PDE2D, A General-Purpose PDE Solver Granville Sewell Mathematics Dept.

University of Texas El Paso

PDE2D History

• • • • • • • • Work began 1974 in Caracas, Venezuela Sold as TWODEPEP by IMSL, 1980-1984 Sold as PDE/PROTRAN by IMSL, 1984-1991 “Analysis of a Finite Element Method: PDE/PROTRAN,” Springer Verlag, 1985 Sold as PDE2D by Granville Sewell, 1991-2007 “The Numerical Solution of Ordinary and Partial Differential Equations, second edition” John Wiley & Sons, 2005 Sold as PDE2D by VNI/Rogue Wave, 2007 2011: Free versions for Windows and Linux, for small to moderate size problems, now downloadable at www.roguewave.com/pde2d

PDE2D User Interfaces

• • • • A GUI interface can be used to access the collocation (0D,1D,2D,3D) finite element methods An Interactive Driver interface can be used to access the collocation and Galerkin (1D,2D) finite element methods PDE2D produces its own graphics, but also automatically generates a MATLAB program to produce MATLAB graphics PDE2D has all the flexibility of FORTRAN, for example, you can write FORTRAN functions to define any PDE or BC coefficients, or write your own postprocessing code

Galerkin Method Handles General 2D Regions

• • • User-supplied initial triangulation can be refined adaptively or graded according to user-supplied specifications Curved boundaries can be defined by parametric equations, or a cubic spline can be drawn through user-supplied boundary points Interactive driver must be used to access Galerkin methods

Other Applications

Algorithms Used

• • • • • • The Galerkin options use up to 4 th degree isoparametric elements, thus up to O(h 5 ) accuracy, even with curved boundaries The collocation options use 3 rd degree elements, thus O(h 4 ) accuracy, even with curved boundaries Newton’s method is used to solve the algebraic equations, for nonlinear PDEs Shifted inverse power method is used to find a single eigenvalue (with eigenfuction), for eigenvalue PDEs. If all eigenvalues are desired (without eigenfunctions), a shifted QR iteration is used from EISPACK Adaptive time step control is available for time-dependent problems

Linear System Solver Options

• • • • • Harwell sparse direct solvers, MA27/MA37, for 1D, 2D and 3D problems Frontal methods, for 2D and 3D problems (slow but minimal memory requirements) Preconditioned conjugate gradient iterative solvers, for 2D and 3D problems.

MPI-based parallel band solvers available on parallel systems, for 2D and 3D problems Easy to plug in user supplied linear system solvers

Links

• • www.roguewave.com/pde2d – Download free versions or purchase PDE2D www.pde2d.com

or www.roguewave.com/pde2d – Video – List of >200 journal publications using PDE2D to general numerical results – Appendix A of 2005 John Wiley book, with most complete documentation