pathsim.solvers.gear module

pathsim.solvers.gear.compute_bdf_coefficients(order, timesteps)[source]

Computes the coefficients for backward differentiation formulas for a given order. The timesteps can be specified for variable timestep BDF methods.

For m-th order BDF we have for the n-th timestep:

sum(alpha_i * x_i; i=n-m,…,n) = h_n * f_n(x_n, t_n)

or

x_n = beta * h_n * f_n(x_n, t_n) - sum(alpha_j * x_{n-1-j}; j=0,…,order-1)

Parameters:

  • order (int) – order of the integration scheme

  • timesteps (list[float]) – timestep buffer (h_{n-j}; j=0,…,order-1)

Returns:

  • beta (float) – weight for function

  • alpha (array[float]) – weights for previous solutions

class pathsim.solvers.gear.GEAR(*solver_args, **solver_kwargs)[source]

Bases: ImplicitSolver

Base class for GEAR-type integrators that defines the universal methods.

Numerical integration method based on BDFs (linear multistep methods). Uses n-th order BDF for timestepping and (n-1)-th order BDF coefficients to estimate a lower ordersolutuin for error control.

The adaptive timestep BDF coefficients are dynamically computed at the beginning of each timestep from the buffered previous timsteps.

Notes

Not to be used directly!!!

reset()[source]

“Resets integration engine to initial state.

buffer(dt)[source]

Buffer the state and timestep. Dynamically precompute the variable timestep BDF coefficients on the fly for the current timestep.

Parameters:

dt (float) – integration timestep

revert()[source]

Revert integration engine to previous timestep, this is only relevant for adaptive methods where the simulation timestep ‘dt’ is rescaled and the engine step is recomputed with the smaller timestep.

error_controller(tr)[source]

Compute scaling factor for adaptive timestep based on absolute and relative tolerances for local truncation error.

Checks if the error tolerance is achieved and returns a success metric.

Parameters:

tr (array[float]) – truncation error estimate

Returns:

  • success (bool) – True if the timestep was successful

  • error (float) – estimated error of the internal error controller

  • scale (float) – estimated timestep rescale factor for error control

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

step(f, dt)[source]

Finalizes the timestep by resetting the solver for the implicit update equation and computing the lower order estimate of the solution for error control.

Parameters:

  • f (array_like) – evaluation of function

  • dt (float) – integration timestep

Returns:

  • success (bool) – True if the timestep was successful

  • error (float) – estimated error of the internal error controller

  • scale (float) – estimated timestep rescale factor for error control

class pathsim.solvers.gear.GEAR21(*solver_args, **solver_kwargs)[source]

Bases: GEAR

Adaptive GEAR integrator with 2-nd order BDF for timestepping and 1-st order BDF (euler backward) for truncation error estimation.

class pathsim.solvers.gear.GEAR32(*solver_args, **solver_kwargs)[source]

Bases: GEAR

Adaptive GEAR integrator with 3-rd order BDF for timestepping and 2-nd order BDF for truncation error estimation.

class pathsim.solvers.gear.GEAR43(*solver_args, **solver_kwargs)[source]

Bases: GEAR

Adaptive GEAR integrator with 4-th order BDF for timestepping and 3-rd order BDF for truncation error estimation.

class pathsim.solvers.gear.GEAR54(*solver_args, **solver_kwargs)[source]

Bases: GEAR

Adaptive GEAR integrator with 5-th order BDF for timestepping and 4-th order BDF for truncation error estimation.

class pathsim.solvers.gear.GEAR52A(*solver_args, **solver_kwargs)[source]

Bases: GEAR

Adaptive order adaptive stepsize GEAR integrator.

Adaptively selects the order (BDF coefficients) for timestepping between 2 and 5 depending on which method yields the lower truncation error. This balances the stability of the lower order methods with the accuracy of higher order methods.

Previous solutions and the variable timestep BDF coefficients are used to estimate a lower and a higher order solution from the solution of the timestepping method. This gives two separate estimates for the local tuncation error.

If the error is dominated by stability (lte of lower order method is lower), the order of the stepping method is decreased for the next timestep.

If the error is dominated by the accuracy of the method (higher order error is lower), the order of the stepping method is increased for the next timestep.

This means the integrator can take larger steps in regions where the solution is smooth using a higher order method and use more stable lower order methods in regions where the system exhibits stiffness or discontinuities.

buffer(dt)[source]

Buffer the state and timestep. Dynamically precompute the variable timestep BDF coefficients on the fly for the current timestep.

Parameters:

dt (float) – evaluation time

error_controller(tr_m, tr_p)[source]

Compute scaling factor for adaptive timestep based on absolute and relative tolerances of the local truncation error estimate obtained from esimated lower and higher order solution.

Checks if the error tolerance is achieved and returns a success metric.

Adapts the stepping order such that the normalized error is minimized and larger steps can be taken by the integrator.

Parameters:

  • tr_m (array[float]) – lower order truncation error estimate

  • tr_p (array[float]) – higher order truncation error estimate

Returns:

  • success (bool) – True if the timestep was successful

  • error (float) – estimated error of the internal error controller

  • scale (float) – estimated timestep rescale factor for error control

step(f, dt)[source]

Finalizes the timestep by resetting the solver for the implicit update equation and computing the lower and higher order estimate of the solution.

Then calls the error controller.

Parameters:

  • f (array_like) – evaluation of function

  • dt (float) – integration timestep

Returns:

  • success (bool) – True if the timestep was successful

  • error (float) – estimated error of the internal error controller

  • scale (float) – estimated timestep rescale factor for error control