^2-AC>0$ the equation has two characteristics and is called hyperbolic (wave eq.)
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

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