FiPy: Partial Differential Equations with Python
of a Python-based PDE solver in these pages.12 FiPy has proven to be a powerful teaching tool as ... have worked
Solving Partial Differential Equations with Python - Tentative
FiPY ( FiPy: A Finite Volume PDE Solver Using Python) is an open source python program that solves numerically partial differential equations. ?. It uses the
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26 ago 2020 Keywords: Partial differential equation Mathematics
A Finite Volume PDE Solver Using Python
FiPy is a computer program written in Python to solve partial differential equations (PDEs) using the Finite Volume method. Python is a powerful object
FiPy: Partial Differential Equations with Python
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py-pde: A Python package for solving partial differential equations
3 apr 2020 The py-pde python package presented in this paper allows researchers to quickly and conve- niently simulate and analyze PDEs of the general ...
Coupling kinetic models and advection–diffusion equations. 1
KEYWORDS: Advection–diffusion–reaction; partial differential equations; phloem flow; simulation; sugarcane. FiPy is a PDE solver written in Python using.
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6 giorni fa FiPy: A Finite Volume PDE Solver Using. Python. FiPy is an object oriented partial differential equation (PDE) solver
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https://arxiv.org/pdf/1111.6583
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(PDF) FiPy: Partial Differential Equations with Python - ResearchGate
9 déc 2019 · PDF Partial differential equations (PDEs) are ubiquitous to the mathematical description of physical phenomena
[PDF] FiPy: Partial Differential Equations with Python - ResearchGate
Many existing partial differential equation solver packages focus on the important but arcane task of numerically solving the linearized set of algebraic
[PDF] Solving Partial Differential Equations with Python - Fjordforsk AS
FiPY ( FiPy: A Finite Volume PDE Solver Using Python) is an open source python program that solves numerically partial differential equations
FiPy: Partial Differential Equations with Python - Academiaedu
We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials
FiPy: Partial Differential Equations with Python - IEEE Xplore
17 avr 2009 · Many existing partial differential equation solver packages focus on the important but arcane task of numerically solving the linearized
FiPy: A Finite Volume PDE Solver Using Python
15 jui 2022 · FiPy is an object oriented partial differential equation (PDE) solver written in Python based on a standard finite volume (FV) approach
[PDF] A Finite Volume PDE Solver Using Python - math NIST
FiPy is a computer program written in Python to solve partial differential equations (PDEs) using the Finite Volume method Python is a powerful object
FiPy: Partial Differential Equations with Python - Semantic Scholar
1 mai 2009 · py-pde: A Python package for solving partial differential equations J Open Source Softw 2020 Partial differential equations (PDEs) play a
[PDF] Using Python to Solve Partial Differential Equations
This article describes two Python modules for solving partial differential equations (PDEs): PyCC is designed as a Matlab-like environment for writing
A short course on solving partial differential equations in Python
Partial differential equations (PDEs) are of significant interest to scientists FiPy and other PDE solver codes in Python do well in all four aspects
Can Python solve PDEs?
py-pde is a Python package for solving partial differential equations (PDEs). The package provides classes for grids on which scalar and tensor fields can be defined. The associated differential operators are computed using a numba-compiled implementation of finite differences.What is FiPy?
FiPy is an object oriented, partial differential equation (PDE) solver, written in Python, based on a standard finite volume (FV) approach.15 jui. 2022How do you plot a partial differential equation in Python?
First import numpy and matplotlib. Create a numpy linspace array for the X values and a np. zeros with the exact values as X for the U function. Declare the initial condition and boundary conditions.- Differential equations are solved in Python with the Scipy. integrate package using function odeint or solve_ivp. t: Time points at which the solution should be reported. Additional internal points are often calculated to maintain accuracy of the solution but are not reported.
Solving Partial Differential Equations
with Python - Tentative application to Rogue WavesSergio Manzetti1,2
1. Institute for Cellular and Molecular Biology, Uppsala University, Uppsala, Sweden.
2. Fjordforsk AS, Midtun, 6894 Vangsnes, Norway.
FipY FiPY ( FiPy: A Finite Volume PDE Solver Using Python) is an open source python program that solves numerically partial differential equations. It uses the solvers PySparse, SciPy, PyAMG, Trilinos and mpi4py. Visualization is done using Matplotlib and Mayavi FipY can solve in parallel mode, reproduce the numerical in graphical viewers, and include boundary conditions, initialconditions and solve higher order PDEs (i.e. d /dx by ⁴ ⁴splitting them into two second order PDEs).
FiPY is also incorporated with GIT and can be run onWindows, Linux or other conventional OS.
FiPY: https://www.ctcms.nist.gov/fipy/
Topic of research
Rogue waves are rare phenomena occurring in optic
fibers, in the atmosphere and most importantly in the oceans. Rogue waves can be represented using the Non-linear The analytical solutions to the NLSE are conjugates of polynomials with a complex exponential term1. Many recent papers have solved both inhomogenous and homogenous variants of the NLSE, both analytically and numerically.1. Manzetti S. (2017). Mathematical modelling Rogue Waves. Subm. Comp. And Appl. Mathematics.
Figure. Top: Rogue wave recorded at the North Alwyn field, 500km east of Shetland islands, reaching 3X of the surrounding waves. . Bottom: The Draupner Wave
recorded on 31 Dec 1995 on the Norwegian Draupner Platform, reaching 27 meters.FiPY approach
FiPy is called in a Python2.7 script using the command "from fipy import *"The interest is to solve first a simple equation:
which describes diffusion of some particles along the x- axis and tentatively try to work towards the NLSE:∂ϕ ∂t-∂2ϕ ∂x2=0Python script structure - Part 1First we need to define a domain (with solution points) for the PDE. This requires definition of an object, Grid1D, which represents the linear structure grid.
