Finite-Difference Schemes for Interval Advection Equations Within a New Interval-Calculus Framework

This paper focuses on three finite difference schemes for the interval advection equation based on a novel interval calculus framework. Different from classical interval arithmetic, this innovative framework equips the interval number space with a rigorous Hilbert space structure and enables a critical decoupling of the derivative for interval-valued functions, where the center component obeys classical differentiation rules and the radius component complies with multiplicative differentiation principles. Using this unique decoupling property, we construct interval counterparts of three representative classical finite difference schemes, namely the Upwind, Lax–Friedrichs, and Lax–Wendroff schemes, and conduct a comprehensive and rigorous theoretical assessment of their numerical properties. Utilizing the inherent isometric isomorphism between the interval space and ${\mathbb{R}}^2$, we rigorously establish the consistency of the proposed schemes and adopt the von Neumann method for systematic stability analysis. A sharp, explicit Courant–Friedrichs–Lewy (CFL) condition is derived, which jointly accounts for the center velocity and logarithmic radius velocity of the interval advection field, and the scheme convergence is strictly guaranteed via the Lax equivalence theorem. Extensive numerical experiments are carried out to validate the theoretical conclusions and verify the practical merits of the developed interval schemes. The numerical results demonstrate that the proposed methods retain nearly constant interval width in long-duration simulations, completely bypass the switching-point complexity that intrinsically exists in traditional generalized Hukuhara (gH)-based interval approaches, and deliver competitive computational efficiency. This work corroborates that the newly proposed interval calculus framework serves as an elegant, solid, and versatile foundation for the numerical computation and analysis of interval partial differential equations.