time discretization (it’s about time…) sauro succi
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Time Discretization(It’s about time…)
Sauro Succi
Time Evolution
Initial and Boundary Conditions
Time Marching: Formal
Time Marching: Matrix Exponential
L banded b, L*L banded 2b+1, L*L*L banded 3b+2 …. UNPRACTICAL
Numerical Marching
On the lattice x_j=j*d:
Numerical Marching
On the lattice x_j=j*d:
Space-Time Connectivity
Numerical Marching
Explicit
Crank-Nicolson
Fully Implicit
Unconditionally stable --> large dt
Non-local, matrix algebra, expensive
Time marching: Explicit/Implicit methods
Conditionally stable (|1+Ldt|<1)---> small dt
Matrix-free (local in spacetime)Simple and fast
X ----------X----------X
X ----------X----------X
X ----------X----------X X ----------X----------X
X ----------X----------X
Euler fwd
- t -
- t+dt -
- t+2dt -
Cranck-Nicolson
Implicit=Non-local
3 spatial connections
5 spatial connections
2p+1 spatial connections
Implicit Time-Marching
Matrix problem
Implicit Time-Marching
Implicit Time-Marching
Numerical Marching
Explicit
Crank-Nicolson
Forward Euler
Stability (for stable phys)
Crank-Nicolson
Trapezoidal rule
Matrix Algebra
Direct Methods
Iterative Methods
Algebraic Methods
Matrix Inversion
Direct Methods
1. Forward sweep: y
2. Backward sweep: x
Gaussian elimination
Zeroes included!
Multiple Dimensions
Memory is 1d: Addressing
Sequential index:
Physical vs Logical indexing
Bandwidth minimization (d>1)
Splitting
Relaxation
Gradient
Conjugate Gradient
Iterative: Methods
No matrix inversion: only matrix*vector products
Iterative: Jacobi splitting
Stopping criteria:
Gauss-Seidel: aggressive Jacobi
Convergence criteria:
Jacobi: does it converge?
Diagonal dominant:
Good matrix condition:
Optimum condition=1 Bad condition >>1
Alternating Direction Implicit
Directional Splitting
A sequence of Ny one-dimensionalProblems of size NX:
Over/Under relaxation methods
Accelerated Relaxation
Conservative, oscillating convergence
Aggressive, montonic convergence
Gradient methods
Relaxation methods
Fictitious time
Iteration loop
Convergence criterion:
Usually very slow!
Steepest descent: gradient
Residual=Gradient of the energy
r is the gradient ofE=1/2<x,Ax>-<b,x>
Steepest Descent
Convergent guaranteed only for SPD matrices [(x,Ax)>0]
Long-term slow because r-->0 as Ax-->b
Round-off sensitive, needle-sensitive
Conjugate Gradient
The steepest descent is not the shortest path to the minimum
g=r= Ax-b; <g,t>=0
c defined by <c,At> = 0
One shot on the local minimum at each plane x_k,x_(k+1)
tgc
x_k
x_(k+1)
x
Conjugate Gradient
The steepest descent is not the shortest path to the minimum!
0. x0, p0=r0=A*x0-b The new search direction is between r and old p1. a0=<r0,p0>/<r0,A*p0>
2. x1=x0+a0*p0; r1=r0-a0*A*p0 Out? 3. b0=<r1*r1>/<r0*r0> p1 = r1+b0*p0 next search direction based upon <p1,A*p0>=0 4. Go to 1 with p1
t
gc
x
{p0,A*p0,A^2*p0 … A^N p0} is a Krylov sequence: N step convergent for SPD
End of Lecture
Time Marching: Spectral
Hermitian Stable/Unstable
Example: diffusion
Operator Splitting (useful in d>1)
Non-commuting propagators
Trotter splitting