First and second order error estimates for the Upwind Source at Interface method

Abstract

Abstract. The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced by Perthame and Simeoni is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove L p -error estimates, 1 ≤ p <+∞, in the case of a uniform spatial mesh, for which an optimal result can be obtained. We thus conclude that the same convergence rates hold as for the corresponding homogeneous problem. To improve the numerical accuracy, we develop two different approaches of dealing with the source term and we discuss the question of deriving second order error estimates. Numerical evidence shows that those techniques produce high resolution schemes compatible with the U.S.I. method.

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

“…In particular, according to [20], [19], [7], since not only the reconstructed values U i,r at i + 1/2−, and U i+1,l at i + 1/2+ need be defined but also z i,r , z i+1,l , a cell-centered source term S ci must by added to preserve the consistency. We remark that even if z i do not depend on time, the reconstructed values z i,l , z i,r could depend on time via a coupling with U i in the reconstruction step.…”