Grid Points Research Articles

Optimizing Interpolation Methods and Point Distances for Accurate Earthquake Hazard Mapping

Earthquake hazard mapping assesses and visualizes seismic hazards in a region using data from specific points. Conducting a seismic hazard analysis for each point is essential, while continuous assessment for all points is impractical. The practical approach involves identifying hazards at specific points and utilizing interpolation for the rest. This method considers grid point spacing and chooses the right interpolation technique for estimating hazards at other points. This article examines different point distances and interpolation methods through a case study. To gauge accuracy, it tests 15 point distances and employs two interpolation methods, inverse distance weighted and ordinary kriging. Point distances are chosen as a percentage of longitude and latitude, ranging from 0.02 to 0.3. A baseline distance of 0.02 is set, and other distances and interpolation methods are compared with it. Five statistical indicators assess the methods. Ordinary kriging interpolation shows greater accuracy. With error rates and hazard map similarities in mind, a distance of 0.14 points seems optimal, balancing computational time and accuracy needs. Based on the research findings, this approach offers a cost-effective method for creating seismic hazard maps. It enables informed risk assessments for structures spanning various geographic areas, like linear infrastructures.

Investigation of free vibration of piezoelectric actuator reinforced with functionally graded Boron Nitride nanotube using timoshenko beam model and differential quadrature method

The vibrational characteristics and natural frequencies of Boron Nitride Nanotube (BNNT)-reinforced piezoelectric beams, based on the Timoshenko model, have been studied for the first time in this paper. Four types of functionally graded (FG) distribution patterns of BNNT reinforcements were considered for the FG-BNNT-reinforced piezoelectric beam. It is assumed that the material properties of FG-BNNT-reinforced beams are estimated using the Representative Volume Element (RVE) micromechanical model. The FG-BNNT-reinforced piezoelectric beam with different Boundary Conditions (BCs) is exposed to electrical voltage. Coupled partial differential equations of the system have been derived with the aid of Hamilton's principle and then converted into corresponding dimensionless equations using dimensionless variables. The differential quadrature method (DQM) has been used to obtain the natural frequencies and shape modes. According to the convergence analysis, twelve grid points were enough to obtain an accurate result for the analysis. To confirm the accuracy and reliability of the present study, the obtained results were validated with frequencies from the literature. Also, the effects of BNNT volume fraction and its reinforcement pattern, as well as applied voltage, on dimensionless natural frequencies have been investigated for different BCs. The results show that applying electric loadings can be utilized as well as controlling parameters to modify the resonance frequency of the composite actuator. More specifically, it is shown that applying voltages to a FG-BNNT-reinforced piezoelectric beam changes the natural frequency for each reinforcement pattern, BCs, and mode numbers. Such ability could be employed when designing reinforced piezoelectric composites for vibrational applications.

Tropical cyclone signatures in SAR ocean radial Doppler Velocity

Ocean surface radial Doppler Velocity (DV) signatures of Tropical Cyclones (TC) at moderate incidence angles are analyzed using a semi-empirical DV model. This model, originally named KaDOP, is based on the physics of Ka-band Doppler radar backscattering from the sea surface and represents the DV as a sum of components due to surface currents, long surface waves, Bragg scattering, and wave breaking. The latter two components were modified in this study to fit the model to C-band observations at strong winds. This modified model, referred to as a TC-DOP, requires several environmental input parameters, including TC winds and translation velocity, waves, and currents. For known winds and TC translation velocity, the surface currents are simulated using a model of the upper ocean response to TC passage. Waves are modeled using a self-similar description for the energy-containing TC-generated local wind waves and emanating swell. Given known winds, waves, and currents, the DV is calculated by the TC-DOP independently at each grid point. Calculations are performed for scanner-like observations that detect only the radial (cross-track) DV component. It is shown that at storm-strength winds and radar incidence angle, θ=45°, the main contribution to the total DV is provided by the surface current and wave breaking DV components (∼1 m s−1). The latter component has strong azimuth asymmetry and maximizes in the upwind sector. Next in magnitude is the contribution from wind waves (∼0.5 m s−1), followed by Bragg waves (∼0.2 m s−1), while the contribution from swell is less important. Comparisons of the TC-DOP calculations with the C-band Sentinel-1A SAR radial DV data show qualitatively good agreement. All observed radial velocity images in TC storm areas have a dipole-like feature similar to that predicted by the simulations, with a zero DV area roughly aligned with the cross-wind direction. The TC-DOP model prediction is found generally consistent with calculations utilizing an empirical geophysical model function (known as the CDOP). The residual differences are attributed to specific features of radar scattering mechanisms in different radar wavelength bands (Ka- vs C-band), as well as possible shortcomings of the Ka-band Modulation Transfer Function (MTF) originally derived from platform-based measurements.

