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P DE’s Discretization. Sauro Succi. Now to actual PDE’s. PDE’s Family. Diffusion Equation. Advection-Diffusion Equation. Advection-Diffusion-Reaction Equation. Diffusion Equation. DE: Dispersion Relation. Dispersion Relation: Fourier. Fourier Transform: plane-wave superposition. - PowerPoint PPT Presentation
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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
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