Lighthill-Whitham-Richards
The Lighthill-Whitham-Richards model, the conservation of cars with the Greenshields speed law.
- Name:
lighthill_whitham_richards - Field:
engineering/traffic - Formalism: ordinary differential equations
- References: Lighthill and Whitham 1955; Richards 1956; Chowdhury, Santen, and Schadschneider 2000
Factory
@register_model(
field='engineering/traffic',
tags=('traffic', 'field', 'conservation law', 'finite volume', 'shocks', 'pde'),
references=('Lighthill and Whitham 1955', 'Richards 1956', 'Chowdhury, Santen, and Schadschneider 2000'),
defaults={'until': 1.0, 'dt': 0.05, 'solver': SSPRKSolver()},
)
def lighthill_whitham_richards(
n: int = 200,
*,
upstream: float = 0.3,
downstream: float = 0.8,
vmax: float = 1.0,
rho_max: float = 1.0,
) -> System:
"""Return the Lighthill-Whitham-Richards model, the conservation of cars with the Greenshields speed law.
The density follows `rho_t + (rho V(rho))_x = 0` with `V(rho) = vmax (1 - rho / rho_max)` on a road of length 1
with open edges. Free traffic at density `upstream` runs into a queue at density `downstream` in the second half of
the road, and the shock between them moves at `vmax (1 - (upstream + downstream) / rho_max)`.
Args:
n:
The number of cells.
upstream:
The density in the first half of the road.
downstream:
The density in the second half of the road.
vmax:
The speed on an empty road.
rho_max:
The jam density.
Returns:
system:
The model, with the density `road.rho`.
Examples:
>>> from skcomplex.models.engineering.traffic import lighthill_whitham_richards
>>> float(lighthill_whitham_richards(10).elements[0].values['rho'][-1])
0.8
"""
density = np.where(np.arange(n) < n // 2, upstream, downstream).astype(float)
values = {'rho': density, 'vmax': vmax, 'rho_max': rho_max}
space = LatticeSpace(n, spacing=1 / n)
road = FieldElement(
'road',
element=LighthillWhithamRichardsElement,
space=space,
boundary={'rho': 'zero-flux'},
values=values,
discretization='finite-volume',
)
flux = _get_lighthill_whitham_richards_flux
density = ConservationInteraction('density', target='road', state='rho', flux=flux, riemann='godunov')
return System(elements=[road], interactions=[density])
Components
class LighthillWhithamRichardsElement(Element):
"""Cell of a road with the traffic density `rho`, the free speed `vmax`, and the jam density `rho_max`."""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
rho: Annotated[object, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.3,
vmax: Annotated[object, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
rho_max: Annotated[object, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
) -> None:
super().__init__(label, visual=visual)
self.rho = rho
self.vmax = vmax
self.rho_max = rho_max
class ConservationInteraction(Interaction):
"""Conservation law `du/dt + divergence(flux(ns)) = source(ns)` of a state `u` of a field.
The flux is one expression for a field of one dimension, or one expression per dimension, of the states of the
field, and the interaction lowers it to a finite-volume flux law with a slope limiter and a numerical flux, or,
without a numerical flux, to the rate `-divergence(flux)` that the field discretizes by its own operators, as a
spectral or finite-difference field does. Without a source the total of the state on a periodic field stays
constant. The conservation laws of several states of one field form one system, whose numerical flux reads the
wave speeds of all of them, and a flux may read parameters that vary across the cells. A flux or a source function
takes the namespace of the interaction, with the namespace `target` of the field, the time `t`, and the symbols of
the parameters a subclass declares.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the field whose state is conserved.
state:
The name of the conserved state of the field.
flux:
A function of the namespace that returns the flux, or one flux per dimension.
source:
A function of the namespace that returns the source, or `None` for none.
limiter:
The slope limiter of the reconstruction, `'minmod'`, `'van-leer'`, or `None` for first order.
riemann:
The numerical flux of a finite-volume field, `'rusanov'` or `'godunov'`, the exact solver for one state, or
`None` for the divergence of the flux on a field of another discretization.
sources:
The paths of other elements the flux and the source read, as the namespaces `sources`. A field over the
position lattice of a target on a phase space is read broadcast along the class axes.