Sea surface temperature prediction by stacked generalization ensemble of deep learning

Accurate sea surface temperature (SST) prediction is of great significance for fishery farming, marine ecological protection, and planning of maritime activities. In this paper, the stacked generalization ensemble is demonstrated for SST prediction to improve every single deep-learning model. Long-term high-resolution satellite-derived SST is used with the sub-regions of the Taiwan Strait and East China Sea taken as the study area. We select the Multilayer Perceptron, Long short-term memory (LSTM), Convolutional Neural Networks (CNN), CNN-LSTM as individual learners, and Convolutional LSTM as the meta-learner. The individual learners are trained and validated on the retained data subset I, while the meta-learner is trained and validated by constructing the samples with the predictions of validated individual learners on the retained data subset II. The two types of models are evaluated on the same test dataset with root-mean-square error and coefficient of efficiency as the scoring criteria. We find that the meta-model outperforms any individual model and other baselines for the one-day-/three-day-ahead forecasts in the Taiwan Strait and one-day-/three-day-/five-day-ahead predictions in the East China Sea. Furthermore, when the lead time is 1 day and 3 days, the meta-model has a better spatial distribution of prediction metrics across all grid points in the Taiwan Strait sub-area. For the East China Sea sub-region, the meta-model advantage is extended to the lead time of 5 days. Probably due to the higher quality of offshore satellite data, the prediction ability enhancement of the stacked ensemble applied in the East China Sea is better than that in the Taiwan Strait. The better-performing meta-model prediction suggests that the stacked generalization ensemble is encouraging and promising for improving the short-term prediction of the daily SST field.

A non-hierarchical multi-layer multi-configurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces.

The correlation discrete variable representation (CDVR) enables multi-layer multi-configurational time-dependent Hartree (MCTDH) quantum dynamics simulations on general potential energy surfaces. In a recent study [R. Ellerbrock and U. Manthe, J. Chem. Phys. 156, 134107 (2022)], an improved CDVR that can account for the symmetry properties of a tree-shaped wavefunction representation has been introduced. This non-hierarchical CDVR drastically reduces the number of grid points required in the time-dependent quadrature used to evaluate all potential energy matrix elements. While the first studies on the non-hierarchical CDVR approach have been restricted to single-layer calculations, here the complete theory required for the implementation of the non-hierarchical CDVR approach in the multi-layer MCTDH context will be presented. Detailed equations facilitating the efficient recursive computation of all matrix elements are derived, and a new notation adapted to the symmetry properties of the tree-shaped representation is introduced. Calculations studying the non-adiabatic quantum dynamics of photoexcited pyrazine in 24 dimensions illustrate the properties of the non-hierarchical multi-layer CDVR.

Regional ionospheric correction generation for GNSS PPP-RTK: theoretical analyses and a new interpolation method

The regional ionospheric corrections have become one of the critical parts of Precise Point Positioning (PPP)-based Real-Time-Kinematic (RTK) services to achieve fast positioning convergence. Several methods for regional ionospheric corrections supporting PPP-RTK have been developed and implemented over recent years, but little attention is given to the theoretical foundation of existing ionospheric correction methods and their performance comparison to find an optimal method in some sense. The optimality criterion of such methods should not only be based on the precision of the ionospheric correction itself, but also on its broadcasting strategies, and implementation aspects. This contribution studies ionospheric correction generation methods within the best linear unbiased predictor (BLUP) framework. Comparing the accuracy performances of the methods, we demonstrate that the Kriging method with trend, as a special case of BLUP, is the most appropriate method for large-scale networks (above 500 km). A strategy for the evaluation of the uncertainty of the grid-interpolated ionospheric corrections is also developed. In contrast to other empirical methods, this new method is rigorous in the sense that it avoids the underestimation of the uncertainty of predicted ionospheric corrections, especially when reference stations are close to a grid point.

