Advantages of the Samarskii-type schemes on the Shishkin mesh

Journal Article · Mon Dec 01 00:00:00 EST 2025 · Journal of Computational and Applied Mathematics

DOI:https://doi.org/10.1016/j.cam.2025.116688· OSTI ID:3004771

Vulanović, Relja; Nhan, Thái Anh

The schemes of the Samarskii type are simple modifications of the upwind scheme. We use them on the Shishkin mesh and discuss their advantages over the upwind scheme when applied to the linear one-dimensional singularly perturbed convection–diffusion problem. One of the advantages is that the Samarskii-type schemes have exact first-order accuracy uniform in the perturbation parameter, as opposed to the upwind scheme which is almost first-order uniformly accurate because its accuracy is diminished by logarithmic factors. Although this is not a new result, we re-emphasize it in the paper. We also demonstrate another advantage, that the Samarskii-type schemes are almost second-order uniformly accurate on the layer component of the solution. Motivated by this fact, we present a further improvement of the numerical method.

• Research Organization: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
• Sponsoring Organization: US Department of Energy; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
• Grant/Contract Number: AC02-05CH11231
• OSTI ID: 3004771
• Journal Information: Journal of Computational and Applied Mathematics, Journal Name: Journal of Computational and Applied Mathematics Vol. 470
• Country of Publication: United States
• Language: English

Related Subjects

Convection–diffusion
Finite differences
Shishkin mesh
Singular perturbation
Uniform convergence