Automatic generation of finite difference equations and fourier stability analyses

ABSTRACT

Recently several software tools based on the algebraic manipulation system MACSYMA have been implemented which facilitate the design, analysis and construction of finite difference programs for the numerical solution of systems of partial differential equations. Two of them are described here. The FDIFF package converts scalar, non-linear partial differential equations into linear, finite difference approximations. It includes tools for discretization of the domain of the PDE's dependent variables, linearization of non-linear terms and conversion of derivative terms into finite difference expressions. A notation and algebra for building arbitrary finite difference operators is provided. The FSTAB package automatically performs local Fourier stability analyses on sets of finite difference equations by deriving amplification matrices.

REFERENCES

• 1: Fateman, R. J., "MAC-ED, An Interactive Expression Editor for MACSYMA," Proc. of the 1979 MACSYMA Users' Conference, (MIT Laboratory for Computer Science, Washington, D.C., June 20-22, 1979).
• 2: Lanam, D.H., A Package for Generating and Executing FORTRAN Programs with MACSYMA, MS Project Report, Univ. of California, Berkeley (November 1980).
• 3: MACSYMA Reference Manual, Laboratory for Computer Science, MIT, Cambridge, Version Nine (December 1977).
• 4: Joel Moses, Algebraic simplification: a guide for the perplexed, Communications of the ACM, v.14 n.8, p.527-537, Aug. 1971 [doi>10.1145/362637.362648]
• 5: Edward W. Ng, Symbolic-Numeric Interface: a review (in absentia), Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, p.330-345, June 01, 1979
• 6: Richtmyer, R. D. and K. W. Morton, Difference Methods for Initial-Value Problems, (John Wiley & Sons, New York, 1967), Second edition.
• 7: Wirth, M. C., "Symbolic Vector and Dyadic Analysis," SIAM Journal On Computing, 8(3), 306-319 (August 1979).
• 8: Michael Clarence Wirth, On the automation of computational physics, 1980

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.8 Partial Differential Equations
          Subjects: Finite difference methods

Additional Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.1 Expressions and Their Representation
          Subjects: Simplification of expressions
      I.1.3 Languages and Systems

          Nouns: MACSYMA

General Terms:
Algorithms, Languages, Performance, Theory