A deep learning operator-based numerical scheme method for solving 1D wave equations

Abstract In this paper, we introduce the deep numerical technique DeepNM, which is designed for solving one-dimensional (1D) hyperbolic conservation laws, particularly wave equations. By creatively integrating traditional numerical schemes with deep learning techniques, the method yields improvements over conventional approaches. Specifically, we compare this approach against two established classical numerical methods: the discontinuous Galerkin method (DG) and the Lax–Wendroff correction method (LWC). While maintaining a comparable level of accuracy, DeepNM significantly improves computational speed, surpassing conventional numerical methods in this aspect by more than tenfold, and reducing storage requirements by over 1000 times. Furthermore, DeepNM facilitates the utilization of higher-order numerical schemes and allows for an increased number of grid points, thereby enhancing precision. In contrast to the more prevalent PINN method, DeepNM optimally combines the strengths of conventional mathematical techniques with deep learning, resulting in heightened accuracy and expedited computations for solving partial differential equations. Notably, DeepNM introduces a novel research paradigm for numerical equation-solving that can be seamlessly integrated with various traditional numerical methods.

Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows

In numerical simulations, achieving high accuracy without significantly increasing computational cost is often a challenge. To address this issue, this paper proposes an improved finite volume Weighted Essentially Non-Oscillatory (WENO) scheme for structured grids. By employing a single-point quadrature rule to perform flux integration on the control volume faces, this scheme is designed for use in NUAA-Turbo three-dimensional fluid solvers based on structured grids, utilizing RANS and RANS/LES coupling to simulate turbomachinery flows. Firstly, the new WENO scheme is validated against classical numerical test cases to evaluate its stability and reliability in handling discontinuities, double Mach reflection problems, and Rayleigh–Taylor (RT) instability. Compared to the original scheme, this improved finite-volume WENO scheme demonstrates better stability near discontinuities and more effectively resolves flow features at the same grid resolution. Next, for engineering applications related to turbomachinery, such as compressor and turbine characteristics, calculations using RANS are performed and the results obtained with WENO-ZQ3 and WENO-JS3 are compared. Finally, the new fifth-order WENO scheme is applied to RANS/LES coupling simulations of turbine wake and film cooling. The results indicate that the improved finite-volume WENO scheme provides better stability and accuracy in engineering applications. For instance, the average error in calculating compressor efficiency characteristics is reduced from 0.76% to 0.05%, the error in turbine vane pressure distribution compared to the experimental values is within 1%, and the error in film cooling efficiency centerline distribution compared to the experimental values is within 3%. Additionally, the qualitative results of turbine wake and film cooling show that even with a small number of grid points, more detailed flow physics can be captured, thereby reducing computational costs in aerodynamic applications.

The influence of water vapour on winter humidity‐related extreme cold events in China

AbstractWinter extreme cold events (ECEs) have profound impacts on human health, transportation, energy consumption and the economy, particularly associated with high‐humidity conditions. Recent research has indicated a growing interest in extreme temperature events that account for the presence of water vapour. However, the characteristics of humidity‐related ECEs (HECEs) remain unclear. Given the rise in atmospheric moisture due to global warming, understanding the influence of water vapour on HECEs becomes crucial. This study examined variations in HECEs identified using the wet‐bulb globe temperature (WBGT) in China during 1959–2020 and quantified the impacts of water vapour. The results revealed a decrease in the intensity and frequency of HECEs in China during this period, except for in northern China, where the intensity increased, which contrasts with the trends in traditional ECEs. Furthermore, strong HECEs were found to exhibit greater frequency, duration and intensity in northern China than weak HECEs. Additionally, the specific humidity for all categories of HECEs in China exhibited an upward trend, with a more rapid increase for strong HECEs. The findings indicated that the intensity of HECEs is positively correlated with the increase in water vapour. Defining a water vapour amplification factor revealed amplified intensity and duration at most grid points, particularly across northern China. Therefore, while winter warming reduces traditional ECEs, HECEs persist and even intensify as humidity increases, confirming significant amplification by water vapour. Our results emphasized the importance of increasing water vapour, driven by global warming, which may increase the risk of HECEs in the future.

