Abstract
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.
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