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DG_Approximation.py 10.41 KiB
# -*- coding: utf-8 -*-
"""
@author: Laura C. Kühle
"""
import json
import numpy as np
from sympy import Symbol
import math
import seaborn as sns
from . import Troubled_Cell_Detector
from . import Initial_Condition
from . import Limiter
from . import Quadrature
from . import Update_Scheme
from .Basis_Function import OrthonormalLegendre
from .encoding_utils import encode_ndarray
from .projection_utils import Mesh
x = Symbol('x')
sns.set()
class DGScheme:
"""Class for Discontinuous Galerkin Method.
Approximates linear advection equation using Discontinuous Galerkin Method
with troubled-cell-based limiting.
Attributes
----------
basis : Basis object
Basis for calculation.
mesh : Mesh
Mesh for calculation.
Methods
-------
approximate()
Approximates projection.
save_plots()
Saves plots generated during approximation process.
build_training_data(x_shift, stencil_length, init_cond=None)
Builds training data set.
"""
def __init__(self, detector, **kwargs):
"""Initializes DGScheme.
Parameters
----------
detector : str
Name of troubled cell detector class.
Other Parameters
----------------
wave_speed : float, optional
Speed of wave in rightward direction. Default: 1.
polynomial_degree : int, optional
Polynomial degree. Default: 2.
cfl_number : float, optional
CFL number to ensure stability. Default: 0.2.
num_mesh_cells : int, optional
Number of cells in the mesh. Usually exponential of 2. Default: 64.
final_time : float, optional
Final time for which approximation is calculated. Default: 1.
left_bound : float, optional
Left boundary of interval. Default: -1.
right_bound : float, optional
Right boundary of interval. Default: 1.
verbose : bool, optional
Flag whether commentary in console is wanted. Default: False.
history_threshold : float, optional
Threshold when history will be recorded.
Default: math.ceil(0.2/cfl_number).
detector_config : dict, optional
Additional parameters for detector object. Default: {}.
init_cond : str, optional
Name of initial condition for evaluation. Default: 'Sine'
init_config : dict, optional
Additional parameters for initial condition object. Default: {}.
limiter : str, optional
Name of limiter for evaluation. Default: 'ModifiedMinMod'.
limiter_config : dict, optional
Additional parameters for limiter. object. Default: {}:
quadrature : str, optional
Name of quadrature for evaluation. Default: 'Gauss'.
quadrature_config : dict, optional
Additional parameters for quadrature object. Default: {}.
update_scheme : str, optional
Name of update scheme for evaluation. Default: 'SSPRK3'.
"""
# Unpack keyword arguments
self._wave_speed = kwargs.pop('wave_speed', 1)
self._cfl_number = kwargs.pop('cfl_number', 0.2)
self._final_time = kwargs.pop('final_time', 1)
self._verbose = kwargs.pop('verbose', False)
self._history_threshold = kwargs.pop('history_threshold',
math.ceil(0.2/self._cfl_number))
self._detector = detector
self._detector_config = kwargs.pop('detector_config', {})
self._init_cond = kwargs.pop('init_cond', 'Sine')
self._init_config = kwargs.pop('init_config', {})
self._limiter = kwargs.pop('limiter', 'ModifiedMinMod')
self._limiter_config = kwargs.pop('limiter_config', {})
self._quadrature = kwargs.pop('quadrature', 'Gauss')
self._quadrature_config = kwargs.pop('quadrature_config', {})
self._update_scheme = kwargs.pop('update_scheme', 'SSPRK3')
self._basis = OrthonormalLegendre(kwargs.pop('polynomial_degree', 2))
# Initialize mesh with two ghost cells on each side
self._mesh = Mesh(num_cells=kwargs.pop('num_mesh_cells', 64),
left_bound=kwargs.pop('left_bound', -1),
right_bound=kwargs.pop('right_bound', 1),
num_ghost_cells=2)
# print(len(self._mesh.cells))
# print(type(self._mesh.cells))
# Throw an error if there are extra keyword arguments
if len(kwargs) > 0:
extra = ', '.join('"%s"' % k for k in list(kwargs.keys()))
raise ValueError('Unrecognized arguments: %s' % extra)
# Make sure all classes actually exist
if not hasattr(Troubled_Cell_Detector, self._detector):
raise ValueError('Invalid detector: "%s"' % self._detector)
if not hasattr(Initial_Condition, self._init_cond):
raise ValueError('Invalid initial condition: "%s"'
% self._init_cond)
if not hasattr(Limiter, self._limiter):
raise ValueError('Invalid limiter: "%s"' % self._limiter)
if not hasattr(Quadrature, self._quadrature):
raise ValueError('Invalid quadrature: "%s"' % self._quadrature)
if not hasattr(Update_Scheme, self._update_scheme):
raise ValueError('Invalid update scheme: "%s"'
% self._update_scheme)
self._reset()
# Replace the string names with the actual class instances
# (and add the instance variables for the quadrature)
self._init_cond = getattr(Initial_Condition, self._init_cond)(
config=self._init_config)
self._limiter = getattr(Limiter, self._limiter)(
config=self._limiter_config)
self._quadrature = getattr(Quadrature, self._quadrature)(
config=self._quadrature_config)
self._detector = getattr(Troubled_Cell_Detector, self._detector)(
config=self._detector_config, mesh=self._mesh, basis=self._basis)
self._update_scheme = getattr(Update_Scheme, self._update_scheme)(
polynomial_degree=self._basis.polynomial_degree,
num_cells=self._mesh.num_cells, detector=self._detector,
limiter=self._limiter, wave_speed=self._wave_speed)
def approximate(self, data_file):
"""Approximates projection.
