pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual

pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path for elliptic systems of PDEs over bounded 2D domains, based on the Matlab pdetoolbox. The new features include a more efficient use of FEM, easier switching between different single parameter continuations, genuine multi-parameter continuation (e.g., fold continuation), more efficient implementation of nonlinear boundary conditions, cylinder and torus geometries (i.e., periodic boundary conditions), and a general interface for adding auxiliary equations like mass conservation or phase equations for continuation of traveling waves. The package (library, demos, manuals) can be downloaded at www.staff.uni-oldenburg.de/hannes.uecker/pde2pat

pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems

pde2path is a free and easy to use Matlab continuation/bifurcation package for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on the FEM of the Matlab pdetoolbox, and is explained by a number of examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard convection, and von Karman plate equations. These serve as templates to study new problems, for which the user has to provide, via Matlab function files, a description of the geometry, the boundary conditions, the coefficients of the PDE, and a rough initial guess of a solution. The basic algorithm is a one parameter arclength continuation with optional bifurcation detection and branch-switching. Stability calculations, error control and mesh-handling, and some elementary time-integration for the associated parabolic problem are also supported. The continuation, branch-switching, plotting etc are performed via Matlab command-line function calls guided by the AUTO style. The software can be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where also an online documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems

Convergence Proof of Some Generalised Backward Differentiation (GBDF) Methods to Solve the General Second Order Ordinary Differential Equations (ODE).

Methods for the solution of the general second order ODE are discussed. Stiffness and convergence are also defined. Finally proof of convergence is given for certain cases of the GBDF methods for stiff pro blems

Partitioning Techniques and Their Parallelization for Stiff System of Ordinary Differential Equations

A new code based on variable order and variable stepsize component wise partitioning is introduced to solve a system of equations dynamically. In previous partitioning technique researches, once an equation is identified as stiff, it will remain in stiff subsystem until the integration is completed. In this current technique, the system is treated as nonstiff and any equation that caused stiffness will be treated as stiff equation. However, should the characteristics showed the elements of nonstiffness, and then it will be treated again with Adam method. This process will continue switching from stiff to nonstiff vice versa whenever it is necessary until the interval of integration is completed.Next, a block method with R-points generate R new approximate solution values;is a strategy for solving a system and also for parallelizing ODEs. Partitioning this block method to solve stiff differential equations is a new strategy; it is more efficient and takes less computational time compared to the sequential methods. Two partitioning techniques are constructed, Intervalwise Block Partitioning (IBP) and Componentwise Block Partitioning (CBP). Numerical results are compared as validation of its effectiveness. Intervalwise block partitioning will initially treat the systems of equations as nonstiff and solve them using Adams method, by switching to the Backward Differentiation formula when there is a step failure and indication of stiffness. Componentwise block partitioning will place the necessary equations that cause instability and stiffness into the stiff subsystem and solve using Backward Differentiation Formula, while all other equations will still be treated as non-stiff and solved using Adams formula. Parallelizing the partitioning strategies using Message Passing Interface (MPI) is the most appropriate method to solve large system of equations. Parallelizing the right algorithm in the partitioning code will give a better perfonnance with shorter execution times. The graphs of its performance and execution time, visualize the advantages of parallelizing

The use of approximate factorization in stiff ODE solvers

AbstractWe consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration

The Exponentially Faster Stick-Slip Dynamics of the Peeling of an Adhesive Tape

The stick-slip dynamics is considered from the nonlinear differential-algebraic equation (DAE) point of view and the peeling dynamics is shown to be a switching differential index DAE model. In the stick-slip regime with bifurcations, the differential index can be arbitrarily high. The time scale of the peeling velocity, the algebraic variable, in this regime is shown to be exponentially faster compared to the angular velocity of the spool and/or the stretch rate of the tape. A homogenization scheme for the peeling velocity which is characterized by the bifurcations is discussed and is illustrated with numerical examples.Comment: 7 figures, 24 page