"""Timestepping routines for low-storage variants of explicit Runge-Kutta methods.
.. autofunction:: euler_step
.. autofunction:: lsrk54_step
.. autofunction:: lsrk144_step
"""
__copyright__ = """
Copyright (C) 2020 University of Illinois Board of Trustees
"""
__license__ = """
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
"""
from dataclasses import dataclass
import numpy as np
@dataclass(frozen=True)
class LSRKCoefficients:
"""Dataclass which defines a given low-storage Runge-Kutta (LSRK) scheme.
The methods are determined by the provided `A`, `B` and `C` coefficient arrays.
"""
A: np.ndarray
B: np.ndarray
C: np.ndarray
def lsrk_step(coefs, state, t, dt, rhs):
"""Take one step using a low-storage Runge-Kutta method."""
k = 0.0 * state
for i in range(len(coefs.A)):
k = coefs.A[i]*k + dt*rhs(t + coefs.C[i]*dt, state)
state = state + coefs.B[i]*k
return state
EulerCoefs = LSRKCoefficients(
A=np.array([0.]),
B=np.array([1.]),
C=np.array([0.]))
[docs]
def euler_step(state, t, dt, rhs):
"""Take one step using the explicit, 1st-order accurate, Euler method."""
return lsrk_step(EulerCoefs, state, t, dt, rhs)
LSRK54CarpenterKennedyCoefs = LSRKCoefficients(
A=np.array([
0.,
-567301805773/1357537059087,
-2404267990393/2016746695238,
-3550918686646/2091501179385,
-1275806237668/842570457699]),
B=np.array([
1432997174477/9575080441755,
5161836677717/13612068292357,
1720146321549/2090206949498,
3134564353537/4481467310338,
2277821191437/14882151754819]),
C=np.array([
0.,
1432997174477/9575080441755,
2526269341429/6820363962896,
2006345519317/3224310063776,
2802321613138/2924317926251]))
[docs]
def lsrk54_step(state, t, dt, rhs):
"""Take one step using an explicit 5-stage, 4th-order, LSRK method.
Coefficients are summarized in [Hesthaven_2008]_, Section 3.4.
"""
return lsrk_step(LSRK54CarpenterKennedyCoefs, state, t, dt, rhs)
LSRK144NiegemannDiehlBuschCoefs = LSRKCoefficients(
A=np.array([
0.,
-0.7188012108672410,
-0.7785331173421570,
-0.0053282796654044,
-0.8552979934029281,
-3.9564138245774565,
-1.5780575380587385,
-2.0837094552574054,
-0.7483334182761610,
-0.7032861106563359,
0.0013917096117681,
-0.0932075369637460,
-0.9514200470875948,
-7.1151571693922548]),
B=np.array([
0.0367762454319673,
0.3136296607553959,
0.1531848691869027,
0.0030097086818182,
0.3326293790646110,
0.2440251405350864,
0.3718879239592277,
0.6204126221582444,
0.1524043173028741,
0.0760894927419266,
0.0077604214040978,
0.0024647284755382,
0.0780348340049386,
5.5059777270269628]),
C=np.array([
0.,
0.0367762454319673,
0.1249685262725025,
0.2446177702277698,
0.2476149531070420,
0.2969311120382472,
0.3978149645802642,
0.5270854589440328,
0.6981269994175695,
0.8190890835352128,
0.8527059887098624,
0.8604711817462826,
0.8627060376969976,
0.8734213127600976]))
[docs]
def lsrk144_step(state, t, dt, rhs):
"""Take one step using an explicit 14-stage, 4th-order, LSRK method.
This method is derived by Niegemann, Diehl, and Busch (2012), with
an optimal stability region for advection-dominated flows. The
LSRK coefficients are summarized in [Niegemann_2012]_, Table 3.
"""
return lsrk_step(LSRK144NiegemannDiehlBuschCoefs, state, t, dt, rhs)