Fine Grid Spacing Research Articles

Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient

In this paper, a fully discrete finite element scheme is established with L1 approximation for solving the nonlinear time-fractional wave equation with a variable coefficient. The stability of the scheme is analyzed and the optimal error estimates are derived. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete finite element scheme. With the proposed technique, a small-scale nonlinear problem is calculated by iteration in a coarse-grid space with mesh size H, and then a linearized problem is solved in a fine-grid space with mesh size h, H≫h. This technique not only maintains the numerical precision, but also saves a lot of computing time. The stability and optimal convergence order are considered. Using the Taylor expansion, the second two-grid algorithm is constructed to improve the optimal convergence order. The similar properties are discussed in detail. Numerical experiments are provided to illustrate the theoretical analysis and test the performance of the proposed algorithms.

Greenland ice sheet rainfall climatology, extremes and atmospheric river rapids

AbstractGreenland rainfall has come into focus as a climate change indicator and from a variety of emerging cryospheric impacts. This study first evaluates rainfall in five state‐of‐the‐art numerical prediction systems (NPSs) (CARRA, ERA5, NHM‐SMAP, RACMO, MAR) using in situ rainfall data from two regions spanning from land onto the ice sheet. The new EU Copernicus Climate Change Service (C3S) Arctic Regional ReAnalysis (CARRA), with a relatively fine (2.5 km) horizontal grid spacing and extensive within‐model‐domain observational initialization, has the lowest average bias and highest explained variance relative to the field data. ERA5 inland wet bias versus CARRA is consistent with the field data and other research and is presumably due to more ERA5 topographic smoothing. A CARRA climatology 1991–2021 has rainfall increasing by more than one‐third for the ice sheet and its peripheral ice masses. CARRA and in situ data illuminate extreme (above 300 mm per day) local rainfall episodes. A detailed examination CARRA data reveals the interplay of mass conservation that splits flow around southern Greenland and condensational buoyancy generation that maintains along‐flow updraft ‘rapids’ 2 km above sea level, which produce rain bands within an atmospheric river interacting with Greenland. CARRA resolves gravity wave oscillations that initiate as a result of buoyancy offshore, which then amplify from terrain‐forced uplift. In a detailed case study, CARRA resolves orographic intensification of rainfall by up to a factor of four, which is consistent with the field data.

Multi-adaptive spatial discretization of bond-based peridynamics

Peridynamic (PD) models are commonly implemented by exploiting a particle-based method referred to as standard scheme. Compared to numerical methods based on classical theories (e.g., the finite element method), PD models using the meshfree standard scheme are typically computationally more expensive mainly for two reasons. First, the nonlocal nature of PD requires advanced quadrature schemes. Second, non-uniform discretizations of the standard scheme are inaccurate and thus typically avoided. Hence, very fine uniform discretizations are applied in the whole domain even in cases where a fine resolution is per se required only in a small part of it (e.g., close to discontinuities and interfaces). In the present study, a new framework is devised to enhance the computational performance of PD models substantially. It applies the standard scheme only to localized regions where discontinuities and interfaces emerge, and a less demanding quadrature scheme to the rest of the domain. Moreover, it uses a multi-grid approach with a fine grid spacing only in critical regions. Because these regions are identified dynamically over time, our framework is referred to as multi-adaptive. The performance of the proposed approach is examined by means of two real-world problems, the Kalthoff–Winkler experiment and the bio-degradation of a magnesium-based bone implant screw. It is demonstrated that our novel framework can vastly reduce the computational cost (for given accuracy requirements) compared to a simple application of the standard scheme.

DownScaleBench for developing and applying a deep learning based urban climate downscaling- first results for high-resolution urban precipitation climatology over Austin, Texas

Cities need climate information to develop resilient infrastructure and for adaptation decisions. The information desired is at the order of magnitudes finer scales relative to what is typically available from climate analysis and future projections. Urban downscaling refers to developing such climate information at the city (order of 1 – 10 km) and neighborhood (order of 0.1 – 1 km) resolutions from coarser climate products. Developing these higher resolution (finer grid spacing) data needed for assessments typically covering multiyear climatology of past data and future projections is complex and computationally expensive for traditional physics-based dynamical models. In this study, we develop and adopt a novel approach for urban downscaling by generating a general-purpose operator using deep learning. This ‘DownScaleBench’ tool can aid the process of downscaling to any location. The DownScaleBench has been generalized for both in situ (ground- based) and satellite or reanalysis gridded data. The algorithm employs an iterative super-resolution convolutional neural network (Iterative SRCNN) over the city. We apply this for the development of a high-resolution gridded precipitation product (300 m) from a relatively coarse (10 km) satellite-based product (JAXA GsMAP). The high-resolution gridded precipitation datasets is compared against insitu observations for past heavy rain events over Austin, Texas, and shows marked improvement relative to the coarser datasets relative to cubic interpolation as a baseline. The creation of this Downscaling Bench has implications for generating high-resolution gridded urban meteorological datasets and aiding the planning process for climate-ready cities.