Magneto Axisymmetric Vibration of FG-GPLs Reinforced Annular Sandwich Plates with an FG Porous Core Using DQM and a New Shear Deformation Theory

Based on the differential quadrature procedure (DQP), the vibrational response of functionally graded (FG) sandwich annular plates enhanced with graphene platelets (GPLs) and with an FG porous core is illustrated in this paper. The current annular plate is assumed to deform axisymmetrically and expose to a radial magnetic field. The Lorentz magnetic body force is deduced via Maxwell’s relations. The effective physical properties of the upper and lower layers of the sandwich plate are obtained by employing the Halpin–Tsai model. Our technique depends on a new four-unknown shear deformation theory to depict the displacements. In addition, the motion equations are established via Hamilton’s principle. The motion equations are solved by employing the DQP. In order to study the convergence of the DQ method, the minimum number of grid points needed for a converged solution is ascertained. In addition, the current theory’s outcomes are compared with those of previous higher-order theories. The effects of the porosity distribution type, porosity factor, GPLs distribution pattern, GPLs weight fraction, inner-to-outer radius ratio, outer radius-to-thickness ratio, magnetic field parameters, core thickness, and elastic substrate parameters on the nondimensional vibration frequencies are discussed.

Simplified Approximations of Direct Cumulus Entrainment and Detrainment

Abstract In recent years, direct calculations of simulated cumulus entrainment and detrainment have facilitated new physical insights into these highly elusive but critically important processes. However, these calculations require substantial computational resources that may limit their widespread usage. To facilitate such calculations, two simplified approximations of direct cumulus entrainment and detrainment are examined herein. The first approximation, termed the “semidirect” method, follows a standard bulk approach but makes more realistic assumptions about the sources of entrained and detrained air near the cloud edges. In contrast, the second approximation (the “projection” method) uses the governing equations of motion to project whether grid points near the cloud edge will entrain or detrain as the mean cloud ascends by one grid point. Verification exercises using large-eddy simulations reveal that both methods generally agree better with corresponding direct entrainment/detrainment estimates than the traditional bulk formulation, with the projection method outperforming the semidirect method. The two methods can be used in a synergistic fashion, with the semidirect method helping to optimize the projection method, to suit a wide range of applications. Because the latter incorporates the essential dynamics of entrainment and detrainment at the local scale, it can be used to gain physical insight into the causal mechanisms regulating these complex processes.

High-efficiency sound source localization using data-driven sparse sampling with validation using monopole laser sound source

This study proposes a framework that reduces the calculation cost of sound source localization with the Amiet model, using a data-driven sparse sampling method. This method accelerates the calculation of the steering vector used in conventional beamforming. An aeroacoustic wind tunnel test was conducted in a 2 × 2 m2 low-speed wind tunnel, and the proposed method was verified. During the test, a monopole laser sound source, which does not interfere with the flow, was used, and its acoustic signals were measured using a microphone array. Next, steering vectors were reconstructed by discovering dominant modes and optimized sampling points from the training data based on the Amiet model and the modified data-driven sparse sampling method. Finally, the sound-source positions when the steering vector of the proposed model was used were compared with the positions observed when the steering vector of which all the grid points were calculated was used. The error was less than 2 mm when 16 modes were used, and the calculation time was reduced to ∼1/33 of that of the previous Amiet model.

Lattice Boltzmann simulation for wake interactions of aligned wind turbines using actuator line model with turbine control

A wind turbine wake causes a decrease in wind speed and an increase in turbulence intensity. The wind turbine wake interaction is essential for predicting the power output of a wind farm consisting of many wind turbines. This research proposes a CFD method able to reproduce wake interactions and power outputs of multiple wind turbines with high speed and accuracy. Large eddy simulations with the lattice Boltzmann method are used for fluid calculations, specifically for large-scale CFD simulations. The wind turbines are represented using an actuator line model. Optimal power generation efficiency is achieved by controlling the rotor speed and blade pitch angle. Large-scale simulations of eight aligned wind turbines are conducted using 1.75 billion grid points and 40 GPUs. We compare two cases with and without control to investigate the effect of turbine control on wake and power output. Both the instantaneous and mean streamwise velocities confirm that the turbine control reduces the wake velocity deficit of the downwind wind turbine. High-speed inflow of wind to the downstream turbines augments their power output. With implementation of turbine control, the power outputs of the downstream turbines agree well with the observation data obtained in an earlier study. The results demonstrate the importance of controlling the rotational speed and pitch angle for actuator line simulations.

