In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed fromthe backward Euler scheme to amonotone hybrid scheme. Three kinds of nonuniform grids are considered, especially the standard Shishkin mesh, the Bakhavalov–Shishkinmesh and the adaptive grid. Themethods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and second order for themonotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal.

中文翻译：

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一类参数化奇异摄动问题的参数统一数值方法

在本文中，提出了一种加权有限差分方案来解决一类参数化的奇异摄动问题（SPPs）。取决于权重参数的选择，该方案自动从反向欧拉方案转换为单调混合方案。考虑了三种非均匀网格，特别是标准的Shishkin网格，Bakhavalov-Shishkinmesh和自适应网格。对于所有三种类型的网格，该方法相对于摄动参数均显示为一致收敛。对于后向欧拉方案，收敛速度为一阶，对于单调混合方案，收敛率为二阶。此外，所提出的方法被扩展到具有混合类型边界条件的参数化问题，并且被证明是均匀收敛的。