PDE’s Discretization

Sauro Succi

Now to actual PDE’s

Diffusion Equation

Advection-Diffusion Equation

Advection-Diffusion-Reaction Equation

PDE’s Family

Diffusion Equation

DE: Dispersion Relation

Dispersion Relation: Fourier

Fourier Transform: plane-wave superposition

DE: Dispersion Relations

Continuum

Discrete

Dispersion Relation: general

Propagation/Dispersion:

Diffusion/Hyperdiffusion

Numerical Diffusion: positive = overdamping

Numerical dispersion: Gibbs phenomena

Diffusion Equation: Forward Euler

DE: centered differences

Computational molecules

Elementary trimer (stencil) builds up a crystal (matrix)

DE: Centered-Euler

DE: Centered-Euler

DE: Stability analysis

DE: Stability analysis

DE: Linear stability analysis

DE: Discrete DR

Square and Sum:

Stable: 2nd order accurate

Discrete DR: Dispersion

Divide:

Exact

Entry Exercise: Heat diffusion in 1d wire

Advection Equation

Dispersion Relation

AE: Euler centered

1st order centered: unconditionally unstable

AE: Centered-Euler

Never between [0,1] together >>>>> Unstable!

AE: DDR

Unconditionally unstable!

AE: phase errors

Dispersion errors: 3rd order

Add artificial diffusion

Lax-Wendroff

Upwind

Time centered

……

Stabilize AE centered

Stabilize AE: Add artificial diffusion

ADE: Centered-Euler

ADE: Centered-Euler

Stability vs Realizability

Stability analysis: general

Propagation

Stability analysis: general

(<1) (>0)

General: Stability and dispersion

Infrared limit: k,omega to zero

ADE: Stability analysis

Conditionally stable!

Lax-Wendroff

LW: Stability analysis

Unconditionally stable!

AE: Upwind

One sided: first order in BOTH space and time

AE: Upwind

For 0<alpha<2 both a,b are in [0,1]: Stable!

Upwind: DDR

DE_centered

AE_centered

ADE_centered

ADE_LW

AE_Upwind

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Summary

End of lecture 6