Quantized tensor networks for solving the Vlasov–Maxwell equations

The Vlasov–Maxwell equations provide an ab initio description of collisionless plasmas, but solving them is often impractical because of the wide range of spatial and temporal scales that must be resolved and the high dimensionality of the problem. In this work, we present a quantum-inspired semi-implicit Vlasov–Maxwell solver that uses the quantized tensor network (QTN) framework. With this QTN solver, the cost of grid-based numerical simulation of size $N$ is reduced from $O(N)$ to $O(\text {poly}(D))$ , where $D$ is the ‘rank’ or ‘bond dimension’ of the QTN and is typically set to be much smaller than $N$ . We find that for the five-dimensional test problems considered here, a modest $D=64$ appears to be sufficient for capturing the expected physics despite the simulations using a total of $N=2^{36}$ grid points, which would require $D=2^{18}$ for full-rank calculations. Additionally, we observe that a QTN time evolution scheme based on the Dirac–Frenkel variational principle allows one to use somewhat larger time steps than prescribed by the Courant–Friedrichs–Lewy constraint. As such, this work demonstrates that the QTN format is a promising means of approximately solving the Vlasov–Maxwell equations with significantly reduced cost.

RaFSIP: Parameterizing Ice Multiplication in Models Using a Machine Learning Approach

AbstractAccurately representing mixed‐phase clouds (MPCs) in global climate models (GCMs) is critical for capturing climate sensitivity and Arctic amplification. Secondary ice production (SIP), can significantly increase ice crystal number concentration (ICNC) in MPCs, affecting cloud properties and processes. Here, we introduce a machine‐learning (ML) approach, called Random Forest SIP (RaFSIP), to parameterize SIP in stratiform MPCs. RaFSIP is trained on 16 grid points with 10‐km horizontal spacing derived from a 2‐year simulation with the Weather Research and Forecasting (WRF) model, including explicit SIP microphysics. Designed for a temperature range of 0 to −25°C, RaFSIP simplifies the description of rime splintering, ice‐ice collisional break‐up, and droplet‐shattering using only a limited set of inputs. RaFSIP was evaluated offline before being integrated into WRF, demonstrating its stable online performance in a 1‐year simulation keeping the same model setup as during training. Even when coupled with the 50‐km grid spacing domain of WRF, RaFSIP reproduces ICNC predictions within a factor of 3 when compared to simulations with explicit SIP microphysics. The coupled WRF‐RaFSIP scheme replicates regions of enhanced SIP and accurately maps ICNCs and liquid water content, particularly at temperatures above −10°C. Uncertainties in RaFSIP minimally impact surface cloud radiative forcing in the Arctic, resulting in radiative biases under 3 Wm−2 compared to simulations with detailed microphysics. Although the performance of RaFSIP in convective clouds remains untested, its adaptable nature allows for data set augmentation to address this aspect. This framework opens possibilities for GCM simplification and process description through physics‐guided ML algorithms.

Numerical study of flow past two square cylinders with horizontal detached control rod through passive control method

A two dimensional (2-D) numerical study is conducted through the lattice Boltzmann method for flow past two square cylinders with a detached control rod placed in the horizontal position of the channel. The range of Reynolds number (Re) is fixed at Re = 150, with the length (l) of the control rod varying from l = 0.1d–21d. First, we check the accuracy of grid points and the validity of the code for present problems by comparing the results with already available data in the literature. After that, we discussed the obtained results in terms of vorticity contour, drag and lift coefficients, and force statistics. In our study of flow structure mechanisms, we examined four types of flow regimes: a small length control rod, a small and moderate length control rod, a moderate length control rod, and a larger length control rod according to flow behavior. In force statistics, we calculated the values of CDmean, CDrms, CLrms, CDamp, CLamp, and St, respectively. The mean drag coefficient contains the smallest values at the largest length, i.e., l = 19 and 21d. The maximum value of CDmean is examined at the first cylinder (C1) as compared to the second cylinder (C2) at l = 2.25d, and it is 1.2789. The values of CDmean for C2 at the length of the control rod, l = 0.1d–1.25d, and at l = 21d are negative due to the effect of thrust. The value of the Strouhal number is the same for both cylinders C1 and C2, and its highest value is 0.1036. The pressure amplitude of the drag coefficient is maximum for C1, while the pressure amplitude of the lift coefficient is maximum for C2, and its value is 2.1569.

