Numerical Errors Research Articles

Retracted: Safety and efficacy of titrated oral misoprostol solution versus vaginal dinoprostone for induction of labor: A single-center randomized control trial.

X. Wang, C. Zhang, Xia Li, H. Qi, Q. Liu, and J. Lei, "Safety and Efficacy of Titrated Oral Misoprostol Solution Versus Vaginal Dinoprostone for Induction of Labor: A Single-Center Randomized Control Trial," International Journal of Gynecology & Obstetrics 154, no. 3 (2021): 436-443. https://doi.org/10.1002/ijgo.13546. The above article, published online on 17 December 2020, in Wiley Online Library (wileyonlinelibrary.com), and has been retracted by agreement between the journal Editor-in-Chief, Professor Michael Geary; the International Federation of Gynecology and Obstetrics; and John Wiley and Sons Ltd. A third party raised concerns regarding errors in the reported data and statistics of the article. An external statistician reviewed the data and was not able to reproduce the listed results for the Bishop score, the post-term pregnancy variable, the gestational hypertension variable, the not using insulin variable, the vaginal delivery variable, and the failed induction variable. In addition, the journal detected numerical errors in Table4, with contradictions between frequencies and reported percentages. The authors did not respond to multiple requests for an explanation and the original data. The editors have determined that the large number of errors identified and lack of original data provided fundamentally compromise the results presented, which necessitates retraction. The authors did not respond to our notice regarding the retraction.

Numerical Solutions of Volterra Integral Equations with Subdivision-Based Collocation Approach

Integral equations play a vital role in the fields of applied and computational mathematics. This paper introduces a unique subdivision-based collocation technique for solving the initial value problems (IVPs) arising from Volterra integral equations (VIEs). A suitable subdivision scheme is employed to develop the numerical algorithm for solving the VIEs. Several numerical illustrations are tested and compared with the existing methods for assessing the efficiency of the proposed numerical algorithm and error estimation.

A gas-surface interaction algorithm for discrete velocity methods in predicting rarefied and multi-scale flows: For Maxwell boundary model

The discrete velocity method (DVM) for rarefied flows and the unified methods (based on the DVM framework) for flows in all regimes, from continuum one to free molecular one, have worked well as precise flow solvers over the past decades and have been successfully extended to other important physical fields. Both DVM and unified methods endeavor to model the gas-gas interaction physically. However, for the gas-surface interaction (GSI) at the wall boundary, they have only use the full accommodation boundary up to now, which can be viewed as a rough Maxwell boundary with a fixed accommodation coefficient (AC) at unity, deviating from the real value. For example, the AC for metal materials typically falls in the range of 0.8 to 0.9. To overcome this bottleneck and extend the DVM and unified methods to more physical boundary conditions, an algorithm for Maxwell boundary with an adjustable AC is established into the DVM framework. The Maxwell boundary model splits the distribution of the bounce-back molecules into specular ones and Maxwellian (normal) ones. Since the bounce-back molecules after the spectral reflection does not math with the discrete velocity space (DVS), both macroscopic conservation (from numerical quadrature) and microscopic consistency in the DVS are hard to achieve in the DVM framework. In this work, this problem is addressed by employing a combination of interpolation methods for mismatch points in DVS and an efficient numerical error correction method for micro-macro consistency. On the other hand, the current Maxwell boundary for DVM takes the generality into consideration, accommodating both the recently developed efficient unstructured velocity space and the traditional Cartesian velocity space. Moreover, the proposed algorithm allows for calculations of both monatomic gases and diatomic gases with internal degrees in DVS. Finally, by being integrated with the unified gas-kinetic scheme within the DVM framework, the performance of the present GSI algorithm is validated through a series of benchmark numerical tests across a wide range of Knudsen numbers.

A dynamic linearized wall model for turbulent flow simulation: towards grid convergence in wall-modeled simulations

This paper addresses the common issue of (no-)grid convergence in wall-modeled numerical simulations and proposes a dynamic linearization technique applied to the Spalart-Allmaras wall model to achieve a proper behavior on fine grids and low-friction areas. A theoretical analysis of the numerical error committed on the shear stress balance close to the walls is performed. It shows that the error is due to the inappropriate imposition of too steep wall-normal velocity gradients that cannot be properly accounted for on the typical grids used for wall-modeled simulations. Based on this error quantification, a dedicated wall model linearization technique is proposed, following the approach developed by Tamaki, Harada and Imamura in 2017. In the proposed modified linearization method, the linearization distance is modified and adjusted dynamically. This is done according to the theoretical shear stress error estimate, in order to keep the numerical error below a user-defined threshold. The method is applied to well-referenced test cases of increasing complexity from the Turbulence Modeling Resource. Overall, the proposed wall model clearly exhibits appropriate grid convergence properties and is also able to predict accurately non-equilibrium boundary layers and flow separation using proper grid refinement.

