In this paper, an adaptive grid method for singularly perturbed Volterra integro-differential equations exhibiting a boundary layer is studied. At first, based on an arbitrary nonuniform mesh, we use the classical backward Euler formula to discrete the first-order derivative part and the left rectangle formula to approximate the integral part, respectively. Meanwhile, it is shown from the mesh equidistribution principle and the truncation error analysis that the presented adaptive grid method is first-order uniformly convergent with respect to the perturbation parameter. Furthermore, the Richardson extrapolation technique is utilized to improve the uniform accuracy of the presented adaptive grid method in the discrete supremum norm form $O(N^{-1})$ to $O(N^{-2}),$ where $N$ is the mesh parameter. Finally, some numerical results are given to support the theoretical results.

In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.

Parameters estimation of Muskingum model is very significative in both exploitation and utilization of water resources and hydrological forecasting. The optimal results of parameters directly affect the accuracy of flood forecasting. This paper considers the parameters estimation problem of Muskingum model from the following two aspects. Firstly, based on the general trapezoid formulas, a class of new discretization methods including a parameter θ to approximate Muskingum model is presented. The accuracy of these methods is second‐order, when θ ≠ 1/3. Particularly, if we choose θ = 1/3, the accuracy of the presented method can be improved to third‐order. Secondly, according to the Newton‐type trust region algorithm, a new Newton‐type trust region algorithm is given to obtain the parameters of Muskingum model. This method can avoid high dependence on the initial parameters. The average absolute errors (AAE) and the average relative errors (ARE) of the proposed algorithm of parameters estimation for Muskingum model are 8.208122 and 2.462438%, respectively, where θ = 1/3. It is shown from these results that the presented algorithm has higher forecasting accuracy and wider practicability than other methods.