SPHERA, a new convection‐permitting regional reanalysis over Italy: Improving the description of heavy rainfall

AbstractRegional reanalyses allow us to better describe weather patterns related to rapidly evolving high‐impact events thanks to substantially finer detailing than global datasets. However, most regional datasets still do not permit the explicit representation of deep convection. SPHERA (High rEsolution ReAnalysis over Italy) is a new high‐resolution convection‐permitting reanalysis centred over Italy. It covers 26 years (1995–2020), is based on the non‐hydrostatic limited‐area model COSMO, and is produced by dynamically downscaling the global reanalysis ERA5. A nudging data assimilation scheme steers the model toward observations. The fine horizontal grid spacing of 2.2 km allows us to switch off deep‐convection parametrization. This study reports the added value of SPHERA over ERA5 in representing rainfall over Italy, particularly for severe precipitation, using rain‐gauge observations during 2003–2017 as reference. Concerning the 95th percentile of spatial rainfall distributions, ERA5 presents dry estimates with biases reaching −12 mm·day−1 over mountainous regions. At the same time, the enhanced locally driven effects of SPHERA produce seasonal biases ranging from wet in JJA (up to +12 mm·day−1) to dry in DJF (down to −9 mm·day−1). For daily maximum rates, the regional reanalysis shows better skill in detecting occurred events (with hit rates higher than ERA5 by roughly 0.4 points in the range of 15–80 mm·day−1) and frequency biases closer to 0 at all intensities when coming to daily averages. Similarly, for hourly maximum accumulations, improved adherence to observations is detected for SPHERA at all intensities, conversely to the underprediction of the global driver (with frequency biases <1 starting from 1.5 mm·hr−1). Additionally, the analyses of two specific events reveal the enhancements of SPHERA in simulating extreme precipitation, with a maximum intensity underestimation on the order of 24% versus the 73% detected for ERA5. Further improvements include the spatial detailing, timing, and temporal evolution of the events.

ICON-Sapphire: simulating the components of the Earth system and their interactions at kilometer and subkilometer scales

Abstract. State-of-the-art Earth system models typically employ grid spacings of O(100 km), which is too coarse to explicitly resolve main drivers of the flow of energy and matter across the Earth system. In this paper, we present the new ICON-Sapphire model configuration, which targets a representation of the components of the Earth system and their interactions with a grid spacing of 10 km and finer. Through the use of selected simulation examples, we demonstrate that ICON-Sapphire can (i) be run coupled globally on seasonal timescales with a grid spacing of 5 km, on monthly timescales with a grid spacing of 2.5 km, and on daily timescales with a grid spacing of 1.25 km; (ii) resolve large eddies in the atmosphere using hectometer grid spacings on limited-area domains in atmosphere-only simulations; (iii) resolve submesoscale ocean eddies by using a global uniform grid of 1.25 km or a telescoping grid with the finest grid spacing at 530 m, the latter coupled to a uniform atmosphere; and (iv) simulate biogeochemistry in an ocean-only simulation integrated for 4 years at 10 km. Comparison of basic features of the climate system to observations reveals no obvious pitfalls, even though some observed aspects remain difficult to capture. The throughput of the coupled 5 km global simulation is 126 simulated days per day employing 21 % of the latest machine of the German Climate Computing Center. Extrapolating from these results, multi-decadal global simulations including interactive carbon are now possible, and short global simulations resolving large eddies in the atmosphere and submesoscale eddies in the ocean are within reach.

