Structural schemes for one dimension stationary equations

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 equa-tions 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.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Automated summary

In this paper, a new paradigm for finite differences numerical methods is proposed, based on compact schemes that provide high order accurate approximations of a smooth solution. The method involves approximations of derivatives and the construction of structural equations derived from a matrix that collects variables belonging to a small stencil. Numerical schemes combine physical equations and structural relations. The spectral resolution of common structural equations is analyzed, and numerical tests are performed to address stability and accuracy issues for linear and non-linear problems.