EasyStab

This document “easystab”, is a door open for you to study the stability and bifurcation of physical systems using octave/matlab, mainly in the domain of fluid mechanics. It may help you for instance to answer the questions:

• What is the velocity of a water wave?
• When there is a heavy fluid on top af a light fluid with a horizontal interface, what will be the most growing wavelength?
• What is the height of a capilary meniscus?
• When does a pendent drop fall?

To find the solution of nonlinear equations we use the Newton iteration, for the stability we compute eigenmodes of matrices. Except on a few instances, it is not presented as a tool for simulation of these systems (time marching). When we need to do time marching, we will use Gerris or Basilisk.

To get a better idea of the technical choices and notations in Easystab, please read pedagogy.

It was initiated by Jerome Hoepffner at Université Pierre et Marie Curie to help sharing his computational codes and know-how. It is thought to become a collaborative tool where you can profit and contribute. The project is developed for research, education and popularization purposes. As for education the project has served as the support for two classes:

All of the codes are based on the idea of the “differentiation matrix”, used as the building block to describe many linear and nonlinear equations.

Most of the pages you will visit on this wiki are effective octave/matlab scripts that you can run for yourself. The presentation follows the idea of Donald Knuth of the “Litterate programming”: the comments included in the code are not mere indications on the functonalities of the code, there are a detailed description of the both the physical theory behind the equations and the description of the functionality of the code. This is why the comments are long. The wiki tool that displays these pages have the hability to properly render the comments for readability. This document is not a theoretical essay in which some codes are included, it is a library of codes in which the theory is described. At the end what matters is to keep together the effective codes and the ideas behind them.

The standard

Numerical results are never given bare: - they are compared to formulas that can be obtained by hand in simplified configurations, - they are compared to results published in the litterature, - different models are compared to each-other in the asymptotical limits where they should be equivalent. - The results are compared to numerical simulations. This continuity between all levels of representations (models) of a same phenomenon is the core of easystab.

• Litterate programming: we should not separate coding from litterature. Books have too much theory, codes have too much technicality.
• Learn by doing: you find here many examples that you can copy and paste and modify, and understand them progressivelly as you see them succeed and fail.
• Domain specific language: inventing a syntax for coding that is built upon the way you think and talk in a given scientific community, thus most of the conceptual steps are already done. Here everything is built upon linear algebra.
• Collaboration and open-source: by having common tools, we can save a lot of technical work spent by PhD students and Post-docs. Easystab is a wiki.
• Education: the production of students is typically graded then trashed. We should instead invite them to contribute.
• Reproducibilty: if most of the technical details of papers are embeded in common collaborative tools like Gerris, Basilisk, and esaystab, you can include in your papers the files that will allow the reader to reproduce your results. This was not possible when you used a wind-tunnel. It is now possible.

How can I learn all this?

To get familiar with easystab, the first thing to do is to read and run and modify diffmat.m where you see how to build differentiation matrices (using the simplest finite-difference method) and how to use them to compute derivatives. Then you can continue with differential_equation.m where we use the differentiation matrices to write and solve differential equations in 1D. This is also the place where you will see how to impose boundary conditions. To get yet more ease you can continue with vibrating_string.m where we do the marching in time of a vibrating string. Once you are familiar with differenciation matrices optained using the simplest finite difference method, you may try other more powerful methods such as the collocation Checyshev method. See an example here : differential_equation_fd_cheb.m.

Then it’s probably time to read the description of how things are done in pedagogy.

Then you will be ripe to read/run about the assumption of wavelike behaviour which we use almost all the time to transform 2D or 3D flow problems into 1D problems. Ths is done in wave_like.m. To practice this, have then a look at the computation of the stability of the Poiseuille flow (flat channel flow) in poiseuille_uvp.m and the other codes for stability in 1D.

If you want to continue, you can learn about how to solve nonlinear 1D differential equations for instance in meniscus.m, and then nonlinear 2D differential equations like for instance in venturi.m.

The codes

Before all, you should get the basic functions, they are in easypack.zip

Here are all the file and the folders for navigation: /sandbox/easystab/

The differentiation matrix is at the heart of all these codes. We sometimes build them based on finite differences, but most of the time we use the ones provided in the Matlab Differentiation Matrix Suite by Weideman and Reddy. The link to their paper on their web page is broken so I put it here: matlabdifmatsuite.pdf.

Computing and using the derivatives

Here are the pedagogical introductions to the methods. In pedagogy, you will find all the ideas and methods described.