Accurate simulations of moving flexible objects with an improved immersed boundary-lattice Boltzmann method

Accurately capturing boundaries is crucial for simulating fluid–structure interaction (FSI) problems involving flexible objects undergoing large deformations. This paper presents a coupling of the immersed boundary-lattice Boltzmann method with a node-based partly smoothed point interpolation method (NPS-PIM) to enhance the accuracy of simulating moving flexible bodies in FSI problems. The proposed method integrates a multiple relaxation time scheme and employs a force correction technique to address boundary capturing inaccuracies. The effect of virtual fluid is accounted for through a Lagrangian point approximation, ensuring precise FSI force calculations for unsteady solid motions. NPS-PIM is utilized as the solid solver, constructing a moderately softened model stiffness by combining the finite element method (FEM) with the node-based smoothed PIM (NS-PIM). Simulations of flow fields near flexible objects with large deformations demonstrate that the proposed approach reduces numerical errors, improves computational efficiency compared to traditional FSI models using FEM and NS-PIM, and accurately captures the behavior of moving flexible bodies and detailed flow fields.

Discontinuous galerkin method of the first order parabolic differential equation

The objective of this paper is to propose a modest numerical error analysis by applying DG finite element method for the parabolic differential equation. The DG method is an imperative numerical method with much mass compensation and more flexible meshing than other numerical methods. The DG method starts by discretizing the domain into a set of non-overlapping elements. This study gives a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Temporally and spatially segregated discretization for a coupled electromechanical myocardium model

Abstract In this paper, we propose a novel temporally and spatially segregated numerical scheme to discretize the coupled electromechanical model of myocardium. We perform several numerical experiments with activation of a myocardial slab with structural inhomogeneity and evaluate the dependence of numerical errors on the size of spatial and temporal discretization steps. In our study, we show that the spatial step for the mechanical equations h m ⩽2.5 mm yields reasonable results with noticeable errors only in the region of myocardial inhomogeneity. We also show that time step τ m ⩽1 ms can be used for temporal discretization of mechanical equations, and the propagation velocity of the activation and contraction fronts differs from the reference one by no more than 1.3%for such time step. Finally, we show that the increase of time discretization steps of the mechanical equations τ m and the monodomain equation τ e leads to phase errors with opposite signs, and we can compensate these errors by tuning the relationship between the time steps.

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

ABSTRACTUnsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers.

Efficient Environment Map Rendering Based on Decomposition

AbstractThis paper presents an efficient environment map sampling algorithm designed to render high‐quality, low‐noise images with only a few light samples, making it ideal for real‐time applications. We observe that bright pixels in the environment map produce high‐frequency shading effects, such as sharp shadows and shading, while the rest influence the overall tone of the scene. Building on this insight, our approach differs from existing techniques by categorizing the pixels in an environment map into emissive and non‐emissive regions and developing specialized algorithms tailored to the distinct properties of each region. By decomposing the environment lighting, we ensure that light sources are deposited on bright pixels, leading to more accurate shadows and specular highlights. Additionally, this strategy allows us to exploit the smoothness in the low‐frequency component by rendering a smaller image with more lights, thereby enhancing shading accuracy. Extensive experiments demonstrate that our method significantly reduces shadow artefacts and image noise compared to previous techniques, while also achieving lower numerical errors across a range of illumination types, particularly under limited sample conditions.

Temporal error monitoring: Does agency matter?

Error monitoring is the ability to report one's errors without relying on feedback. Although error monitoring is investigated mostly with choice tasks, recent studies have discovered that participants parametrically also keep track of the magnitude and direction of their temporal, spatial, and numerical judgment errors. We investigated whether temporal error monitoring relies on internal generative processes that lead to the to-be-judged first-order timing performance. We hypothesized that if the endogenous processes underlie temporal error monitoring, one can monitor timing errors in emitted but not observed timing behaviors. We conducted six experiments to test this hypothesis. The first two experiments showed that confidence ratings were negatively related to error magnitude only in emitted behaviors, but error directionality judgments of observed behaviors were more precise. Experiment 3 replicated these effects even after controlling for the motor aspects of first-order timing performance. The last three experiments demonstrated that belief of agency (i.e., believing that the error belongs to the self or someone else) was critical in accounting for the confidence rating effects observed in the first two experiments. The precision of error directionality judgments was higherin thenon-agency condition. These results show that confidence is sensitive to belief, and short-long judgment is sensitive to the actual agency of timing behavior (i.e., whether the behavior was emitted by the self or someone else).

