Semi-Linear Difference Schemes for Singular Perturbation Problems in One Dimension.
Abstract
Numerical differential formulas play a very important role in constructing difference schemes of differential equations. Usual numerical differentation formulas based on polynomial approximations are derived for smooth functions without large derivatives, it is possible for these formulas to lead to very poor results when the functions are not smooth. There are usually two ways to avoid this trouble: refine the mesh, or use higher order polynmial interpolation. Sometimes they are called h-version and p-version, respectively. The approach presented in this paper is quite different. The main reason why the usual linear schemes lead to worse results for problems with large derivatives, especially those with singularity, is that the usual numerical differentation formulas based on polynomial approximation is not accurate enough in this case, e.g., an asymptotic behavior near singularity is exponential. It seems hard to get high precision numerical differentiation formula near singularity if we restrict ourselves to use only piecewise polynomial approximations or other linear functional space. Hence, in this paper, we try to look for some new numerical differentiation formulas beyond linear functional space.
Document Details
- Document Type
- Technical Report
- Publication Date
- May 17, 1982
- Accession Number
- ADA121568
Entities
People
- Jiachang Sun
Organizations
- Yale University