Raises:
ValidationError:
The state is not a name, a function is not callable, or the limiter or the numerical flux is unknown.
Examples:
>>> from typing import Annotated
>>> import numpy as np
>>> from skcomplex.core.elements import Element, FieldElement
>>> from skcomplex.core.interactions import ConservationInteraction
>>> from skcomplex.core.spaces import LatticeSpace
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> from skcomplex.specs import StateSpec
>>> class CellElement(Element):
... def __init__(self, label, u: Annotated[float, StateSpec()] = 0.0):
... super().__init__(label)
... self.u = u
>>> space = LatticeSpace((50,), spacing=0.1, boundary='periodic')
>>> values = {'u': 1.0 + np.sin(2 * np.pi * np.arange(50) / 50)}
>>> volumes = 'finite-volume'
>>> field = FieldElement('wave', element=CellElement, space=space, values=values, discretization=volumes)
>>> burgers = ConservationInteraction('burgers', target='wave', state='u', flux=lambda s: s.target.u**2 / 2)
>>> system = System(elements=[field], interactions=[burgers])
>>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=0.5).trajectory_
>>> round(float(trajectory.values('wave.u')[-1].sum()), 6)
50.0
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str = 'u',
flux: Callable[[SimpleNamespace], object] | None = None,
source: Callable[[SimpleNamespace], object] | None = None,
limiter: str | None = 'minmod',
riemann: str | None = 'rusanov',
sources: Annotated[Sequence[str], RoleSpec(variadic=True)] = (),
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.flux = flux
self.source = source
self.limiter = limiter
self.riemann = riemann
self.sources = sources
def _check(self) -> None:
"""Raise an error for a state that is not a name, a function that is not callable, or an unknown scheme.
Raises:
ValidationError:
The type declares states, the state is not a name, a function is not callable, or the limiter or the
numerical flux is unknown.
"""
super()._check()
if not isinstance(self.state, str) or not self.state.isidentifier():
error_msg = f'{type(self).__name__} {self.label!r} conserves the state {self.state!r}, which is no name.'
raise ValidationError(error_msg)
for name in ('flux', 'source'):
value = getattr(self, name)
if value is not None and not callable(value):
error_msg = f'The {name} of {self.label!r} is {value!r}. Give a function of the namespace.'
raise ValidationError(error_msg)
if self.limiter not in LIMITERS:
error_msg = f'Unknown limiter {self.limiter!r} of {self.label!r}. Use minmod, van-leer, or None.'
raise ValidationError(error_msg)
if self.riemann not in RIEMANN_SOLVERS:
error_msg = f'Unknown numerical flux {self.riemann!r} of {self.label!r}. Use rusanov, godunov, or None.'
raise ValidationError(error_msg)
def _get_flux(self, namespace: SimpleNamespace) -> object:
"""Return the flux, or `None` for none, the hook a subclass overrides to derive it from its arguments."""
return None if self.flux is None else self.flux(namespace)
def _get_transport_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the laws of the divergence of the flux, the hook a subclass overrides to lower its own flux."""
flux = self._get_flux(namespace)
if flux is None:
return []
state = getattr(self._get_member(namespace, 'target'), self.state)
if self.riemann is None:
components = list(flux) if isinstance(flux, FreeModuleElement | list | tuple) else [flux]
return [RateLaw(state, -divergence(components))]
return [FluxLaw(state, flux, limiter=self.limiter, riemann=self.riemann)]
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the laws of the divergence of the flux and the rate of the source.
Args:
namespace:
The namespace of the symbols the laws can use, with the namespace of the field.
Returns:
laws:
The flux law, and the rate law of the source when there is one.
"""
laws = self._get_transport_laws(namespace)
if self.source is not None:
laws.append(RateLaw(getattr(self._get_member(namespace, 'target'), self.state), self.source(namespace)))
return laws
Equations
road.rho[i]: diff(rho_road(t), t) == 0