Initializes projection and evolves it in time. Each time step consists
of three parts: A projection update, a troubled-cell detection,
and limiting based on the detected cells.
At final time, results are saved in JSON file.
Attributes
----------
data_file: str
Path to file in which data will be saved.
"""
projection = do_initial_projection(
init_cond=self._init_cond, mesh=self._mesh,
basis=self._basis, quadrature=self._quadrature)
time_step = abs(self._cfl_number * self._mesh.cell_len /
self._wave_speed)
current_time = 0
iteration = 0
troubled_cell_history = []
time_history = []
while current_time < self._final_time:
# Adjust for last cell
cfl_number = self._cfl_number
if current_time+time_step > self._final_time:
time_step = self._final_time-current_time
cfl_number = self._wave_speed * time_step / self._mesh.cell_len
# Update projection
projection, troubled_cells = self._update_scheme.step(projection,
cfl_number)
iteration += 1
if (iteration % self._history_threshold) == 0:
troubled_cell_history.append(troubled_cells)
time_history.append(current_time)
current_time += time_step
# Save detector-specific data in dictionary
approx_stats = self._detector.create_data_dict(projection)
# Save approximation results in dictionary
approx_stats['wave_speed'] = self._wave_speed
approx_stats['final_time'] = self._final_time
approx_stats['time_history'] = time_history
approx_stats['troubled_cell_history'] = troubled_cell_history
# Encode all ndarrays to fit JSON format
approx_stats = {key: encode_ndarray(approx_stats[key])
for key in approx_stats.keys()}
# Save approximation results in JSON format
with open(data_file + '.json', 'w') \
as json_file:
json_file.write(json.dumps(approx_stats))
def _reset(self):
"""Resets instance variables."""
# Set additional necessary config parameters
self._limiter_config['cell_len'] = self._mesh.cell_len
def do_initial_projection(init_cond, mesh, basis, quadrature,
x_shift=0):
"""Calculates initial projection.
Calculates a projection at time step 0 and adds ghost cells on both
sides of the array.
Parameters
----------
init_cond : InitialCondition object
Initial condition used for calculation.
mesh : Mesh
Mesh for calculation.
basis: Basis object
Basis used for calculation.
quadrature: Quadrature object
Quadrature used for evaluation.
x_shift: float, optional
Shift in x-direction for each evaluation point. Default: 0.
Returns
-------
ndarray
Matrix containing projection of size (N+2, p+1) with N being the
number of mesh cells and p being the polynomial degree of the basis.
"""
# Initialize matrix and set first entry to accommodate for ghost cell
output_matrix = [0]
for eval_point in mesh.non_ghost_cells:
new_row = []
for degree in range(basis.polynomial_degree + 1):
new_row.append(np.float64(sum(init_cond.calculate(
x=eval_point + mesh.cell_len/2
* quadrature.nodes[point] - x_shift, mesh=mesh)
* basis.basis[degree].subs(
x, quadrature.nodes[point])
* quadrature.weights[point]
for point in range(quadrature.num_nodes))))
new_row = np.array(new_row)
output_matrix.append(basis.inverse_mass_matrix @ new_row)
# Set ghost cells to respective value
output_matrix[0] = output_matrix[mesh.num_cells]
output_matrix.append(output_matrix[1])
# print(np.array(output_matrix).shape)
return np.transpose(np.array(output_matrix))