Computing water flow through complex landscapes – Part 3: Fill–Spill–Merge: flow routing in depression hierarchies

Depressions – inwardly draining regions – are common to many landscapes. When there is sufficient moisture, depressions take the form of lakes and wetlands; otherwise, they may be dry. Hydrological flow models used in geomorphology, hydrology, planetary science, soil and water conservation, and other fields often eliminate depressions through filling or breaching; however, this can produce unrealistic results. Models that retain depressions, on the other hand, are often undesirably expensive to run. In previous work we began to address this by developing a depression hierarchy data structure to capture the full topographic complexity of depressions in a region. Here, we extend this work by presenting the Fill–Spill–Merge algorithm that utilizes our depression hierarchy data structure to rapidly process and distribute runoff. Runoff fills depressions, which then overflow and spill into their neighbors. If both a depression and its neighbor fill, they merge. We provide a detailed explanation of the algorithm and results from two sample study areas. In these case studies, the algorithm runs 90–2600 times faster (with a reduction in compute time of 2000–63 000 times) than the commonly used Jacobi iteration and produces a more accurate output. Complete, well-commented, open-source code with 97 % test coverage is available on GitHub and Zenodo.

Wavelet Element Modelling for Inviscid Fluid–Solid Coupling Problem based on Partitioned Approach

To provide a simple numerical formulation based on fixed grids, a wavelet element method for fluid–solid modelling is introduced in this work. Compared with the classical wavelet finite element method, the presented method can potentially handle more complex shapes. Considering the differences between the solid and fluid regions, a damping-like interface based on wavelet elements is designed, in order to ensure consistency between the two parts. The inner regions are constructed with the same wavelet function in space. In the time and spatial domains, a partitioned approach based on Jacobi iteration is combined with the pseudo-parallel calculation method. Numerical convergence analyses show that the method can serve as an alternative choice for fluid–solid coupling modelling.

Perbandingan Metode Iterasi Jacobi dan Metode Iterasi Gauss-Seidel dalam Menyelesaikan Sistem Persamaan Linear Fuzzy

Penelitian ini mengkaji tentang menyelesaian Sistem Persamaan Linear Fuzzy dengan Membanding kan Metode Iterasi Jacobi dan Metode Iterasi Gauss-Seidel. Metode iterasi Jacobi merupakan salah satu metode tak langsung, yang bermula dari suatu hampiran Metode iterasi Jacobi ini digunakan untuk menyelesaikan persamaan linier yang proporsi koefisien nol nya besar. Iterasi dapat diartikan sebagai suatu proses atau metode yang digunakan secara berulang-ulang (pengulangan) dalam menyelesaikan suatu permasalahan matematika ditulis dalam bentuk . Pada metode iterasi Gauss-Seidel, nilai-nilai yang paling akhir dihitung digunakan di dalam semua perhitungan. Jelasnya, di dalam iterasi Jacobi, menghitung dalam bentuk . Setelah mendapatkan Hasil iterasi kedua Metode tersebut maka langkah selanjutnya membandingkan kedua metode tersebut dengan melihat jumlah iterasinya dan nilai Galatnya manakah yang lebih baik dalam menyelesaikan Sistem Persamaan Linear Fuzzy.Kata kunci: Sistem Persamaan Linear Fuzzy, Metode Itersi Jacobi, Metode Iterasi Gauss-Seidel. This study examines the completion of the Linear Fuzzy Equation System by Comparing the Jacobi Iteration Method and the Gauss-Seidel Iteration Method. The Jacobi iteration method is one of the indirect methods, which stems from an almost a method of this Jacobi iteration method used to solve linear equations whose proportion of large zero coefficients. Iteration can be interpreted as a process or method used repeatedly (repetition) in solving a mathematical problem written in the form . In the Gauss-Seidel iteration method, the most recently calculated values are used in all calculations. Obviously, inside Jacobi iteration, counting in form After obtaining the result of second iteration of the Method then the next step compare both methods by seeing the number of iteration and the Error value which is better in solving Linear Fuzzy Equation System.Keywords: Linear Fuzzy Equation System, Jacobi Itersi Method, Gauss-Seidel Iteration Method.

Computing water flow through complex landscapes, Part 3: Fill-Spill-Merge: Flow routing in depression hierarchies

Depressions – inwardly-draining regions – are common to many landscapes. When there is sufficient moisture, depressions take the form of lakes and wetlands; otherwise, they may be dry. Hydrological flow models used in geomorphology, hydrology, planetary science, soil and water conservation, and other fields often eliminate depressions through filling or breaching; however, this can produce unrealistic results. Models that retain depressions, on the other hand, are often undesirably expensive to run. In previous work we began to address this by developing a depression hierarchy data structure to capture the full topographic complexity of depressions in a region. Here, we extend this work by presenting a Fill-Spill-Merge algorithm that utilizes our depression hierarchy to rapidly process and distribute runoff. Runoff fills depressions, which then overflow and spill into their neighbors. If both a depression and its neighbor fill, they merge. We provide a detailed explanation of the algorithm as well as results from two sample study areas. In these case studies, the algorithm runs 90–2600× faster (with a 2000–63 000× reduction in compute time) than the commonly-used Jacobi iteration and produces a more accurate output. Complete, well-commented, open-source code is available on Github and Zenodo.