SYLLABUS Previous: 1.2.3 Boundary / initial conditions
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Slide : [ characteristics elliptic - parabolic - hyperbolic || local dispersion relation || VIDEO login]
The characteristics of a PDE can loosely be defined as the trajectories 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
Three categories of equations depend on the sign of the discriminant:
The local properties of a linear equation are conveniently investigated by introducing the harmonic ansatz , which transforms differential operators in into algebraic expression in . In effect, you substitute and
to produce a dispersion relation (or Von Neuman stability relation), relating the PHYSICAL phase velocity or the growth rate to the typical scale of the solution . Assuming a homogeneous grid in space and time , 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 or temporal resolutions .
For example, take the effective wavenumber that results when the mid-point rule (1.2.1#eq.3) is used to approximate the first derivative of the spatial function :
The wave number is under-estimated for poor resolution and even changes sign with less than two mesh points per wavelength. The forward difference (1.2.1#eq.2)
has an imaginary part, showing that small wavelengths will be strongly damped .
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