Structural schemes for one dimension stationary equations

• Published in: Applied mathematics and computation Vol. 457; p. 128207
• Main Authors: Clain, Stéphane, Pereira, Rui M.S., Pereira, Paulo A., Lopes, Diogo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-11-2023
• Subjects: Compact scheme; Finite difference; Structural equation; Very high-order; Finite difference; Compact scheme; Structural equation; Very high-order
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.128207

•The introduction is shorten and a historical note has been added in the appendix.•Section 2 is the critical part of the paper and has been deeply reformulated.
In this paper, we propose a new paradigm for finite differences numerical methods, based on compact schemes to provide high order accurate approximations of a smooth solution. The method involves its derivatives approximations at the grid points and the construction of structural equations deriving from the kernels of a matrix that gathers the variables belonging to a small stencil. Numerical schemes involve combinations of physical equations and the structural relations. We have analysed the spectral resolution of the most common structural equations and performed numerical tests to address both the stability and accuracy issues for popular linear and non-linear problems. Several benchmarks are presented that ensure that the developed technology can cope with several problems that may involve non-linearity.