Accurate and efficient quadrature for volterra integral equations

Journal Article · Sun Aug 01 04:00:00 EDT 1976 · J. Comput. Phys.; (United States)

DOI:https://doi.org/10.1016/0021-9991(76)90037-1· OSTI ID:7256112

Knirk, D L

Four quadrature schemes were tested and compared in considerable detail to determine their usefulness in the noniterative integral equation method for single-channel quantum-mechanical calculations. They are two forms of linear approximation (trapezoidal rule) and two forms of quadratic approximation (Simpson's rule). Their implementation in this method is shown, a formal discussion of error propagation is given, and tests are performed to determine actual operating characteristics on various bound and scattering problems in different potentials. The quadratic schemes are generally superior to the linear ones in terms of accuracy and efficiency. The previous implementation of Simpson's rule is shown to possess an inherent instability which requires testing on each problem for which it is used to assure its reliability. The alternative quadratic approximation does not suffer this deficiency, but still enjoys the advantages of higher order. In addition, the new scheme obeys very well an h/sup 4/ Richardson extrapolation, whereas the old one does so rather poorly. 6 figures, 11 tables.

• Research Organization: Univ. of California, Berkeley
• OSTI ID: 7256112
• Journal Information: J. Comput. Phys.; (United States), Journal Name: J. Comput. Phys.; (United States) Vol. 21:4; ISSN JCTPA
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ACCURACY
BOUND STATE
COMPARATIVE EVALUATIONS
EFFICIENCY
EIGENVALUES
EQUATIONS
ERRORS
INTEGRAL EQUATIONS
LENNARD-JONES POTENTIAL
MECHANICS
NUMERICAL SOLUTION
PHASE SHIFT
QUADRATURES
QUANTUM MECHANICS
SCATTERING
VOLTERRA INTEGRAL EQUATIONS