TY - JOUR
T1 - A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity
AU - Hsieh , Po-Wen
AU - Yang , Suh-Yuh
AU - You , Cheng-Shu
JO - Advances in Applied Mathematics and Mechanics
VL - 5
SP - 637
EP - 662
PY - 2014
DA - 2014/06
SN - 6
DO - http://doi.org/10.4208/aamm.2014.5.s4
UR - https://global-sci.org/intro/article_detail/aamm/40.html
KW - Reaction-convection-diffusion equation, incompressible Navier-Stokes equations, boundary layer, interior layer, finite difference scheme.
AB - <p style="text-align: justify;">This paper is devoted to a new high-accuracy finite difference
scheme for solving reaction-convection-diffusion problems
with a small diffusivity $\varepsilon$.
With a novel treatment for the reaction term, we first derive a difference scheme
of accuracy $\mathcal{O}(\varepsilon^2 h + \varepsilon h^2 + h^3)$ for the 1-D case.
Using the alternating direction technique, we then extend the scheme
to the 2-D case on a nine-point stencil.
We apply the high-accuracy finite difference scheme to solve the 2-D steady
incompressible Navier-Stokes equations in the stream function-vorticity formulation.
Numerical examples are given to illustrate the effectiveness
of the proposed difference scheme.
Comparisons made with some high-order compact difference schemes
show that the newly proposed scheme can achieve good accuracy
with better stability.</p>