An Evaluation of Surface Wind and Gust Forecasts from the High-Resolution Rapid Refresh Model

Abstract We utilized high temporal resolution, near-surface observations of sustained winds and gusts from two networks, the primarily airport-based Automated Surface Observing System (ASOS) and the New York State Mesonet (NYSM), to evaluate forecasts from the operational High-Resolution Rapid Refresh (HRRR) model, versions 3 and 4. Consistent with past studies, we showed the model has a high degree of skill in reproducing the diurnal variation of network-averaged wind speed of ASOS stations, but also revealed several areas where improvements could be made. Forecasts were found to be underdispersive, deficient in both temporal and spatial variability, with significant errors occurring during local nighttime hours in all regions and in forested environments for all hours of the day. This explained why the model overpredicted the network-averaged wind in the NYSM because much of that network’s stations are in forested areas. A simple gust parameterization was shown not only to have skill in predicting gusts in both networks but also to mitigate systemic biases found in the sustained wind forecasts. Significance Statement Many users depend on forecasts from operational models and need to know their strengths, weaknesses, and limitations. We examined generally high-quality near-surface observations of sustained winds and gusts from the nationwide Automated Surface Observing System (ASOS) and the New York State Mesonet (NYSM) and used them to evaluate forecasts from the previous (version 3) and current (version 4) operational High-Resolution Rapid Refresh (HRRR) model for a selected month. Evidence indicated that the wind forecasts are excellent yet imperfect and areas for further improvement remain. In particular, we showed there is a high degree of skill in representing the diurnal variation of sustained wind at ASOS stations but insufficient spatial and temporal forecast variability and overprediction at night everywhere, in forested areas at all times of day, and at NYSM sites in particular, which are more likely to be sited in the forest. Gusts are subgrid even at the fine grid spacing of the HRRR (3 km) and thus must be parameterized. Our simple gust algorithm corrected for some of these systemic biases, resulting in very good predictions of the maximum hourly gust.

A two-grid method for discontinuous Galerkin approximations to compressible miscible displacement problems

Discontinuous Galerkin approximations are employed to the compressible miscible displacement problem. A two-grid algorithm is proposed, which is based on one coarse grid space and one fine grid space. The H1 error estimate of the concentration and the L2 error estimate of the velocity are presented for the proposed two-grid method, which show that the two-grid method achieves optimal approximations if the mesh sizes satisfy h=O(H2). The numerical results are presented to confirm that the algorithm is effective.

Comparison between Observed and Simulated AgI Seeding Impacts in a Well-Observed Case from the SNOWIE Field Program

AbstractA dry-air intrusion induced by the tropopause folding split the deep cloud into two layers resulting in a shallow orographic cloud with a supercooled liquid cloud top at around −15°C and an ice cloud above it on 19 January 2017 during the Seeded and Natural Orographic Wintertime Clouds: The Idaho Experiment (SNOWIE). The airborne AgI seeding of this case was simulated by the WRF Weather Modification (WRF-WxMod) Model with different configurations. Simulations at different grid spacing, driven by different reanalysis data, using different model physics were conducted to explore the ability of WRF-WxMod to capture the properties of natural and seeded clouds. The detailed model–observation comparisons show that the simulation driven by ERA5 data, using Thompson–Eidhammer microphysics with 30% of the CCN climatology, best captured the observed cloud structure and supercooled liquid water properties. The ability of the model to correctly capture the wind field was critical for successful simulation of the seeding plume locations. The seeding plume features and ice number concentrations within them from the large-eddy simulations (LES) are in better agreement with observations than non-LES runs mostly due to weaker AgI dispersion associated with the finer grid spacing. Seeding effects on precipitation amount and impacted areas from LES seeding simulations agreed well with radar-derived values. This study shows that WRF-WxMod is able to simulate and quantify observed features of natural and seeded clouds given that critical observations are available to validate the model. Observation-constrained seeding ensemble simulations are proposed to quantify the AgI seeding impacts on wintertime orographic clouds.Significance StatementRecent observational work has demonstrated that the impact of airborne glaciogenic seeding of orographic supercooled liquid clouds is detectable and can be quantified in terms of the extra ground precipitation. This study aims, for the first time, to simulate this seeding impact for one well-observed case. The stakes are high: if the model performs well in this case, then seasonal simulations can be conducted with appropriate configurations after validations against observations, to determine the impact of a seeding program on the seasonal mountain snowpack and runoff, with more fidelity than ever. High–resolution weather simulations inherently carry uncertainty. Within the envelope of this uncertainty, the model compares very well to the field observations.

Two-grid finite volume element method for the time-dependent Schrödinger equation

In this paper, we construct a backward Euler full-discrete two-grid finite volume element scheme for solving time-dependent Schrödinger equation. Combining the idea of the two-grid discretization, the original coupling system is solved in the coarse grid space, and the decoupling system with two independent Poisson problems is solved in the fine grid space, which ensures the accuracy and improves the computational efficiency. We further prove the optimal error estimate of the scheme rigorously. Finally, the numerical simulation results show that the two-grid method is more effective than the standard finite volume element method in solving coupled partial differential problems.

