P DE’s Discretization
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Transcript of P DE’s Discretization
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PDE’s Discretization
Sauro Succi
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Now to actual PDE’s
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Diffusion Equation
Advection-Diffusion Equation
Advection-Diffusion-Reaction Equation
PDE’s Family
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Diffusion Equation
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DE: Dispersion Relation
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Dispersion Relation: Fourier
Fourier Transform: plane-wave superposition
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DE: Dispersion Relations
Continuum
Discrete
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Dispersion Relation: general
Propagation/Dispersion:
Diffusion/Hyperdiffusion
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Numerical Diffusion: positive = overdamping
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Numerical dispersion: Gibbs phenomena
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Diffusion Equation: Forward Euler
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DE: centered differences
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Computational molecules
Elementary trimer (stencil) builds up a crystal (matrix)
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DE: Centered-Euler
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DE: Centered-Euler
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DE: Stability analysis
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DE: Stability analysis
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DE: Linear stability analysis
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DE: Discrete DR
Square and Sum:
Stable: 2nd order accurate
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Discrete DR: Dispersion
Divide:
Exact
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Entry Exercise: Heat diffusion in 1d wire
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Advection Equation
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Dispersion Relation
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AE: Euler centered
1st order centered: unconditionally unstable
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AE: Centered-Euler
Never between [0,1] together >>>>> Unstable!
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AE: DDR
Unconditionally unstable!
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AE: phase errors
Dispersion errors: 3rd order
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Add artificial diffusion
Lax-Wendroff
Upwind
Time centered
……
Stabilize AE centered
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Stabilize AE: Add artificial diffusion
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ADE: Centered-Euler
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ADE: Centered-Euler
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Stability vs Realizability
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Stability analysis: general
Propagation
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Stability analysis: general
(<1) (>0)
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General: Stability and dispersion
Infrared limit: k,omega to zero
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ADE: Stability analysis
Conditionally stable!
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Lax-Wendroff
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LW: Stability analysis
Unconditionally stable!
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AE: Upwind
One sided: first order in BOTH space and time
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AE: Upwind
For 0<alpha<2 both a,b are in [0,1]: Stable!
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Upwind: DDR
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DE_centered
AE_centered
ADE_centered
ADE_LW
AE_Upwind
========= ===========
Summary
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End of lecture 6