A new class of finite difference methods: The zigzag schemes

Abstract: We introduce a novel class of finite difference approximations, termed zigzag schemes, that employ a hybrid stencil that is neither symmetrical, nor fully one-sided. These zigzag schemes often enjoy more permissive stability constraints and see their coefficients vanish as the order tends to infinity. This property permits the formulation of higher order schemes. An explicit formula is given for both collocated and staggered grids for an arbitrary order and a closed-form expression for the infinite-order scheme is also provided. A linear stability analysis indicates that the zigzag scheme offer a broader range of conditional stability compared to the centred and upwind schemes, sometimes being the only stable scheme. Additionally, the asymmetrical structure of the stencil of zigzag schemes prevents some issues such as the formation of ``ghost solutions''. Moreover, implementing zigzag schemes is relatively easy when a code using classical finite differences is available, that is an important feature for well-tested legacy codes. Overall, zigzag schemes provide a compelling alternative for finite differences methods by enabling faster and more stable numerical simulations without sacrificing accuracy or ease of use.