A new class of finite difference methods: The zigzag schemes

TL;DR

The paper introduces zigzag finite difference schemes with hybrid, asymmetric stencils that offer enhanced stability, higher order accuracy, and ease of implementation, providing a promising alternative for numerical simulations.

Contribution

It presents a new class of finite difference schemes with hybrid stencils, including explicit formulas and stability analysis, expanding the toolkit for numerical approximation methods.

Findings

Zigzag schemes have broader stability ranges than centered and upwind schemes.

They enable higher order accuracy with vanishing coefficients at high orders.

Implementation is straightforward in existing finite difference codes.

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.