We consider a second order,two-point,singularly perturbed boundary value problem,of reaction-convection-diffusion type with two small parameters,and we obtain analytic regularity results for its solution,under the ass...We consider a second order,two-point,singularly perturbed boundary value problem,of reaction-convection-diffusion type with two small parameters,and we obtain analytic regularity results for its solution,under the assumption of analytic input data.First,we establish classical differentiability bounds that are explicit in the order of differentiation and the singular perturbation parameters.Next,for small values of these parameters we show that the solution can be decomposed into a smooth part,boundary layers at the two endpoints,and a negligible remainder.Derivative estimates are obtained for each component of the solution,which again are explicit in the differentiation order and the singular perturbation parameters.展开更多
This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivityε.With a novel treatment for the reaction term,we first derive a differ...This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivityε.With a novel treatment for the reaction term,we first derive a difference scheme of accuracy O(ε^(2)h+εh^(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 a better stability。展开更多
文摘We consider a second order,two-point,singularly perturbed boundary value problem,of reaction-convection-diffusion type with two small parameters,and we obtain analytic regularity results for its solution,under the assumption of analytic input data.First,we establish classical differentiability bounds that are explicit in the order of differentiation and the singular perturbation parameters.Next,for small values of these parameters we show that the solution can be decomposed into a smooth part,boundary layers at the two endpoints,and a negligible remainder.Derivative estimates are obtained for each component of the solution,which again are explicit in the differentiation order and the singular perturbation parameters.
基金the National Science Council of Taiwan under the grants NSC 101-2811-M-008-032 and NSC 102-2115-M-033-007-MY2the National Science Council of Taiwan under the grants NSC 99-2115-M-008-012-MY2 and NSC 101-2115-M-008-008-MY2.
文摘This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivityε.With a novel treatment for the reaction term,we first derive a difference scheme of accuracy O(ε^(2)h+εh^(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 a better stability。
