Algorithm 688: EPDCOL: a more efficient PDECOL code

ABSTRACT

The software package PDECOL [7] is a popular code among scientists wishing to solve systems of nonlinear partial differential equations. The code is based on a method-of-lines approach, with collocation in the space variable to reduce the problem to a system of ordinary differential equations. There are three principal components: the basis functions employed in the collocation; the method used to solve the system of ordinary differential equations; and the linear equation solver which handles the linear algebra. This paper will concentrate on the third component, and will report on the improvement in the performance of PDECOL resulting from replacing the current linear algebra modules of the code by modules which take full advantage of the special structure of the equations which arise. Savings of over 50 percent in total execution time can be realized.

REFERENCES

• 1: G. Bader , U. Ascher, A new basis implementation for a mixed order boundary value ODE solver, SIAM Journal on Scientific and Statistical Computing, v.8 n.4, p.483-500, July 1, 1987 [doi>10.1137/0908047]
• 2: Carl de Boor , Richard Weiss, SOLVEBLOK: A Package for Solving Almost Block Diagonal Linear Systems, ACM Transactions on Mathematical Software (TOMS), v.6 n.1, p.80-87, March 1980 [doi>10.1145/355873.355880]
• 3: J. C. Diaz , G. Fairweather , P. Keast, FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination, ACM Transactions on Mathematical Software (TOMS), v.9 n.3, p.358-375, Sept. 1983 [doi>10.1145/356044.356053]
• 4: HINDMARSH, A C. Preliminary documentation of GEARIB: Solution of implicit systems of ODE with banded Jacobian. Rep UCID-30130, Lawrence Livermore Lab, Livermore, Calif., 1976.
• 5: HINDMARSH, A.C. LSODE and LSODI, two new initial value ODE solvers. ACM SIGNUM Newsl. 15, 4 (Dec. 1980), 10-11.
• 6: HINDMARSH, A.C. private communication, 1984.
• 7: N. K. Madsen , R. F. Sincovec, Algorithm 540: PDECOL, General Collocation Software for Partial Differential Equations [D3], ACM Transactions on Mathematical Software (TOMS), v.5 n.3, p.326-351, Sept. 1979 [doi>10.1145/355841.355849]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.8 Partial Differential Equations

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
      G.1.7 Ordinary Differential Equations
  G.4 MATHEMATICAL SOFTWARE

General Terms:
Algorithms, Performance

Keywords:
collocation, method of lines, numerical software, partial differential equations