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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Evolutionary PDE
Formal: big time
Formal: small time
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Evolutionary PDE’s
Advection
Diffusion
Reaction
Self-advection (fluids)
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Finite-Difference Schemes
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Finite-Difference Schemes
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The guiding principles of ComPhys
Stability/Conservativeness 1st and 2nd principle, error decay
Efficiency Cost per unit update
AccuracyFast error decay with increasing resolution
ConsistencyRecover the continuum limit at infinite resolution (no anomaly)
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The guiding principles of ComPhys
LocalityComputational density independent of system size (Feynman)
CausalityNo simultaneous interactions (Present-->Future)
Reversibility No burnt-bridges doors, exact roll-back, very long time integration
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Jump to actual PFDE’s
(with apologies to the theory-inclined)
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Consistency
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Finite-Difference Schemes
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Consistency
Consistent
Forward Euler
Centered
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Accuracy
Reproduce poly(p) at x=xj
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Stability
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Courant Numbers
Faster than light?
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Linear instability (early) Non-linear instability (long-term)
Short/Long term instability
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Stability: spectral analysis
Spectral Deformations:
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Lax equivalence Theorem
Consistent schemes for well-posedLinear PDE’s are convergent if theyare stable
Stability is easier to prove than convergence!
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Scaling limits
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Transfer Operator
First order in time:
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Efficiency
Slow diffusion
Diffusion
Advection
Acceleration
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Computer metrics
1 Petaflop
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Computer metrics
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Locality
Differential Operators
Sparse matrices
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Causality
Present depends on past
NO simultaneous dependence
No inverse time depenedence
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Reversibility(Hamiltonian)
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Reversibility: Euler
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Reversibility: Crank-Nicolson
Pade’:
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Now to actual PDE’s