Runge-Kutta adaptive-step solvers for nonlinear PDEs

# rkstiff

Runge-Kutta nonlinear-PDE solver for equations of the form ut = Lu + NL(u) with adaptive-step Runge-Kutta methods using either integrating factor (IF) or exponential time-differencing (ETD). Some examples of non-linear PDES that can be numerically solved using these methods are:

• Nonlinear Schrodinger equation (NLS)
• Kuramoto-Sivashinsky (KS)
• Korteweg-de Vries (KdV)
• Burgers
• Allen-Cahn
• Sine-Gordon

The adaptive step solver options provided in this package are

1. ETD35 (5th order ETD with 3rd orderembedding)
2. ETD34 (4th order ETD with 3rd order embedding)
3. IF34 (4th order IF with 3rd order embedding)
4. IF45DP (5th order IF with 4th order embedding)

In general, one should prefer ETD35 as it often has the best speed and stability for diagonal systems or diagonalized non-diagonal systems. Because the RK coefficients can be costly to compute IF34 can also be a good option in certain cases. A detailed discussion of these solvers is provided in the journal article Exponential time-differencing with embedded Rungeâ€“Kutta adaptive step control .

# Dependencies

Package requires

• numpy
• scipy
Tested with versions
• numpy = 1.19.2
• scipy = 1.6.0

# Usage

Each of the solvers is a python class (UPPERCASE) stored in a module of the same name (lowercase). Initializing each class requires two arguments, a linear operator L in the form of a numpy array, and a nonlinear function NL(u). The solvers can then be proagated either by using the solver.step function (user steps through time) or using the solver.evolve function (stepping handled internally). For example

from rkstiff import etd35
L = # some linear operator
def NL(u): #  nonlinear function defined here
solver = etd35.ETD35(linop=L,NLfunc=NL)
u0 = # initial field to be propagated
t0 =  # initial time
tf = # final time
uf = solver.evolve(u0,t0=t0,tf=tf)


By default, when using the function evolve, the field is stored at each step in a python list: u0,u1,...,uf are stored in solver.u. The corresponding times t0,t1,...,tf are stored in solver.t.

# Example

Consider the Kuramoto-Sivashinsky (KS) equation:
ut = -uxx - uxxxx - uux.

Converting to spectral space using a Fourier transform (F) we have
vt = kx2(1- kx2)v - F { F-1 {v} F-1{ i kx v} }
where v = F{u}. We can then plug L = kx2(1- kx2), and NL(u) = - F { F-1 {v} F-1{ i kx v} } into an rkstiff solver and propagate the field u in spectral space, converting back to real space when desired. For exampe, the python code may look something like this

import numpy as np
from rkstiff import grids
from rkstiff import if34

# uniform grid spacing, real-valued u -> construct_x_kx_rfft
N = 8192
a,b = 0,32*np.pi
x,kx = grids.construct_x_kx_rfft(N,a,b)

L = kx**2*(1-kx**2)
def NL(uFFT):
u = np.fft.irfft(uFFT)
ux = np.fft.irfft(1j*kx*uFFT)
return -np.fft.rfft(u*ux)

u0 = np.cos(x/16)*(1.+np.sin(x/16))
u0FFT = np.fft.rfft(u0)
solver = if34.IF34(linop=L,NLfunc=NL)
ufFFT = solver.evolve(u0FFT,t0=0,tf=50,store_freq=20) # store every 20th step in solver.u and solver.t
U = []
for uFFT in solver.u:
U.append(np.fft.irfft(uFFT))
U = np.array(U)
t = np.array(solver.t)


The grid module in rkstiff has several useful helper functions for setting up spatial and spectral grids. Here we used it to construct grids for a real-valued u utilizing the real-valued numpy Fourier transform (rfft). The results of the KS 'chaotic' propagation are shown below.

# Installation

From github download the rkstiff package. Once downloaded, one can install the package such that the modules can be imported outside the top level rkstiff folder. From the terminal the commands required are (assumes pip previously installed)

git clone https://github.com/whalenpt/rkstiff.git
cd rkstiff
python3 -m pip install numpy scipy
python3 -m pip install .


For use with Anaconda (see the Getting started with conda guide), from the terminal

git clone https://github.com/whalenpt/rkstiff.git
cd rkstiff
conda create --name rkstiff-env numpy scipy pip
conda activate rkstiff-env
pip install .


For use with a virtualenv the install looks like (see the Python Packaging Authority guide)

git clone https://github.com/whalenpt/rkstiff.git
cd rkstiff
python3 -m venv env
source env/bin/activate
python3 -m pip install numpy scipy wheel
python3 -m pip install .


The demos require installation of the python matplotlib and jupyter packages in addition to numpy and scipy.

# Contact

Patrick Whalen - whalenpt@gmail.com

## Project details

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