3D limit analysis of reinforced concrete with sliding along smeared cracks

Limit analysis (LA) is successfully used for investigating the bearing capacity of reinforced concrete (RC) structures. Some cautions must be taken when using this method for RC since the concrete component exhibits a softening behavior with decreasing strength and limited ductility. A commonly adopted provision consists of considering isotropic reduced values of concrete strength to be input in the analysis (empirical effectiveness factors). In this paper, an alternative and completely new approach is proposed and investigated in which concrete strength is weakened in a more targeted manner. To that purpose, the commonly used 3D truncated Mohr-Coulomb (TMC) criterion is adopted to classically describe the compressive, tensile, and shear failure of concrete. However, TMC is here in an original way enriched by additional constraints that allow to account for weakness and anisotropy induced by preferential failure patterns, assumed a priori. The limit analysis approach is then formulated for two dual analyses in the convex optimization framework, making it possible to quantify the numerical error and obtain a lower and an upper bound of the limit load. Numerical examples illustrate the agreement of the formulation with academic results and laboratory tests.

Structural damage detection based on transmissibility functions with unsupervised domain adaptation

Deep learning (DL) has been used for structural damage detection by training neural network models with a large amount of data. These trained models perform well when the test samples are from the data with an identical distribution of the training data. The distributions are always different due to numerical modelling errors and operational environmental varieties, and the data for damage scenarios is difficult to be obtained in practice. A novel method based on the joint maximum discrepancy and adversarial discriminative domain adaptation (JMDAD) for structural damage detection without labelled measurement data has been developed to address the above issues. To reduce the influence of external excitations in practice, the transmissibility function of measured structural responses is used for structural damage detection. The proposed network includes a feature generator, two classifiers and one discriminator. Firstly, the feature generator and domain discriminator are to extract features and merge their distributions at the domain level to overcome the issue of insufficient data from the target real structure. Secondly, the generator and two classifiers are optimised to align their distributions at the class level using the classification discrepancy between the two classifiers. As a result, the damage-sensitive features are extracted and aligned to eliminate modelling errors and operational environmental varieties between the source and target structures. Three case studies have been conducted in this paper, e.g., one case between two building structures with different storeys, another case between the numerical and experimental structures subjected to random and earthquake excitations and the application of Canton Tower. The results show that the proposed method is much robust to the measurement noise and operational environment and the structural damage is identified accurately without labelled measurement data from the target structure in operational environments.

Comparisons of experimentally and numerically determined statistics for predicting low-occurrence wind speeds around a 1:1:2 block model

Accurate prediction of low-occurrence wind speeds around urban structures is crucial for safe building design. Although Large-eddy simulation (LES) is commonly used as a high-fidelity model as compared with the Reynolds-Averaged Navier–Stokes (RANS) simulations, the present validation process relies on the comparison of fundamental statistics of the mean and standard deviations. The discrepancies in LESs and wind-tunnel experiments (WTEs) are unclear in terms of physical quantities characterizing the unsteadiness of the simulated turbulent flow such as probability density and power spectral densities, and low-occurrence winds speeds. Therefore, this study aims to evaluate the effectiveness of LES in predicting unsteady wind behavior around a 1:1:2 block model. The study identifies prominent differences to improve the accuracy of unsteady numerical simulations especially for the purpose of predicting low-occurrence wind speeds. Various advection schemes in LESs were investigated, including first-order upwind, second-order linear, and dynamic interpolation schemes. The results show significant discrepancies, particularly in higher-order statistics and low-occurrence wind speeds, with WTE consistently exhibiting higher energy levels across all frequencies. These findings highlight the need to refine advection schemes to enhance their predictive accuracy. LESs with minimal numerical errors from discretization schemes can substantially improve urban wind assessments and contribute to the design of safer structures.

