integration

Time stepping (integration) tools.

FD_Jac(func, eps=1e-07)

Finite-diff approx. of Jacobian of func.

The function func(x) must be compatible with x.ndim == 1 and 2, where, in the 2D case, each row is seen as one function input.

Returns:

Type	Description
function	The first input argument is that of which the derivative is taken.

Examples:

>>> dstep_dx = FD_Jac(step)

integrate_TLM(TLM, dt, method='approx')

Compute the resolvent.

The resolvent may also be called

• the Jacobian of the step func.
• the integral of (with M as the TLM):

\[ \frac{d U}{d t} = M U, \quad U_0 = I . \]

The tangent linear model (TLM)

is assumed constant (for each method below).

Parameters:

Name	Type	Description	Default
method	str	'approx' : derived from the forward-euler scheme. 'rk4' : higher-precision approx. 'analytic': exact. Warning "'analytic' typically requries higher inflation in the ExtKF."	'approx'

See Also

FD_Jac.

rk4(f, x, t, dt, stages=4, s=0)

Runge-Kutta (explicit, non-adaptive) numerical (S)ODE solvers.

For ODEs, the order of convergence equals the number of stages.

For SDEs with additive noise (s>0), the order of convergence (both weak and strong) is 1 for stages equal to 1 or 4. These correspond to the classic Euler-Maruyama scheme and the Runge-Kutta scheme for S-ODEs respectively, see grudzien2020a for a DA-specific discussion on integration schemes and their discretization errors.

Parameters:

Name	Type	Description	Default
f	function	The time derivative of the dynamical system. Must be of the form f(t, x)	required
x	ndarray or float	State vector of the forcing term	required
t	float	Starting time of the integration	required
dt	float	Integration time step.	required
stages	int	The number of stages of the RK method. When stages=1, this becomes the Euler (-Maruyama) scheme. Default: 4.	4
s	float	The diffusion coeffient (std. dev) for models with additive noise. Default: 0, yielding deterministic integration.	0

Returns:

Type	Description
ndarray	State vector at the new time, t+dt

with_recursion(func, prog=False)

Make function recursive in its 1^st arg.

Return a version of func whose 2^nd argument (k) specifies the number of times to times apply func on its output.

!!! warning Only the first argument to func will change, so, for example, if func is step(x, t, dt), it will get fed the same t and dt at each iteration.

Parameters:

Name	Type	Description	Default
func	function	Function to recurse with.	required
prog	bool or str	Enable/Disable progressbar. If str, set its name to this.	False

Returns:

Name	Type	Description
fun_k	function	A function that returns the sequence generated by recursively running func, i.e. the trajectory of system's evolution.

Examples:

>>> def dxdt(x):
...     return -x
>>> step_1  = with_rk4(dxdt, autonom=True)
>>> step_k  = with_recursion(step_1)
>>> x0      = np.arange(3)
>>> x7      = step_k(x0, 7, t0=np.nan, dt=0.1)[-1]
>>> x7_true = x0 * np.exp(-0.7)
>>> np.allclose(x7, x7_true)
True

with_rk4(dxdt, autonom=False, stages=4, s=0)

Wrap dxdt in rk4.