论文标题
与不连续的对流系数和源项的奇异扰动的两个参数抛物线问题的参数统一数值方法
Parameter uniform numerical method for singularly perturbed two parameter parabolic problem with discontinuous convection coefficient and source term
论文作者
论文摘要
在本文中，我们考虑了不连续的对流系数和源术语的时间依赖性的两参数奇异的抛物线问题。该问题分别包含参数$ε$和$μ$，分别乘以扩散和对流系数。由于这些参数的结果，边界两侧都会发展边界层。由于对流和源术语中的不连续性，内部层在不连续点附近形成。内部和边界层的宽度取决于扰动参数的比率。我们讨论了比率$ \ displaystyle \ frac {μ^2}ε$的问题。我们在该空间中的Shishkin-Bakhvalov网格上使用了上风有限的差异方法，并在均匀网格上及时使用了Crank-Nicolson方法。在不连续点，使用了三点公式。该方法在空间中与二阶和一阶均匀收敛。 Shishkin-Bakhvalov网格提供一阶收敛；与Shishkin网眼不同，在该网状网中，对数因子会恶化收敛顺序。给出了一些测试示例以验证提出的结果。
In this article, we have considered a time-dependent two-parameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters $ε$ and $μ$ multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss the problem for ratio $\displaystyle\frac{μ^2}ε$. We used an upwind finite difference approach on a Shishkin-Bakhvalov mesh in the space and the Crank-Nicolson method in time on uniform mesh. At the point of discontinuity, a three-point formula was used. This method is uniformly convergent with second order in time and first order in space. Shishkin-Bakhvalov mesh provides first-order convergence; unlike the Shishkin mesh, where a logarithmic factor deteriorates the order of convergence. Some test examples are given to validate the results presented.
