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We propose an adaptive stencil construction for high order accurate finitevolume schemes aposteriori stabilized devoted to solve one-dimensionalsteady-state hyperbolic equations. High-accuracy (up to the sixth-orderpresently) is achieved thanks to polynomial reconstructions while stability isprovided with an aposteriori MOOD method which controls the cell polynomialdegree for eliminating non-physical oscillations in the vicinity ofdiscontinuities. We supplemented this scheme with a stencil constructionallowing to reduce even further the numerical dissipation. The stencil isshifted away from troubles (shocks, discontinuities, etc.) leading to lessoscillating polynomial reconstructions. Experimented on linear, B\"urgers', andEuler equations, we demonstrate that the adaptive stencil technique manages toretrieve smooth solutions with optimal order of accuracy but also irregularones without spurious oscillations. Moreover we numerically show that theapproach allows to reduce the dissipation still maintaining the essentiallynon-oscillatory behavior.

a posteriori stabilized sixth-order finite volume scheme with adaptive stencil construction -- Basics for the 1D steady-state hyperbolic equations