An Investigation of the Impact of Different Turbulence Schemes on the Tropical Cyclone Boundary Layer at Turbulent Gray‐Zone Resolution

AbstractAccurately resolving turbulence in the tropical‐cyclone boundary layer (TCBL) is crucial for realistic simulations of tropical cyclones (TCs), but how well the fine‐scale structure of the simulated TCBL can be reproduced at gray‐zone resolutions by state‐of‐the‐art planetary boundary layer (PBL) parameterization schemes, and why the simulated fine‐scale structure is so sensitive to the PBL scheme, are yet to be evaluated and understood. To address these issues, a series of numerical experiments under idealized conditions were conducted using the advanced research weather research and forecasting (WRF‐ARW) model at 500 and 166 m grid spacing. An 18‐hr WRF‐ARW large‐eddy simulation (LES) with a 55‐m grid spacing based on the nonlinear backscatter and anisotropy subfilter‐scale (NBA‐SFS) stress scheme was used as reference (RLES). Results from experiments using five different PBL schemes, that is, two conventional schemes—the Yonsei University (YSU) and the original MYNN (termed MYNN1)—two scale‐aware schemes—the Shin‐Hong (S‐H) and the revised MYNN (termed MYNN2)—and the coarse‐resolution LES (CLES), were subsequently compared with the RLES. The YSU and S‐H not only produced better simulations of TC intensity and structure than the other three PBL schemes, but also produced a fine‐scale turbulent structure reasonably well compared with the RLES. Discrepancies in the simulated fine‐scale structure among the PBL schemes result primarily from the different strength of vertical mixing, which could be reduced by the scale‐aware option. Results from a series of sensitivity experiments with a 166‐m grid spacing demonstrate that the scale‐aware S‐H can significantly improve the finer resolution simulations than the conventional YSU at relatively finer grid spacing at gray zone.

Using an Artificial Neural Network to improve operational wind prediction in a small unresolved valley

AbstractThis study improves surface wind predictions in an unresolved valley using an artificial neural network (ANN). Forecasting winds in complex terrain with a mesoscale model is challenging. This study assesses the quality of 3-km wind forecasts by the Weather Research and Forecasting (WRF) model and the potential of post-processing by an ANN within the 1-2 km wide Cadarache Valley in southeast France. Operational wind forecasts for 110m above ground level and the near-surface vertical potential temperature gradient with a lead time of 24-48h were used as ANN input. Observed horizontal wind components at 10m within the valley were used as targets during ANN training. We use the Directional ACCuracy (DACC45, wind direction error ≤ 45°) and mean absolute error to evaluate the WRF direct model output and the ANN results. By post-processing, the score for DACC45 improves from 56% in the WRF direct model output to 79% after applying the ANN. Furthermore, the ANN performed well during the day and night, but poorly during the morning and afternoon transitions. The ANN improves the DACC45 at 10m even for poor WRF forecasts (direction bias ≥ 45°) from 42% to 72%. A shorter lead time and finer grid spacing (1 km) showed negligible impact which suggests that a 3 km grid spacing and a 24-48h lead time is effective and relatively cheap to apply. We find that WRF performs well in near-neutral conditions and poorly in other atmospheric stability conditions. The ANN post-treatment consistently improves the wind forecast for all stability classes to a DACC45 of about 80%. The study demonstrates the ability to improve Cadarache valley wind forecasts using an ANN as post-processing for WRF daily forecasts.

A two-grid block-centered finite difference method for the nonlinear regularized long wave equation

In this paper, a Crank-Nicolson block-centered finite difference method is first developed and analyzed for the nonlinear regularized long wave equation. By using a cutoff technique, second-order convergences both in time and space are proved under a suitable time-space stepsize constraint condition. To further improve the computational efficiency, an efficient two-grid block-centered finite difference method is introduced and analyzed, in which a resulting small-scale nonlinear problem is first solved on a coarse grid space of size H, and then a resulting large-scale linear problem is solved on a fine grid space of size h. Under a rough time-space stepsize constraint condition Δt=o(H1/4), optimal-order error estimates for both the primal variable and its flux are derived on non-uniform spatial grids. Thus, the proposed method is competitive both in accuracy and efficiency compared with the fully nonlinear Crank-Nicolson block-centered finite difference scheme. Numerical experiments are presented to verify the theoretical analysis.