A bidirectional power converter connecting electric vehicle battery and DC microgrid

One way to increase electric vehicle (EV) battery utilization is to connect it to a dc microgrid. The EV battery can assume the role of an energy storage from the grid point of view. A bidirectional DC-DC converter will be needed to transfer power between them back and forth. This paper proposes the converter design considering its functional objective, including interleaved phase number determination. Efficiency performance evaluation is presented by power loss analysis with the parasitic-parameters consideration of the components. Finding optimum switching frequency based on power loss analysis is performed independently between input and output sides of the converter. Finally, experiments using a scaled-down prototype are shown to verify the analytical analysis of the converter. The experimental results properly validate the power loss analytic analysis carried out in this paper with a maximum error of 2.04% at 1131-watt, 60 V battery voltage, and 140 V grid voltage. Maximum efficiency 96.97% is obtained at 301-watt, 130 V battery voltage, and 151 V grid voltage. Overall, the converter has a simple structure, capable to be operated in various levels of input and output voltages with a minimum battery side current ripple.

Two-Stage Path Planning for Long-Distance Off-Road Path Planning Based on Terrain Data

In the face of increasing demands for tasks such as mountain rescue, geological exploration, and military operations in complex wilderness environments, planning an efficient walking route is crucial. To address the inefficiency of traditional two-dimensional path planning, this paper proposes a two-stage path planning algorithm. First, an improved Probabilistic Roadmap (PRM) algorithm is used to quickly and roughly determine the initial path. Then, the morphological dilation is applied to process the grid points of the initial path, retaining the surrounding area of the initial path for a precise positioning of the search range. Finally, the idea of the A∗ algorithm is applied to achieve precise path planning in the refined search range. During the process of constructing the topology map, we utilized parallelization acceleration strategies to expedite the graph construction. In order to verify the effectiveness of the algorithm, we used terrain data to construct a wilderness environment model, and tests were conducted on off-road path planning tasks with different terrains and distances. The experimental results show a substantial enhancement in the computational efficiency of the proposed algorithm relative to the conventional A∗ algorithm by 30 to 60 times.

Monitoring spatiotemporal changes in urban flood vulnerability of Peninsular Malaysia from satellite nighttime light data

Urban flood vulnerability monitoring requires a large amount of socioeconomic and environmental data collected at regular time intervals. However, collecting such a large volume of data poses a significant constraint in assessing changes in flood vulnerability. This study proposed a novel method to monitor spatiotemporal changes in urban flood vulnerability from satellite nighttime light (NTL) data. Peninsular Malaysia was chosen as the research region as floods are the most devastating and recurrent phenomena in the region. The study developed a flood vulnerability index (FVI) based on socioeconomic and environmental data from a single year. This FVI was then linked to NTL data using an Adaptive neuro-fuzzy inference system (ANFIS) machine learning algorithm. The model was calibrated and validated with administrative unit scale data and subsequently used to predict FVI at a spatial resolution of 10 km for 2000–2018 using NTL data. Finally, changes in estimated FVI at different grid points were evaluated using the Mann-Kendall trend method to determine changes in flood vulnerability over time and space. Results showed a nonlinear relationship between NTL and flood vulnerability factors such as population density, Gini coefficient, and percentage of foreign nationals. The ANFIS technique performed well in estimating FVI from NTL data with a normalized root-mean-square error of 0.68 and Kling-Gupta Efficiency of 0.73. The FVI revealed a high vulnerability in the urbanized western coastal region (FVI ∼ 0.5 to 0.54), which matches well with major contributing regions to flood losses in Peninsular Malaysia. Trend assessment showed a significant increase in flood vulnerability in the study area from 2000 to 2018. The spatial distribution of the trend indicated an increase in FVI in the urbanized coastal plains, particularly in rapidly developing western and southern urban regions. The results indicate the potential of the technique in urban flood vulnerability assessment using freely available satellite NTL data.

The crucial role of Lagrange multipliers in a space-time symmetry preserving discretization scheme for IVPs

In a recently developed variational discretization scheme for second order initial value problems [1], it was shown that the Noether charge associated with time translation symmetry is exactly preserved in the interior of the simulated domain. The obtained solution also fulfils the naively discretized equations of motions inside the domain, except for the last two grid points. Here we provide an explanation for the deviations at the boundary as stemming from the effect of Lagrange multipliers used to implement initial and connecting conditions. We show explicitly that the Noether charge including the boundary corrections is exactly preserved at its continuum value over the whole simulation domain, including the boundary points.