A two-parameter singularly perturbed problem with discontinuous source and convection coefficient is considered in one dimension. Both convection coefficient and source term are discontinuous at a point in the domain. The presence of perturbation parameters results in boundary layers at the boundaries. Also, an interior layer occurs due to the discontinuity of data at an interior point. An upwind scheme on an appropriately defined Shishkin-Bakhvalov mesh is used to resolve the boundary layers and interior layers. A three-point formula is used at the point of discontinuity. The proposed method has first-order parameter uniform convergence. Theoretical error estimates derived are verified using the numerical method on some test problems. Numerical results authenticate the claims made. The use of the Shiskin-Bakhvalov mesh helps achieve the first-order convergence, unlike the Shishkin mesh, where the order of convergence deteriorates due to a logarithmic term.
翻译:与不连续源和对流系数有关的单数奇相扰的问题在一个维度中加以考虑。对流系数和源术语在域某个点是不连续的。扰动参数的存在导致边界的边界层。此外,内层由于内部点数据的不连续性而出现内层。一个关于适当定义的Shishkin-Bakhvalov网格的上风系统用于解决边界层和内层。在不连续点使用三点公式。拟议方法有第一级参数统一汇合。根据一些测试问题得出的理论错误估计值使用数值方法加以核实。数字结果验证了所提出的主张。使用Shiskin-Bakhvalov网格有助于实现第一级汇合,与Shishkin网格不同,在Shishkin网格不同,由于对数术语而使汇顺序恶化。