Numerical daemons of hydrological models are summoned by extreme precipitation

Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation such as that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation. In this experiment, a large number of hydrographs are generated with the modular modeling framework FUSE (Framework for Understanding Structural Errors), using eight numerical techniques across a variety of forcing data sets. All constructed models are conceptual and lumped. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational cost and numerical error associated with each hydrograph were recorded. Numerical error is assessed via root mean square error and normalized root mean square error. It was found that the root mean square error usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both a small numerical error and a low computational cost. A small literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be suboptimal. We conclude that relatively large numerical errors may be common in current models, highlighting the need for robust numerical techniques, in particular in the face of increasing precipitation extremes.

Exact Analysis of the Finite Precision Error Generated in Important Chaotic Maps and Complete Numerical Remedy of These Schemes

A first aim of the present work is the determination of the actual sources of the “finite precision error” generation and accumulation in two important algorithms: Bernoulli’s map and the folded Baker’s map. These two computational schemes attract the attention of a growing number of researchers, in connection with a wide range of applications. However, both Bernoulli’s and Baker’s maps, when implemented in a contemporary computing machine, suffer from a very serious numerical error due to the finite word length. This error, causally, causes a failure of these two algorithms after a relatively very small number of iterations. In the present manuscript, novel methods for eliminating this numerical error are presented. In fact, the introduced approach succeeds in executing the Bernoulli’s map and the folded Baker’s map in a computing machine for many hundreds of thousands of iterations, offering results practically free of finite precision error. These successful techniques are based on the determination and understanding of the substantial sources of finite precision (round-off) error, which is generated and accumulated in these two important chaotic maps.

Metrics for Performance Quantification of Adaptive Mesh Refinement

Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for tsunami, hurricane or typhoon prediction. Using the example of a shallow water solver, this study explores a set of metrics as a tool to distinguish the performance of numerical methods using adaptively refined versus uniform meshes independent of computational architecture or implementation. These metrics allow us to quantify how a numerical simulation benefits from the use of adaptive mesh refinement. The type of meshes we are focusing on are adaptive triangular meshes that are non-uniform and structured. Refinement is controlled by physics-based indicators that capture relevant physical processes and determine the areas of mesh refinement and coarsening. The proposed performance metrics take into account a number of characteristics of numerical simulations such as numerical errors, spatial resolution, as well as computing time. Using a number of test cases we demonstrate that correlating different quantities offers insight into computational overhead, the distribution of numerical error across various mesh resolutions as well as the evolution of numerical error and run-time per degree of freedom.

Numerical daemons of hydrological models are summoned by extreme precipitation

Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on approximate numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation like that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation events. In this experiment, a large number of hydrographs is generated with the modular modeling framework FUSE, using eight numerical techniques across a variety of forcing datasets. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational expense and numerical error associated with each hydrograph were recorded. It was found that numerical error (root mean square error) usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both low numerical error and low computational cost. A basic literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be sub-optimal. We conclude that relatively large numerical errors might be common in current models, and because these will likely become larger as the climate changes, we advocate for the use of low cost, low error numerical methods.

Getchell’s method for conversion of Cartesian-geocentric to geodetic coordinates – Its properties and Newtonian alternative

The paper concentrates on the iterative Getchell’s method (formulated in 1972) and its alternative Newtonian implementation for conversion of Cartesian geocentric coordinates into geodetic coordinates. The same basic equation formulated in the Getchell’s method is used in both cases. The equation has a stable form in the whole range of argument (latitude) variation \langle -\pi /2,\pi /2\rangle . The original Getchell’s method (somehow “forgotten”) has a simple geometric interpretation and its applications turn out to be particularly effective. Many studies on iterative algorithms usually omit theoretical proofs of convergence replacing them with conclusions based on numerical examples. This paper presents theoretical proofs of algorithms convergence both for the Getchell’s method and the Newton procedure. The convergence parameter and numerical error of results were estimated in each case. Numerical tests were carried out for a set of points distributed on the Earth’s space, also for extreme h values. For typical practical applications of the Getchell’s method, sufficiently accurate results are obtained after 1–3 iterations, while in the Newton procedure already after one iteration, assuming the same numerical error and initial conditions. The accuracy of the geodetic coordinates determinations meets all practical requirements with some margin. For example an absolute numerical error for latitude is approx. 0.4\cdot {10^{-13}} [rad] i. e. about 0.00026 mm in the length of the meridian arc. The proposed methods were compared with other methods (algorithms), including in terms of stability and non-singularity in the entire usable space of the Earth, but excluding the near geocenter, which has no practical significance. Both the modification of the Getchell method and its Newtonian alternative are very good determined in this area (in the Earth’s poles, the final solution is directly the starting value of iterative algorithms). The discussed algorithms were implemented in the form of procedures in DELPHI language.