## pde2path Tutorials and Demos (v3.0)Home |Tutorials and Demos | Applications| Backlog |

Here we collect a number of tutorials dealing with various aspects of pde2path. The current version is 3.0. See also the Quickstart guide and reference card, which contains an overview of all current demo directories.

A collection of movies generated in the demos is available here.

Further tutorials will appear here when available, so please revisit ever once in a while ...

Note (Jan.2021): updates of most tutorials below are due soon.

- H. Uecker, pde2path with higher order FEM; a first version for using of higher order finite elements, following Pozrikidis' FSElib , explained via standard problems (Allen-Cahn, Swift-Hohenberg, Schnakenberg). Download libraries and demos, tar or zip
- H. Uecker, pde2path without FEM, explaining mods to run pde2path on "general" right hand sides.
- J. Rademacher, H. Uecker,
The OOPDE setting of pde2path - a tutorial via some Allen-Cahn models.

This is intended as a soft introduction to pde2path, in particular to the OOPDE setup. It starts with a minimal setup to study a 1D AC with homogeneous Neumann BC, and from there explores step by step issues such as fold and branch point continuation, mesh adaption, various boundary conditions, and quasilinear problems. These are then also taken to 2D and 3D. - H. Uecker,
Tutorial for "Numerical continuation and bifurcation in Nonlinear PDEs - Algorithms,
Applications, and Experiments",
Jahresbericht der DMV, 2021.

Treats two somewhat advanced pattern formation problems, namely the mixing of Turing and Hopf patterns, and the Swift--Hohenberg equation on a disk, and describes experiments with dead-core pattern formation. - H. Uecker,
Pattern formation with pde2path - a tutorial.

This explains some pde2path setups for pattern formation in 1D, 2D and 3D. A focus is on new pde2path functions for branch switching at steady bifurcation points of higher multiplicity, typically due to discrete symmetries, but we also review general concepts of pattern formation and their handling in pde2path including localized patterns and homoclinic snaking, again in 1D, 2D and 3D, based on the demo**sh**(Swift-Hohenberg equation). Moreover, the demos**schnakpat**(a Schnakenberg reaction-diffusion system) and**chemtax**(a quasilinear RD system with cross-diffusion from chemotaxis) simplify and unify previous results in a simple and concise way,**CH**(Cahn-Hilliard) deals with mass constraints,**hexex**deals with (multiple) branch points of higher degeneracy in a scalar problem on a hexagonal domain, and**shgc**illustrates some global coupling. The demos**schnakS**and**schnaktor**(the Schnakenberg model on spheres and tori) consider pattern formation on curved surfaces, and**bruosc**(Brusselator) explains how to augment autonomous systems by a time periodic forcing. Along the way we also comment on the choice of meshes, on time integration, and we give some examples of branch point continuation and Hopf point continuation to approximate stability boundaries. Inter alia, the demo sh also illustrates how to rewrite the (4th order) Swift--Hohenberg equation as a 2-component 2nd-order system in a consistent way.

- H. Uecker,
Using trullekrul in pde2path - anisotropic mesh adaptation for some
Allen-Cahn models in 2D and 3D.

We describe by means of some examples how some functionality of the 2D and 3D mesh adaptation package trullekrul can be used in pde2path. - H. Uecker , D. Wetzel, The ampsys tool of
pde2path.

The computation of coefficients of amplitude systems for Turing bifurcations is a straightforward but sometimes elaborate task, in particular for 2D or 3D wave vector lattices.**ampsys**automates such computations for two classes of problems, namely scalar equations of Swift-Hohenberg (SH) type and generalizations, and reaction-diffusion systems with an arbitrary number of components. The tool is designed to require minimal user input, and for a number of cases can also deal with symbolic computations. After a brief review of the setup of amplitude systems we explain the tool by a number of 1D, 2D and 3D examples over various wave vector lattices. - J. Rademacher, H. Uecker,
Symmetries, freezing, and Hopf bifurcations of modulated traveling waves in pde2path.

We use four 1D model problems to explain the setup of phase conditions to handle continuous symmetries in pde2path. The first is a complex Ginzburg-Landau equation with, inter alia, translational and rotational invariance. The second is a FitzHugh-Nagumo type system, for which we also implement a 'freezing' method to obtain traveling waves and their speed from time integration. Additionally we describe setups to compute branches of relative periodic orbits, namely modulated fronts for a model of autocatalysis, and breathing pulses for another FHN model. - T. Dohnal, H. Uecker,
Periodic boundary conditions in pde2path .

This describes the implementation of periodic boundary conditions in pde2path, and give examples on their usage for some scalar model problems (similar to 1.) in 1D, 2D and 3D. - H. Uecker, D. Wetzel,
Linear system solvers in pde2path - tutorial .

We explain the implementations of two linear system solvers in pde2path, namely the iterative solver lssAMG, and the bordered elimination solver lssbel. We discuss their usage and performance via two tutorial examples. - H.deWitt,
Fold and branch continuation in a Schnakenberg system and details of branch plotting - a pde2path tutorial

This gives an introduction to using OOPDE for a reaction-diffusion system, in 1D and 2D, including foldpoint and branchpoint continuation, and a somewhat comprehensive overview of branch-plotting facilities. A natural next step after the Allen-Cahn tutorial. - D. Wetzel, plot.

This describes, using again the Schnakenberg system in 1D, 2D and 3D as a model problem, details of solution plotting in pde2path, with a focus on isolevel plots in 3D. - H. Uecker,
Hopf bifurcation and periodic orbits with
pde2path - a user guide

Here we describe the setup for using the pde2path libraries for Hopf bifurcation and continuation of branches of periodic orbits, and implementation details of the associated demo directories. See here for a description of the algorithms and the mathematical background of the examples, and here for demo outputs and movies. - H. Uecker, Hannes de Witt
Infinite time horizon
spatially distributed optimal control problems with pde2path -
algorithms and tutorial examples.

We use pde2path to numerically analyze infinite time horizon optimal control problems for parabolic systems of PDEs. The basic idea is a two step approach to the canonical systems, derived from Pontrygin's maximum principle. First we find branches of steady or time--periodic states of the canonical systems, i.e., canonical steady states (CSS) respectively canonical periodic states (CPS), and then use these results to compute time-dependent canonical paths connecting to a CSS or a CPS with the so called saddle point property. This is a (high dimensional) boundary value problem in time, which we solve by a continuation algorithm in the initial states. We explain the algorithms and the implementation via the demos sloc, vegoc, lvoc and pollution.