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

TL;DR

This paper introduces an adaptive stencil construction for high-order finite volume schemes that enhances accuracy and stability in solving 1D steady-state hyperbolic equations, effectively reducing dissipation and oscillations near discontinuities.

Contribution

It presents a novel adaptive stencil method combined with an a posteriori MOOD stabilization for high-order finite volume schemes, improving accuracy and reducing dissipation near discontinuities.

Findings

Achieves up to sixth-order accuracy for smooth solutions.

Effectively suppresses spurious oscillations near shocks.

Reduces numerical dissipation while maintaining stability.

Abstract

We propose an adaptive stencil construction for high order accurate finite volume schemes aposteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an aposteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. We supplemented this scheme with a stencil construction allowing to reduce even further the numerical dissipation. The stencil is shifted away from troubles (shocks, discontinuities, etc.) leading to less oscillating polynomial reconstructions. Experimented on linear, B\"urgers', and Euler equations, we demonstrate that the adaptive stencil technique manages to retrieve smooth solutions with optimal order of accuracy but also irregular ones without spurious oscillations. Moreover we numerically show that the approach allows to reduce the dissipation still maintaining the essentially non-oscillatory behavior.