pde2path: Brief structural and function overview

To implement a continuation problem for pde2path the user needs to supply some variables and functions.
The first step is usually of the form p=[]; p=stanparam(p), which creates a standard pde2path problem struct p, and initializes most fields in p to standard values. Then of course we need to set the problem dependent fields.
We recommend to put this initialization into a function such as acinit
(from ac1D_simple, see Section 2 in the "Allen-Cahn tutorial", for a description)
A minimal setting contains:

• Function handles to routines defining the PDE (sG) and the boundary conditions.
• Domain and mesh initializations, and, in the OOPDE setting, FEM operator initializations via oosetfemops
• Initial conditions and settings for the continuation to be done.
• Additionally, function handles for Jacobians ( sGjac) are recomended and usually simple to implement.

Main fields in the structure p

The data in a pde2path structure p is organized as follows. Most values for stationary problems are initialized to standard values via stanparam, while p.hopf is initialized via hostanparam.

• fuha: struct of function handles; in particular defining the PDE (G=0), boundary condition and Jacobian (Gu).
• nc,sw: structs of numerical controls such as p.nc.tol, p.nc.nq, ..., and switches such as p.sw.bifcheck, ...
• u,np,nu: the solution u ( including all parameters / auxiliary variables in u(p.nu+1:end) ), the number of nodes p.np in the mesh, and the number of nodal values p.nu of PDE-variables
• tau,branch: tangent tau(1:p.nu+p.nc.nq+1), and the branch, filled via bradat.m and p.fuha.outfu.
• sol: struct of other values/fields calculated at runtime
• eqn: the tensors c; a; b for the simple FEM setup
• pdeo: an OOPDE pde object, containing the geometry data, mesh, and FEM data; if the legacy pdetoolbox setup is used, then the geometry and mesh are in p.mesh
• plot,file: structs of switches (and, e.g., figure numbers and directory name) for plotting and file output
• time,pm: structs of timing information, and pmcont switches
• fsol: switches for the interface to fsolve.
• mat: problem matrices, e.g., mass / stiffness matrices M, K for the the simple FEM setting, and drop and fill for periodic BC; by default, the fields mat is not saved to disk.
• hopf: Hopf data, initialized by hoswibra, and further filled at runtime.

Functions typically called during init (in the OOPDE setting)

• p=stanparam(p): sets many parameters to "standard" values; typically called during initialization; also serves as a documentation of the meaning of parameters. See also here.
• pdeo=stanpdeo1D(lx,h): generate pde-object in the OOPDE setting; see also stanpdeo2D, stanpdeo3D
• p=oosetfemops(p): set the FEM operators such as M, K
• p=setfn(p,name): set output directory to name (or "p", if name omitted).

Main functions for user calls

• p=cont(p): continuation of problem p.
• p=pmcont(p): parallel multi-predictor version of cont.
• p=findbif(p,varargin): bifurcation detection via change of stability index; alternative to bifurcation detection in cont or pmcont.
• p=swibra(dir,bpnr,var): branch-switching.
• p=hoswibra(dir,bpnr,var): Hopf branch-switching.
• p=swipar(p,var): switch parametrization, see also swiparf.
• p=spcontini(dir,name,npar): initialization for "spectral continuation".
• p=spcontexit(dir,name): exit spectral continuation.
• p=setpar(p,par): set parameter values, see also par=getpar(p,varargin).

Most basic plotting routines

• plotbra(p,var): plot branch in p, see also plotbraf, and the settings in p.plot field.
• plotsol(p,wnr,cmp,style): plot solution, see also plotsolu and plotsolf.
• hoplot(p,wnr,cnr,var): plot periodic orbit contained in p.hopf.y.

Misc additional routines

• [u,...]=nloop(p,u1) : Newton-loop for (G(u);q(u))=0 (often called for preparation).
• p=loadp(dir,pname,varargin): load point pname from dir; varargin to set new dir.
• p=meshref(p): adaptively refine mesh, see also p=meshadac(p).
• err=errcheck(p): compute error-estimate.
• p=tint(p,dt,nt,pmod): time integration of u_t = -G(u); see also tintx for a version with more output.
• p=tints(p,dt,nt,pmod,nu): time integration for semilinear case, see also tintxs.