Piecewise C~1 Scheme For Solving Thin Boundary Layer Problems

Because of the existence in the higher order derivatives of the perturbation parameter ε, the solution of the singularly perturbed boundary value problems have boundary layers whose width depend on the parameter ε. Using the standard finite difference method or the finite element method always occurs vibrations, leading to the distortion of the approximation. Recent years, some meshless methods and grid methods have been presented to solve these problems. However, when come to the thin boundary layer, vibrations have occurred, especially near the extreme point. Finding a more effective numerical method for the thin boundary problems is more important.Beckett et al. introduced the upwind scheme based on the equidistributed grid, which is the first-order convergent uniformly. Besides, the equidistributed grid makes the boundary layer and outside the boundary layer have the same number of points, which contributes to that half of the points are clustered to the boundary layer. Tang Tao et al. presented the Chebyshev pseudospectral method based on the m-SINE transformation. This coordinate transformation technique can push more collocation points to the boundary layers which is useful for the thin boundary layer problems. Considering the multi-scale feature and the idea of the domain divi-sion, this paper presents a piecewise C1 scheme according to the equidistributed grid and the m-SINE transformation.In chapter 2, we construct the equidistributed grid for the reaction diffusion equation with the convective term; In chapter 3, we give the piecewise C1 scheme, the division point and the iteration parameter m; In chapter 4, we solve different problems with C1 scheme. Through nu-merical experiments, the piecewise C1 scheme shows the following advantages:(1) This scheme is useful for the boundary layer problems with different width, especially valid for the thin layer problems; (2) We give the choice of the iteration parameter m priorily by the classification of the problems; (3) The division point is computed by the equidistributed grid, which leads to the decrease of the errors near the extreme point and the improvement of the approximation results in the whole domain.