The points of the domain, which is repsented by a mesh are defined as: >>> mesh = Grid1D(nx=nx, dx=dx) >>> mesh=mesh, ... value=0.)The points nx
>>> nx = 50 and the grid parameter dx is to to 1: >>> dx = 1 Second, we need to define the initial conditions of the PDE, which are normally given as: >>> valueLeft = 1 >>> valueRight = 0Because the boundary conditions are represented as faces around the exterior of the mesh, we define the constraint on phi by using the names for the left
and right side of the boundary using the phi.constrain() command which calls the values given above: >>> phi.constrain(valueLeft, mesh.facesRight) >>> phi.constrain(valueRight, mesh.facesLeft)We then define the time-steps for the numerical computation, which is defined by 90 percent of the maximum stable timestep, which is given
>>> timeStepDuration = 0.9 * dx**2 / (2 * D) >>> steps = 2000And then give the value of D.
>>> D = 1Python script structure - Part 2
The equation contains a transient term, d/dt, and the diffusion term, d2/dx2.In Fipy, this equation is given by:
>>> eqX = TransientTerm() - ExplicitDiffusionTerm(coeff=D) == 0Then we define the variables of the equation:
>>> x = mesh.cellCenters[0] >>> t = timeStepDuration * steps We then want to view the results and compare the results of the simulation with an analytical solution (if available): >>> phiAnalytical = CellVariable(name="analytical value", ... mesh=mesh) ... if __name__ == '__main__': >>> viewer = Viewer(vars=(phi, phiAnalytical), ... datamin=0., datamax=1.) ... viewer.plot()∂ϕ ∂t-∂2ϕ ∂x2=0Python script structure - Part 3
We then define the analytical solution, in this case, an example:This appears in the script like:
>>> try: from scipy.special import erf phiAnalytical.setValue(1-erf(x / (2 * numerix.sqrt(D * t)))) >>> except ImportError:print ("The Scipy Library is not available to test the solution to the given transient diffusion equation")
The equation is then solved by using eqX.solve repeatedly looping >>> for step in range(steps): eqX.solve(var=phi, dt=timeStepDuration) >>> if __name__ == '__main__': viewer.plot() >>> print (phi.allclose(phiAnalytical, atol = 7e-4) 1 >>> if __name__ == '__main__': raw_input("Explicit transient diffusion. PressApplication to a variant of the NLSE
We try then this structure on one candidate of the NLSE with real coefficients:Which is in FiPY:
>>> eq = TransientTerm(coeff=1., var=phi) + ExponentialConvectionTerm(coeff=(1.,), var=phi) + abs(phi_sq)*(phi) == 0.0 In this test, we simulate the NLSE using a complex exponential function (eix) as a tentative form of phi: >>> phiAnalytical = CellVariable(name="Analytical value", mesh=mesh) >>> phiAnalytical.setValue(numerix.exp(1j*(X)))) Complete python script for the non-complex variant of the NLSE #!/usr/bin/env python # testing a non-complex variant of the NLSE import numpy import cmath as math from fipy import * from fipy import numerix nx = 50 dx = 1. / float(nx) mesh = Grid1D(nx=nx,dx=dx)X = mesh.cellCenters[0]
phi = CellVariable(mesh=mesh, name="Solution") vi = Viewer(vars=phi,datamin=0.0, datamax=1.0) vi.plot() raw_input("Initialization ...") phi.constrain(1., mesh.facesLeft) phi.constrain(0., mesh.facesRight) phi_sq = CellVariable(mesh=mesh) phi_sq.setValue( phi*phi )#We now represent the equation, where u = phi, du/dt + d^2u/dx^2 + |u|^2*u = 0 in fipy commands. du/dx : is a convection term with a unit scalar coefficient, i.e.
# du/dt : is a transient term that one can treat as before, i.e. TransientTerm(var=u). one can add a second order derivative as ExplicitDiffusionTerm(coeff=D)
eq = TransientTerm(coeff=1., var=phi) + ExponentialConvectionTerm(coeff=(1.,), var=phi) + abs((phi_sq))*(phi) == 0.0
dt = 0.01 steps = 100 for step in range(steps): eq.sweep(dt=dt) phi_sq.setValue( phi * phi ) vi.plot() # We then add a trial analytical solution to test for validity in the given PDE above, namely e^ix phiAnalytical = CellVariable(name="Analytical value", mesh=mesh) vi = Viewer(vars=(phi, phiAnalytical)) vi.plot() raw_input("PressOutput from FiPY python script
Fig1.The analytical function eix shows a similar trend as to the numerical solution of the non-complex NLSE shown in the previous slides.
InterpretationThe plot on the previous slide shows that eix is not a bad solution to a one dimensional
case of the given PDE. However, based on the analytical solutions reviewed1 we know that a good solution looks like: where G, H and D are polynomials given by Akhmediev et al 2.The rogue wave plot from this solution
in three dimensions appears as 1:1. Manzetti S. (2017). Mathematical modelling of Rogue Waves. Subm. Comp and App. Mathematics.
Review E 80.2 (2009): 026601.
Conclusions
FiPY has been used to make a test numerical
simulation of a transient/diffusion problem in Python 2.7. Fipy was then tested on a non-complex variant of the NLSE. The result is not applicable to Rogue phenomena, and requires additional development of the FiPY script into a treatment of the complex NLSE.FiPY was successfully implemented as preliminary
testing method for numerically solving PDEs as a basis for further research on the topic of Rogue Waves.Mange takk
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