基于混合有限差分格式的非线性奇异摄动问题的最大范数的后验误差估计

Maximum Norm a Posteriori Error Estimation for Nonlinear Singularly Perturbed Problems Based on a Hybrid Finite Difference Scheme

自适应移动网格算法在奇异摄动微分方程的数值解法中占有非常重要的地位,其关键技术是构造出有效的离散格式和相应的后验误差估计。基于此,对一类带参数的一阶非线性奇异摄动初值问题,给出了其连续解的稳定性估计及相关推论。然后,在任意非均匀网格上,利用向后欧拉公式和一阶中心有限差分格式建立了一个混合有限差分格式,并严格分析了离散解的稳定性。同时,基于连续解的稳定性估计和分段线性插值技术,推导出混合有限差分格式的最大范数的后验误差估计。利用该后验误差估计选择了一个最优的网格控制函数,并结合网格等分布原理设计了一个自适应网格生成算法。最后的数值实验验证了自适应移动网格算法的有效性,且算法的平均收敛阶可达到二阶。数值结果进一步表明自适应移动网格的误差明显小于Shishkin网格的误差,且其收敛阶也高于Shishkin网格计算得到的收敛阶。

The adaptive moving mesh algorithm plays a very important role in the numerical solution of singularly perturbed differential equations.The key technology here is the construction of an effective discrete scheme and the corresponding a posteriori error estimation.Based on this,for a class of nonlinear singularly parameterized problems,the stability estimates of continuous solutions and related corollaries are given.Then,a hybrid finite difference scheme is established by using the backward Euler formula and the first-order central finite difference scheme on an arbitrary nonuniform grid,and the stability of the discrete solution is analyzed.Based on this stability estimation and the piecewise linear interpolation technique,an a posterior error estimation of the maximum norm of the mixed finite difference scheme is given.Using the a posterior error estimation,an optimal grid monitor function is selected,and an adaptive grid generation algorithm is designed based on the mesh equidistribution principle.Finally,numerical experiments verify the effectiveness of the adaptive moving mesh algorithm,and the average convergence order of the algorithm can reach the second order.Furthermore,it is shown from the numerical results that the error of the adaptive moving mesh is obviously smaller than that of the Shishkin mesh,and its convergence order is higher than that of the Shishkin mesh.