PDE 2 D A GeneralPurpose PDE Solver Granville

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PDE 2 D, A General-Purpose PDE Solver Granville Sewell Mathematics Dept. University of Texas

PDE 2 D, A General-Purpose PDE Solver Granville Sewell Mathematics Dept. University of Texas El Paso

PDE 2 D History • • Work began 1974 in Caracas, Venezuela Sold as

PDE 2 D 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 PDE 2 D by Granville Sewell, 1991 -2007 “The Numerical Solution of Ordinary and Partial Differential Equations, second edition” John Wiley & Sons, 2005 Sold as PDE 2 D by VNI/Rogue Wave, 20072011: Free versions for Windows and Linux, for small to moderate size problems, now downloadable at www. roguewave. com/pde 2 d

PDE 2 D User Interfaces • A GUI interface can be used to access

PDE 2 D User Interfaces • A GUI interface can be used to access the collocation (0 D, 1 D, 2 D, 3 D) finite element methods • An Interactive Driver interface can be used to access the collocation and Galerkin (1 D, 2 D) finite element methods • PDE 2 D produces its own graphics, but also automatically generates a MATLAB program to produce MATLAB graphics • PDE 2 D 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 2 D Regions • User-supplied initial triangulation can be refined

Galerkin Method Handles General 2 D 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

Other Applications

Algorithms Used • The Galerkin options use up to 4 th degree isoparametric elements,

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, MA 27/MA 37, for 1

Linear System Solver Options • Harwell sparse direct solvers, MA 27/MA 37, for 1 D, 2 D and 3 D problems • Frontal methods, for 2 D and 3 D problems (slow but minimal memory requirements) • Preconditioned conjugate gradient iterative solvers, for 2 D and 3 D problems. • MPI-based parallel band solvers available on parallel systems, for 2 D and 3 D problems • Easy to plug in user supplied linear system solvers

Links • www. roguewave. com/pde 2 d – Download free versions or purchase PDE

Links • www. roguewave. com/pde 2 d – Download free versions or purchase PDE 2 D • www. pde 2 d. com or www. roguewave. com/pde 2 d – Video – List of >200 journal publications using PDE 2 D to general numerical results – Appendix A of 2005 John Wiley book, with most complete documentation