WRF Precipitation Performance and Predictability for Systematically Varied Parameterizations over Complex Terrain

AbstractPhysics parameterizations in the Weather Research and Forecasting (WRF) Model are systematically varied to investigate precipitation forecast performance over the complex terrain of southwest British Columbia (BC). Comparing a full year of modeling data from over 100 WRF configurations to station observations reveals sensitivities of precipitation intensity, season, location, grid resolution, and accumulation window. The choice of cumulus and microphysics parameterizations is most important. The WSM5 microphysics scheme yields competitive verification scores when compared to more sophisticated and computationally expensive parameterizations. Although the scale-aware Grell–Freitas cumulus parameterization performs better for summertime convective precipitation, the conventional Kain–Fritsch parameterization better simulates wintertime frontal precipitation, which contributes to the majority of the annual precipitation in southwest BC. Finer grid spacings have lower relative biases and a more realistic spread in precipitation intensity distribution, yet higher relative standard deviations of their errors—they produce finer spatial differences and local extrema. Finer resolutions produce the best fraction of correct-to-incorrect forecasts across all precipitation intensities, whereas the coarser 27-km domain yields the highest hit rates and equitable threat scores. Verification metrics improve greatly with longer accumulation windows—hourly precipitation values are prone to double-penalty issues, while longer accumulation windows compensate for timing errors but lose information about short-term precipitation intensities. This study provides insights regarding WRF precipitation performance in complex terrain across a wide variety of configurations, using metrics important to a range of end users.

IMDAA: High Resolution Satellite-era Reanalysis for the Indian Monsoon Region

AbstractA high resolution regional reanalysis of the Indian Monsoon Data Assimilation and Analysis (IMDAA) project is made available to researchers for deeper understanding of the Indian monsoon and its variability. This 12 km resolution reanalysis covering the satellite-era from 1979 to 2018 using 4D-Var data assimilation method and the UK Met Unified Model is presently the highest resolution atmospheric reanalysis carried out for the Indian monsoon region. Conventional and satellite observations from different sources are used, including Indian surface and upper air observations, of which some were not used in any previous reanalyses. Various aspects of this reanalysis, like quality control and bias correction of observations, data assimilation system, land surface analysis, and verification of reanalysis products, are presented in this paper. Representation of important weather phenomena of each season over India in the IMDAA reanalysis verifies reasonably well against India Meteorological Department (IMD) observations and compares closely with ERA5. Salient features of the Indian summer monsoon are found to be well represented in the IMDAA reanalysis. Characteristics of major semi-permanent summer monsoon features (e.g., Low-level Jet and Tropical Easterly Jet) in IMDAA reanalysis are consistent with ERA5. The IMDAA reanalysis has captured the mean, inter-annual, and intra-seasonal variability of summer monsoon rainfall fairly well. IMDAA produces a slightly cooler winter and a hotter summer than the observations; the reverse for ERA5. IMDAA captured the fine-scale features associated with a notable heavy rainfall episode over complex terrain. In this study, the fine grid spacing nature of IMDAA is compromised due to the lack of comparable resolution observations for verification.

An optimal frequency-domain finite-difference operator with a flexible stencil and its application in discontinuous-grid modeling

We have developed an optimal method to determine expansion parameters for flexible stencils in 2D scalar-wave finite-difference frequency-domain (FDFD) simulation. Our stencil only requires the involved grid points to be paired and rotationally symmetric around the central point. We apply this method to the transition zone in discontinuous-grid modeling, in which the key issue is designing particular FDFD stencils to correctly propagate the wavefield passing through the discontinuous interface. Our method can work in an FDFD discontinuous grid with arbitrary integer coarse- to fine-grid spacing ratios. Numerical examples are developed to determine how to apply this optimal method to discontinuous-grid FDFD schemes with spacing ratios of 3 and 5. The synthetic wavefields are highly consistent to those calculated using the conventional dense uniform grid, and the memory requirement and computational costs are greatly reduced. For velocity models with large contrasts, our discontinuous-grid FDFD method can significantly improve the computational efficiency in forward modeling, imaging, and full-waveform inversion.

