Here we collect a number of tutorials dealing with various aspects
of pde2path. The current version is 3.1 (September 2023).
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.
Much of the tutorials (except 1., 2., 3.) has been reorganized
into the pde2path book (June 2021)
Further tutorials will appear here when available,
so please revisit ever once in a while ...
A. Meiners and H. Uecker,
Differential geometric bifurcation problems in pde2path. Explains how to treat bifurcations in differential
geometry in pde2path. Based on Jacobson's gptoolbox , and considers, for instance, spherical caps,
Enneper and Schwarz minimal surfaces, nodoids, and some 4th order biomembrane models.
H. Uecker, pde2path with higher order
FEM; a first version for using of higher order finite elements,
explained via standard problems (Allen-Cahn, Swift-Hohenberg, Schnakenberg).
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 , 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.
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, 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.
Some additional brief comments on (older but compatible) demo directories can
also be found in here ( pde2path2.0 manual )
