PDE’s Discretization

Sauro Succi

Evolutionary PDE

Formal: big time

Formal: small time

Evolutionary PDE’s

Advection

Diffusion

Reaction

Self-advection (fluids)

Finite-Difference Schemes

Finite-Difference Schemes

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)

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

Jump to actual PFDE’s

(with apologies to the theory-inclined)

Consistency

Finite-Difference Schemes

Consistency

Consistent

Forward Euler

Centered

Accuracy

Reproduce poly(p) at x=xj

Stability

Courant Numbers

Faster than light?

Linear instability (early) Non-linear instability (long-term)

Short/Long term instability

Stability: spectral analysis

Spectral Deformations:

Lax equivalence Theorem

Consistent schemes for well-posedLinear PDE’s are convergent if theyare stable

Stability is easier to prove than convergence!

Scaling limits

Transfer Operator

First order in time:

Efficiency

Slow diffusion

Diffusion

Advection

Acceleration

Computer metrics

1 Petaflop

Computer metrics

Locality

Differential Operators

Sparse matrices

Causality

Present depends on past

NO simultaneous dependence

No inverse time depenedence

Reversibility(Hamiltonian)

Reversibility: Euler

Reversibility: Crank-Nicolson

Pade’:

Now to actual PDE’s