Reliability of CPT measurements in sand – influence of spacing

Geotechnical methods, such as cone penetration tests (CPTs), standard penetration tests (SPTs) and seismic cone penetration tests (sCPTs), are widely used to characterise the in situ properties of soil. In studies where more than one in situ method in close spacing is used, it is important for geotechnical engineers to balance between soil heterogeneity and the artefacts of disturbed testing zones. The same applies if detailed information about the subsoil is required and one in situ method is used in close spacing. There is no consensus on how to define the minimum spacing between testing zones and no study on the effect of disturbance caused by in situ tests. In this study, 33 CPTs were performed in natural sediments in northern Germany and a spacing threshold was defined at which cone resistance is affected by soil disturbance from previously performed CPTs. The CPTs were performed sequentially by successively refining the grid spacing, starting with a spacing of 119 cone diameters between CPTs, to a fine grid spacing of seven cone diameters. The cone resistance is affected by previous CPT measurements below a spacing threshold of 24 cone diameters in medium-dense sands. Silt and clay layers showed no reduction in the cone resistance for the minimum grid spacing of seven cone diameters.

Improved Internal Wave Spectral Continuum in a Regional Ocean Model

AbstractRecent work demonstrates that high‐resolution global models forced simultaneously by atmospheric fields and the astronomical tidal potential contain a partial internal (gravity) wave (IW) spectral continuum. Regional simulations of the MITgcm forced at the horizontal boundaries by a global run that carries a partial IW continuum spectrum are performed at the same grid spacing as the global run and at finer grid spacings in an attempt to fill out more of the IW spectral continuum. Decreasing only the horizontal grid spacing from 2 to 0.25 km greatly improves the frequency spectra and slightly improves the vertical wavenumber spectra of the horizontal velocity. Decreasing only the vertical grid spacing by a factor of 3 does not yield any significant improvements. Decreasing both horizontal and vertical grid spacings yields the greatest degree of improvement, filling the frequency spectrum out to 72 cpd. Our results suggest that improved IW spectra in regional models are possible if they are run at finer grid spacings and are forced at their lateral boundaries by remotely generated IWs. Additionally, consistency relations demonstrate that improvements in the spectra are indeed due to the existence of IWs at higher frequencies and vertical wavenumbers when remote IW forcing is included and model grid spacings decrease. By being able to simulate an IW spectral continuum to 0.25 km scales, these simulations demonstrate that one may be able to track the energy pathways of IWs from generation to dissipation and improve the understanding of processes such as IW‐driven mixing.

Sub-radian-accuracy gravitational waves from coalescing binary neutron stars in numerical relativity. II. Systematic study on the equation of state, binary mass, and mass ratio

We report results of numerical relativity simulations for {\it new} 26 non-spinning binary neutron star systems with 6 grid resolutions using an adaptive mesh refinement numerical re\ lativity code {\tt SACRA-MPI}. The finest grid spacing is $\approx 64$--$85$ m, depending on the systems. First, we derive long-term high-precision inspiral gravitational waveforms and show that the accumulated gravitational-wave phase error due to the finite grid resolution is less than $0.5$ rad during more than $200$ rad phase evolution irrespective of the systems. We also find that the gravitational-wave phase error for a binary system with a tabulated equation of state (EOS) is comparable to that for a piecewise polytropic EOS. Then we validate the SACRA inspiral gravitational waveform template, which will be used to extract tidal deformability from gravitational wave observation, and find that accuracy of \ our waveform modeling is $\lesssim 0.1$ rad in the gravitational-wave phase and $\lesssim 20 \%$ in the gravitational-wave amplitude up to the gravitational-wave frequency $1000$ Hz.\ Finally, we calibrate the proposed universal relations between a post-merger gravitational wave signal and tidal deformability/neutron star radius in the literature and show that th\ ey suffer from systematics and many relations proposed as universal are not very universal. Improved fitting formulae are also proposed.

Numerical Treatment of a Two-Dimensional Vertically-Averaged Groundwater Flow Model Using Alternating Direction Explicit Methods

Leachate from a landfill can flow down and contaminate groundwater. Mathematical models are often used to describe groundwater flow, which can help designers to identify an appropriate location for a landfill area. We focused on simulation of hydraulic head of groundwater under landfill construction in a rural area. We used a partial differential equation model called two-dimensional vertically-averaged groundwater flow model to explain groundwater volume, in particular, to explain the hydraulic head of the groundwater which strongly affected the groundwater quality. We applied two alternating direction explicit methods to the groundwater flow model under general boundary conditions and rates of change of boundary conditions to obtain an approximate hydraulic head of groundwater. An alternating direction explicit method for all types of boundaries is used to find approximate solutions. The traditional alternative direction explicit method was modified to form our modified alternative direction explicit method. The results of all method are closed. The traditional alternative direction explicit method and the modified alternative direction explicit method led to more accurate solutions than forward time centered space in heterogeneous aquifer, two methods ensured that a stable solution of all interior points can be obtained by using a fine grid spacing.