Online Evaluation for the POI-Level Inertial Support to the Grid via Ambient Measurements

As renewable energy sources like wind and solar power increasingly replace traditional energy sources and are integrated into the power grid, the issue of insufficient system inertia is becoming more apparent. This paper presents an online adaptive time window inertia constant identification method based on ambient measurements to identify the equivalent inertia constant of the time-varying inertia at Point of Interface (POI) level. The proposed method takes advantage of the online inertia estimation and the data-driven equivalent inertia constant identification techniques to simultaneously achieve online tracking and accuracy. With this regard, this paper first describes the inertia providers in modern system. Then, based on the frequency and power data measured by the Phasor Measurement Unit (PMU), this paper provides an improved data-driven equivalent inertia constant identification method. Subsequently, the paper proposes an ambient data smoothing method to cope with the numerical errors and provides, as a byproduct, an adaptive time window inertia constant identification. The adaptive time window is designed to enhance the accuracy of the method. Finally, the feasibility and accuracy of the proposed method of tracking synthetic inertia are validated by the simulation tests based on a grid in northwest China with high renewable energy penetration and a Virtual Power Plant (VPP). The experimental results show that the accuracy of this method is within 5%.

Accuracy of intraoperative aberrometry versus preoperative biometry for intraocular lens power selection in short and long eyes.

To compare the accuracy of intraoperative wavefront aberrometry using the ORA VLynk system with different biometry-based formulas in short and long eyes after cataract surgery. This prospective study considered 48 eyes with axial lengths of <22.1 mm and 48 eyes with axial lengths of >25.0 mm. All eyes were implanted with the monofocal AcrySof IQ IOL, the power being determined using the ORA VLynk. The postoperative spherical equivalent (SE) at 3 months was compared to that predicted preoperatively using the SRK/T, Hoffer Q, Haigis, Holladay 2, Barrett Universal II, and Barrett True K formulas and intraoperatively using the ORA VLynk. Mean numerical and absolute errors and the percentage of eyes within ±0.50 D/1.00 D of their target were obtained. For long eyes, the mean absolute error values were 0.35, 0.52, 0.34, 0.30, 0.29, 0.27, and 0.24D for SRK/T, Hoffer Q, Haigis, Holladay 2, Barrett Universal II, Barrett True K, and ORA VLynk, respectively (p < 0.001). These values were 0.55, 0.45, 0.49, 0.40, 0.44, 0.44 and 0.50 D for short eyes, respectively (p < 0.001). The proportions of long eyes within ±0.50 D of the target were 77.08, 50, 75, 85.42, 83.33, 79.17, and 87.50%, respectively; and 50, 66.67, 60.42, 66.67, 60.42, 60.42, and 58.33%, respectively, for short eyes. The ORA VLynk performs better than all biometry-based formulas in long eyes and, in short eyes, it is as effective as SRK/T, Haigis, Barrett Universal II, and Barrett true K, with the Hoffer Q and Holladay 2 being the most accurate; however, the differences between the calculation methods were small. Identifier DRKS000028106.

Double-loop importance sampling for McKean–Vlasov stochastic differential equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \ imes d$$\\end{document}-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of OTOLr-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \ extrm{TOL}_{\ extrm{r}}^{-4}\\right) $$\\end{document} with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document}. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document} compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

NUMERICAL ERROR METHOD TO DETERMINE THE EFFICIENCY OF REDUCING VIBRATIONS DUE TO EARTHQUAKE LOADS ON BUILDINGS USING FLUID VISCOUS DAMPER

According to The Indonesian Earthquake Map, Padang City in West Sumatra is in Earthquake zone 6. This indicates that Padang City is very vulnerable to earthquakes. Meanwhile, developments in the construction of high buildings also continue to show progress all the time. The main problem that is often faced is the issue of structural damage due to earthquakes. Efforts are needed in earthquake engineering on buildings so that collapse can be minimized. The earthquake damping system used is FVD (Fluid Viscous Dampers) with different types. The Type-A damper is FVD-750 kN and the Type-B damper is FVD-1000 kN, and without using a damper. This research aims to analyze the efficiency of reducing earthquake loads such as floor displacement and vibration period of an 18-story building structure 18 meters high from the base.The analysis method used is a numerical method by calculating earthquake load reduction based on numerical error analysis from the two types of dampers used on structures without using dampers. Building planning refers to SNI 1727-2013, SNI 1726-2019, and SNI 2847-2019, and is assisted by ETABS software. Based on SNI 1726-2019, earthquake risk category II, soft soil condition type (SE), earthquake acceleration response value SDS = 0.745g and SD1 = 0.784g are obtained. The earthquake load used is a dynamic load, taking into account that the condition of the building is irregular. Based on the results of the analysis, it was found that the displacement between floors using the Type-AFVD Damper could reduce the displacement by up to 45.72% and with the Type-B FVD Damper it could reduce the displacement by up to 92.72%. Meanwhile, the period of vibration natural using a Type-A FVD Damper can be reduced by up to 12.34% and using a Type-B FVD Damper can be reduced by up to 33.21%.

