An Adaptive Method For Non-conservative Linear Singularly Perturbed Two-Point Boundary Value Problems

In this paper,the adaptive upwind finite difference method based on the arc-length equidistribution principle is studied to solve the non-conservative linear singular perturbation two-point boundary value problem.Firstly,the adjoint operator T*of the linear operator T is introduced,and the stability of the linear operator from ‖·‖ to ‖·‖∞ is obtained by means of Green’s function,the barrier function and the comparison principle.Secondly,a new upwind finite difference scheme for the original problem is presented,and the boundedness of the numerical solution obtained on an arbitrary mesh is demonstrated using the properties of the M-matrix and the comparison principle.By introducing a first-order interpolation function for the numerical solution and utilizing the stability of the linear operator,the a posterior error estimation of the above numerical solution is obtained.Thirdly,an adaptive finite difference method based on the principle of equidistribution of arc length is proposed.The existence of the numerical solution is proved by using the fixed-point theorem.Fourthly,the boundedness of the arc-length of the numerical solution is obtained through the discrete Green’s function and the discrete comparison principle,and then the uniform first-order convergence of the numerical solution with respect to the perturbation parameters is obtained by combining the a posterior error estimate.Fifthly,three numerical experiments are conducted to compare the new upwind scheme with the old upwind scheme on the adaptive mesh.The experimental results also show that the adaptive method with the new upwind scheme has smaller errors,better results,and achieve uniform first-order convergence.