In 1D:

• diffmat.m Showing and testing the differentiation matrices for a 1D mesh.
• diffmat_dif1D.m The same thing as diffmat.m but using the function dif1D.m to build the differentiation matrices, like we do in most of the other codes.
• differential_equation.m Showing how to solve 1D linear differential equation with Dirichlet and Neumann boundary conditions, homogeneous and nonhomogeneous.
• differential_equation_fd_cheb.m Showing how to solve a linear problem using Chebyshev method instead of finite-difference method, and comparing the performances of both methods.
• variable_coef.m Solving a differential equation with variable coefficients.
• integration.m Showing the way we do the integration.
• differential_equation_infinitedomain.m Shows you how to solve a problem defined on an infinite (or semi-infinite) domain, and introduces and compares several methods (e.g. Hermite and mapped Chebyshev disccretisations).

In 2D:

In 3D:

Methods

Here we show a few codes that present some methods.

• particles.m showing how to advect tracer particles for a 2D velocity field
• wave_like.m is a pedagogical introduction to the assumption of wavelike behaviour that we use almost all the time for stability of fluid flows. You can also have a look at bruxellator_jerome.m for the comparison of computation and theory on a 1D periodic problem.
• diffusion_eigenmodes.m is a pedagogical introduction to the use of the eigenmodes, here in 1D for the diffusion equation.
• GinsburgLandau.m is another pedagogical illustration of the use of eigenmodes to predict the behaviour of a partial differential equations problem modelling a biological system, including linear and nonlinear regimes.

Dynamical systems

• PhasePortrait_NonLinear.m a pedadogical program drawing phase portraits for a numer of classical 2D dynamical systems, including pendulum, Van Der Pol, Lotka-Volltera, etc…
• PhasePortrait_Linear.m another pedadogical program studying the stability of fixed-points in 2D systems, and classifying them a nodes, foci, saddles, etc…
• lorenz.m The classical Lorenz equation modelling thermal convection.

Stability of classical fluid flows

• poiseuille_uvp.m considers of the stability of the Poiseuille flow in primitive variables, and compares results with litterature data. TS_PlanePoiseuille.m is a more evolved program which computes the neutral curve of the Tollmien-Schlishting instability in the Re/k plane.
• orr_sommerfeld_squire.m Again the stability of the plane channel flow, but using the (v,w) formulation: The Orr-Sommerfeld and Squire equations.
• couette_uvwp.m For the stability of the plane Couette flow in primitive variables with u,v,w, and p as variables, for oblique waves.
• KH_temporal_inviscid.m considers the stability of the tanh shear layer, working with Rayleigh equation. KH_temporal_viscous.m considers the same problem in the viscous case, starting from primitive (u,v,p) equations. Both these programs use stretched chebyshev discretization to fit the infinite domain. Alternatively, kelvin_helmholtz_hermite.m uses differentiation matrices based on Hermite polynomials instead of Chebychev polynomials.
• pipe_sym.m for he stability of the pipe flow, using the symmetry of the variables in the differentiation matrices.
• rayleigh_benard.m considers the classical Rayleigh-Bénard instability of a fluid layer heated fom below, using “slip” boundary conditions, and validates the approach by comparing with theoretical results. The second program RayleighBenard.m considers the more realistic case of no-slip conditions, and draws the neutral curve of the instability in the Ra/k plane.

When the computational domain is itself an unknown

This is an ongoing work where the shape of the computational domain is also an unknown. For instance this is the case for a flow in a domain bounded by a free-surface. Please see unknow-domain. The final result of these codes is free_surface_navier_stokes.m.

Gerris

Connection with the nonlinear code Gerris flow solver

• kelvin_helmholtz_gerris.m For the Kelvin-Helmholtz instability of the shear layer (as in kelvin_helmholtz_hermite.m), using the unstable eigenmode as an initial condition in Gerris.
• rayleigh_taylor_gerris.m For the Rayleigh-Taylor instability of a heavy fluid above a lighter fluid (as in rayleigh_taylor.m), using the unstable eigenmode as an initial condition in Gerris. Here the technical aspect is: how to save the free interface and accounting for the two flow domains.

How can I contribute?

Please contribute to easystab by first using the codes for yourself, playing with them and getting familiar with the way the things are coded and presented in the files. Then a good way to start is to look at the bottom of a code which you use or like, where there is a list of exercices and contributions. You may then add a link in this list to your file.

You will first need to create an account by clicking on the “login/et an account” link on the top right of this page. This is very quick. Then create your file by typing its name in the URL box. Edit it, then preview and save once your done. Then add the link in the Contribution list to your new file.

The comments where you put the theory and explanations are in matlab/octave block comments %{ %} as well as the figures. To get help on how to foramt these comments (putting links, titles, chapter headings…), look at the /Help#markdown page.

If you would like to contribute with something that is not in the contribution suggestions, plese insert it in the README file that you are reading now.

• Please remember that a code that is not validated cannot be trusted.