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 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.S. Clain acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/00324/2020. R. M. S. Pereira acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020. P. A. Pereira acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/00013/2020. Diogo Lopes acknowledges the financial support by national funds (PIDDAC), through the FCT – Fundação para a Ciência e a Tecnologia and FCT/MCTES under the scope of the projects UIDB/05549/2020 and UIDP/05549/2020. S. Clain and R. M.Pereira acknowledge the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operacional Fatores de Competitividade, and the National Funds through FCT, project N◦. POCI-01-0145-FEDER-028118