A QUALITATIVE ANALYSIS ON NONCONSTANT GRAININESS OF THE ADAPTIVE GRIDS VIA TIME SCALES

• Published in: The Rocky Mountain journal of mathematics Vol. 36; no. 1; pp. 115 - 133
• Main Authors: ELOE, P.W., HILGER, S., SHENG, Q.
• Format: Journal Article
• Language: English
• Published: The Rocky Mountain Mathematics Consortium 01-01-2006; Rocky Mountain Mathematics Consortium
• Subjects: 34B10; 35K60; 35K65; 39A10; 65M06; adaptive method; Approximation; Boundary value problems; Calculus; delta and nabla derivatives; Differential calculus; Differential equations; Dynamic and differential equations; Green's functions; Mathematical functions; Mathematical integrals; nonconstant graininess; Riemann sums; Sine function; time scales
• ISSN: 0035-7596; 1945-3795
• DOI: 10.1216/rmjm/1181069491

Calculus on time scales plays a crucial role in unifying the continuous and discrete calculus. In this paper, we apply the time scales calculus methods to study qualitatively properties of the numerical solution of second order ordinary differential equations via different finite difference schemes. The properties become particularly interesting in the case when the computational grids are nonuniform, on which the finite difference operators do not commute. To investigate the solution properties, we introduce the graininess function, and express the numerical solution as functions of the variable grid steps, that is, functions of the graininess and its dynamic derivatives implemented by using the time scales analysis. It is found in the study that a linear combination of the consecutive numerical solutions following the pattern of the nonuniform grid used may improve the accuracy of the numerical solution. We validate our results with several constructive computational experiments.