In this work we deal with the numerical solution of one dimensional semilinear parabolic singularly perturbed systems of convection-diffusion type. We assume that the coupling in the convection terms is weak and also that the coupling reaction terms are nonlinear. In the case of considering different small diffusion parameters at each equation with different orders of magnitude, the exact solution usually shows overlapping boundary layers on the outflow of the spatial domain. To approximate it, we construct a numerical scheme which combines the upwind finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space and a linearized version of the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. Then, the fully discrete method is uniformly convergent with respect to both diffusion parameters, having first order in time and almost first order in space. The choice of this time integrator provokes that only tridiagonal linear systems must be solved at each time step; in this way, the computational cost of the algorithm is considerably lower than this one associated to classical implicit methods. The numerical results obtained for different test problems corroborate in practice the good performance of the numerical algorithm.

中文翻译：

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半线性一维抛物型奇异摄动对流扩散系统的高效数值方法

在这项工作中，我们处理对流扩散型一维半线性抛物线奇摄动系统的数值解。我们假设对流项中的耦合很弱，并且耦合反应项是非线性的。在每个方程考虑不同数量级的不同小扩散参数的情况下，精确解通常在空间域的流出上显示重叠的边界层。为了近似它，我们构建了一个数值方案，该方案结合了在分段均匀希什金网格上定义的迎风有限差分方案以在空间上离散化和分数隐式欧拉方法的线性化版本以及适当的分量分割以在时间上离散化。然后，完全离散方法对于两个扩散参数一致收敛，在时间上具有一阶并且在空间上几乎一阶。该时间积分器的选择使得每个时间步长都必须求解三对角线性系统；通过这种方式，该算法的计算成本比经典隐式方法的计算成本要低得多。针对不同测试问题获得的数值结果在实践中证实了数值算法的良好性能。