Runge-Kutta adaptive-step and constant-step solvers for nonlinear PDEs

## Project description

# rkstiff

Runge-Kutta integrating factor (IF) and exponential time-differencing (ETD) methods
for solving nonlinear-PDE's of the form `u`

.
Some examples of non-linear PDES that can be numerically solved using these methods are:_{t} = Lu + NL(u)

- 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

- ETD35 (5
^{th}order ETD with 3^{rd}orderembedding) - ETD34 (4
^{th}order ETD with 3^{rd}order embedding) - IF34 (4
^{th}order IF with 3^{rd}order embedding) - IF45DP (5
^{th}order IF with 4^{th}order embedding)

The constant step solver options provided are

- ETD4 (4
^{th}order ETD - Krogstad method) - ETD5 (5
^{th}order ETD - same as the 5th order method in ETD35) - IF4 (4
^{th}order IF - same as the 4th order method in IF34)

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 or constant step methods may be preferable in certain settings. A detailed discussion of these solvers is provided in the journal article Exponential time-differencing with embedded Runge–Kutta adaptive step control .

# Dependencies

Package requires

- numpy
- scipy

- 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:

u_{t} = -u_{xx} - u_{xxxx} - uu_{x}.

Converting to spectral space using a Fourier transform (F) we have

v_{t} = k_{x}^{2}(1- k_{x}^{2})v - F { F^{-1} {v} F^{-1}{ i k_{x} v} }

where v = F{u}. We can then plug L = k_{x}^{2}(1- k_{x}^{2}), and NL(u) = - F { F^{-1} {v} F^{-1}{ i k_{x} 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 the github source

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

PyPI install with a virtualenv (see the Python Packaging Authority guide)

```
python3 -m venv env
source env/bin/activate
python3 -m pip install rkstiff
```

For use with Anaconda using the conda-forge channel (see the Getting started with conda guide), from the terminal

conda create --name rkstiff-env conda activate rkstiff-env conda install rkstiff -c conda-forge

The demos require installation of the python `matplotlib`

and `jupyter`

packages in addition to `numpy`

and `scipy`

. The tests require installation of the python package `pytest`

.

# License

This project is licensed under the MIT License - see the LICENSE file for details.

# Citation

@article{WHALEN2015579, title = {Exponential time-differencing with embedded Runge–Kutta adaptive step control}, journal = {Journal of Computational Physics}, volume = {280}, pages = {579-601}, year = {2015}, author = {P. Whalen and M. Brio and J.V. Moloney} }

# Contact

Patrick Whalen - whalenpt@gmail.com

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