The accuracy of flood forecasts generated using spatially lumped hydrological models can be severely affected by errors in the estimates of areal mean rainfall. The quality of the latter depends both on the size and type of errors in point-based rainfall measurements, and on the density and spatial arrangement of rain gauges in the basin. Here we use error feedback correction, based on the dynamic system response curve (DSRC) method, to compute updated estimates of the rainfall inputs. The method is evaluated via synthetic and real-data cases, showing that the method works as theoretically expected. The ability of the method to improve the accuracy of real-time flood forecasts is then demonstrated using 20 basins of different sizes and having different rain gauge densities. We find that the degree of forecast improvement is more significant for larger basins and for basins with lower rain gauge density. The method is relatively simple to apply and can improve the accuracy and stability of real-time model predictions without increasing either model complexity and/or the number of model parameters.

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The fractional reaction-diffusion equations play an important role in dynamical systems. Indeed, it is time consuming to numerically solve differential fractional diffusion equations. In this paper, we present a parallel algorithm for the Riesz space fractional diffusion equation. The parallel algorithm, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel matrix vector multiplication and vector vector addition. As to the authors' knowledge, this is the first parallel algorithm for the Riesz space fractional reaction-diffusion equation. The experimental results show that the parallel algorithm is as accurate as the serial algorithm. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.3-3.4 times faster than the serial algorithm on single CPU core. The parallel efficiency of 64 processes is up to 79.39% compared with 8 processes on a distributed memory cluster system. MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11, 60G22

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The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N 2 M) compared with O(NM) for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.

The dynamic system response curve (DSRC) method has been shown to effectively use error feedback correction to obtain updated areal estimates of mean rainfall and thereby improve the accuracy of real‐time flood forecasts. In this study, we address two main shortcomings of the existing method. First, ridge estimation is used to deal with ill‐conditioning of the normal equation coefficient matrix when the method is applied to small basins, or when the length of updating rainfall series is short. Second, the effects of spatial heterogeneity of rainfall on rainfall error estimates are accounted for using a simple index. The improved performance of the method is demonstrated using both synthetic and real data studies. For smaller basins with relatively homogeneous spatial distributions of rainfall, the use of ridge regression provides more accurate and robust results. For larger‐scale basins with significant spatial heterogeneity of rainfall, spatial rainfall error updating provides significant improvements. Overall, combining the two strategies results in the best performance for all cases, with the effects of ridge estimation and spatially distributed updating complementing each other.