Bounding the Round-Off Error of the Upwind Scheme for Advection

Louise Ben Salem-Knapp (1, 2) , Sylvie Boldo (1) , William Weens (3, 2)

1 TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
2 LDG - Laboratoire de Détection et de Géophysique (CEA)
3 LIHPC - Laboratoire en Informatique Haute Performance pour le Calcul et la simulation

Numerical simulations are carefully-written programs, and their correctness is based on mathematical results. Nevertheless, those programs rely on floating-point arithmetic and the corresponding round-off errors are often ignored. This article deals with a specific simple scheme applied to advection, that is a particular equation from hydrodynamics dedicated to the transport of a substance. It shows a tight bound on the round-off error of the 1D and 2D upwind scheme, with an error roughly proportional to the number of steps. The error bounds are generic with respect to the format and exceptional behaviors are taken into account. Some experiments give an insight of the quality of the bounds.

Mots clés: Round-off error; Floating-Point; Numerical Scheme; Advection; Hydrodynamics

Domaines: Logique en informatique [cs.LO]; Arithmétique des ordinateurs

Louise Ben Salem-Knapp, Sylvie Boldo, William Weens. Bounding the Round-Off Error of the Upwind Scheme for Advection. IEEE Transactions on Emerging Topics in Computing, 2022, 10 (3), ⟨10.1109/TETC.2022.3191472⟩. ⟨hal-03329933⟩