pathsim.solvers.bdf module

class pathsim.solvers.bdf.BDF(*solver_args, **solver_kwargs)[source]

Bases: ImplicitSolver

Base class for the backward differentiation formula (BDF) integrators.

Notes

This solver class is not intended to be used directly

x_0

internal ‘working’ initial value

Type:

numeric, array[numeric]

x

internal ‘working’ state

Type:

numeric, array[numeric]

n

order of integration scheme

Type:

int

s

number of internal intermediate stages

Type:

int

stage

counter for current intermediate stage

Type:

int

eval_stages

rations for evaluation times of intermediate stages

Type:

list[float]

opt

optimizer instance to solve the implicit update equation

Type:

NewtonAnderson, Anderson, etc.

K

bdf coefficients for the state buffer for each order

Type:

dict[int: list[float]]

F

bdf coefficients for the function ‘func’ for each order

Type:

dict[int: float]

B

buffer for previous states

Type:

list[numeric], list[array[numeric]]

reset()[source]

“Resets integration engine to initial state.

buffer(dt)[source]

buffer the state for the multistep method

Parameters:

dt (float) – integration timestep

solve(f, J, dt)[source]

Solves the implicit update equation using the optimizer of the engine.

Parameters:

  • f (array_like) – evaluation of function

  • J (array_like) – evaluation of jacobian of function

  • dt (float) – integration timestep

Returns:

err – residual error of the fixed point update equation

Return type:

float

class pathsim.solvers.bdf.BDF2(*solver_args, **solver_kwargs)[source]

Bases: BDF

Fixed-step 2nd order Backward Differentiation Formula (BDF).

Implicit linear multistep method. Uses the previous two solution points. A-stable, suitable for stiff problems. Uses BDF1 for the first step.

Characteristics:

  • Order: 2

  • Implicit Multistep

  • Fixed timestep only

  • A-stable

class pathsim.solvers.bdf.BDF3(*solver_args, **solver_kwargs)[source]

Bases: BDF

Fixed-step 3rd order Backward Differentiation Formula (BDF).

Implicit linear multistep method. Uses the previous three solution points. A(alpha)-stable, suitable for stiff problems. Uses lower orders for startup.

Characteristics:

  • Order: 3

  • Implicit Multistep

  • Fixed timestep only

  • A(alpha)-stable (\(\alpha \approx 86^\circ\))

class pathsim.solvers.bdf.BDF4(*solver_args, **solver_kwargs)[source]

Bases: BDF

Fixed-step 4th order Backward Differentiation Formula (BDF).

Implicit linear multistep method. Uses the previous four solution points. A(alpha)-stable, suitable for stiff problems. Uses lower orders for startup.

Characteristics:

  • Order: 4

  • Implicit Multistep

  • Fixed timestep only

  • A(alpha)-stable (\(\alpha \approx 73^\circ\))

class pathsim.solvers.bdf.BDF5(*solver_args, **solver_kwargs)[source]

Bases: BDF

Fixed-step 5th order Backward Differentiation Formula (BDF).

Implicit linear multistep method. Uses the previous five solution points. A(alpha)-stable, suitable for stiff problems. Uses lower orders for startup.

Characteristics:

  • Order: 5

  • Implicit Multistep

  • Fixed timestep only

  • A(alpha)-stable (\(\alpha \approx 51^\circ\))

class pathsim.solvers.bdf.BDF6(*solver_args, **solver_kwargs)[source]

Bases: BDF

Fixed-step 6th order Backward Differentiation Formula (BDF).

Implicit linear multistep method. Uses the previous six solution points. Not A-stable, stability region does not contain the entire left half-plane, limiting its use for highly stiff problems compared to lower-order BDFs. Uses lower orders for startup.

Characteristics:

  • Order: 6

  • Implicit Multistep

  • Fixed timestep only

  • Not A-stable (stability angle approx \(18^\circ\))