A design method of large-scale partial similar model test for cementing casing string system

When the cementing casing string system (CCSS) is subjected to a large-scale equal-scale model test, the huge aspect ratio of the system makes the wall thickness of the model casing string too thin to be manufactured. Therefore, this paper proposes a design method for model test with unequal scale of axial size and radial size of the CCSS. Firstly, the dynamic model of the CCSS is established and solved by the transfer matrix method (TMM), and the modal test is carried out and the results show that the numerical calculation error of the first five natural frequencies of the system is less than 4 %. Then, based on the dynamic model, the similarity laws of the complete similarity model and the partial similarity model of the CCSS are derived by equation analysis method, and the design process of the similarity model test is given. The advantages of the research are elaborated in details through examples. To verify the correctness of the similarity model’s similarity laws of the CCSS, similarity models with different parameters are designed, and the prediction errors of prototype response are analyzed based on numerical calculations. The results show that the maximum prediction error of the first eight natural frequencies of the prototype system is less than 3 %, and the response prediction results of the system are also accurate. Finally, considering uncertain factors of the partial similarity test, the similarity model test with the axial size of double scale is designed, and the results show that the maximum prediction error of the first three natural frequencies is 7.06 %, which meets the engineering requirements. This study can provide useful theoretical guidance for the large-scale model test design of the CCSS.

Verification of frequency-dependent impedance boundaries for the Finite-Volume Time-Domain scheme

Numerical simulation methods are widely used to approximate solutions to the wave-equation, taking into account the initial condition ('initial sources') and boundary conditions ('wall materials'). Trustworthy simulations are essential; a primary step is to ensure the correctness of the implementation through a process known as code verification. In this work, we do code verification by studying the convergence of the known finite-volume time-domain (FVTD) method with a frequency-dependent boundary condition modeled as an inductance/resistance/capacitance (LRC) termination. Due to the difficulties of obtaining an analytical solution for frequency-dependent impedance boundaries, we thoroughly derive and implement The Method of Manufactured Solutions tailored to the wave-equation inside a rectangular room for uniform frequency- dependent wall impedance. Asymptotic estimates are found empirically based on a convergence study compared against the manufactured ground truth solution. We investigate the impact of the frequency-dependent boundaries on the convergence affected by the total sum of numerical errors and validate the theoretical properties of the FVTD scheme for various LRC model configurations. The method should also serve as a useful resource for others to verify the correctness of their code implementations.

Should We Conserve Entropy or Energy when Computing CAPE with Mixed-Phase Precipitation Physics?

Abstract The rapidly increasing resolution of global atmospheric reanalysis and climate model datasets necessitates finding methods for computing convective available potential energy (CAPE) both efficiently and accurately. To this end, this article compares two common methods for computing CAPE which conserve either energy or entropy. Inaccuracies in these computations arise from both physical and numerical errors. For instance, computing CAPE with entropy conserved results in physical errors from nonequilibrium phase transitions but minimizes numerical errors because solutions are analytic at each height. In contrast, computing CAPE with energy conserved avoids these physical errors, but accumulates numerical errors that are grid-resolution-dependent because the numerical integration of a differential equation is required. Analysis of CAPE computed with large databases of soundings from the tropical Amazon and midlatitude storm environments shows that physical errors from the entropy method are typically 1%–3% as large as CAPE, which is comparable to the numerical errors from conserving energy with grid spacing of 25 and 250 m using explicit first-order and second-order integration schemes, respectively. Errors in entropy-based CAPE calculations are also insensitive to vertical grid spacing, in contrast to energy-based calculations whose error strongly scales with the grid spacing. It is shown that entropy-based methods are advantageous when intercomparing datasets with differing vertical resolution because they produce accurate and reasonably fast results that are insensitive to grid resolution, whereas a second-order energy-based method is advantageous when analyzing data with a consistent vertical resolution because of its superior computational efficiency. Significance Statement Convective available potential energy (CAPE) is a measure of instability in the atmosphere that helps forecasters and researchers understand when and where thunderstorms will form. The purpose of this article is to identify the most efficient and accurate methods for computing CAPE. Two methods are considered here, one that relates to the entropy (a measure of thermodynamic disorder) of an air parcel and one that relates to the energy of an air parcel. Results indicate that the entropy method is most accurate and insensitive to the resolution of the data used for the calculation (which can vary considerably), whereas the energy method uses the least computation time.