Taylor Series Research Articles

Numerical Solution of Volterra's Integral Equations by Collocation and Taylor Method

In this article, two numerical methods (collocation method and Taylor method) are introduced to solve Volterra integral equation system. In the collocation method, using Bessel polynomials and collocation points, we convert the device into a matrix form and solve the device using the matrix form and obtain an approximate solution for the device. This answer is such that the larger the N becomes, the approximate answer is closer to the exact answer of the device. In Taylor method, With the use of the Taylor series, the system of integral equations is transformed into a matrix equation, and by integrating the output of this new system, a system of algebraic equations is created. By working through this device, we can obtain a rough solution for the Integral Equations device. Using these two methods when the answers are polynomial, we can get real answers.

Optimizing electric vehicle powertrains peak performance with robust predictive direct torque control of induction motors: a practical approach and experimental validation

Enhancing the efficiency of the electric vehicle’s powertrain becomes a crucial focus, wherein the control system for the traction motor plays a significant role. This paper presents a novel electric vehicle traction motor control system based on a robust predictive direct torque control approach, an improved version of the conventional DTC, where the traditional switching table and the hysteresis regulators are substituted with a predictive block based on an optimization algorithm. Additionally, a robust predictive speed loop regulator is employed instead of the proportional-integral regulator, which integrates a new cost function with a finite horizon, incorporating integral action into the control law based on a Taylor series expansion. This technique’s primary benefit is its independence from the necessity to measure and observe external disturbances, as well as uncertainties related to parameters. The effectiveness of the suggested system was confirmed through simulation and experimental results under the OPAL-RT platform. The findings indicate that the proposed approach outperforms the conventional method in terms of rejecting disturbances, exhibiting robustness to variations in parameters, and minimizing torque ripple.

Prediction of the Change Rate of Tumor Cells, Healthy Host Cells, and Effector Immune Cells in a Three-Dimensional Cancer Model using Extended Kalman Filter

In this study, we develop and implement the Extended Kalman Filter (EKF) to forecast the rate of change in tumor cells, healthy host cells, and effector immune cells within the Itik-Banks model. This novel application of EKF in cancer dynamics modeling aims to provide precise real-time estimations of cellular interactions, especially in constructing a new state space representation from the Itik-Banks model. We use a first-order Taylor series to linearize the model. The numerical simulations were performed to analyze the accuracy of this new state space with data from William Gilpin’s GitHub repository. The results show that the EKF predictions strongly align with actual data, i.e., in the prior and posterior steps for tumor and healthy host cells, there is a strong agreement between the predictions and the actual data. The EKF captures the oscillatory nature of the tumor and healthy host cell population well. The peaks and troughs of the predictions align closely with the actual data, indicating the EKF’s effectiveness in modeling the dynamic behavior of the tumor and healthy host cells. However, for effector immune cells, the oscillatory nature of the data in these cells gives rise to slight deviations. This represents a significant challenge in the future for updating the state space representations. Despite minor discrepancies, the EKF demonstrates a strong performance in both the training and testing data, with the posterior step estimates significantly improving the prior step accuracy. This study emphasizes the importance of data availability for accurate predictions, noting a symmetric Mean Absolute Percentage Error (sMAPE) of 35.92% when data is unavailable. Prompt correction with new data is essential to maintain accuracy. This research underscores the EKF’s potential for real-time monitoring and prediction in complex biological systems.

Joint moments of higher order derivatives of CUE characteristic polynomials II: structures, recursive relations, and applications

In a companion paper (Keating and Wei 2023 Int. Math. Res. Not. 2024 9607–32), we established asymptotic formulae for the joint moments of higher order derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed as partition sums of derivatives of determinants of Hankel matrices involving I-Bessel functions, with column indices shifted by Young diagrams. In this paper, we continue the study of these joint moments and establish more properties for their leading order coefficients, including structure theorems and recursive relations. We also build a connection to a solution of the σ-Painlevé III ′ equation. In the process, we give recursive formulae for the Taylor coefficients of the Hankel determinants formed from I-Bessel functions that appear at zero and find some differential equations these determinants satisfy. The approach we establish is applicable to determinants of general Hankel matrices whose columns are shifted by Young diagrams.

Implementation of Small Term Reduction in Taylor Series in Analytical Physics Problem

This research is motivated by the low ability of students to apply mathematical solving methods to analytic physics. The purpose of this research is to show the implementation of small term reduction in Taylor series on analytic physics problems. The analytical physics material described in this study is the pendulum oscillation material, heat conduction and the perturbation quantum state function. This research method involves a mathematical derivation of the Taylor series formation. The epicenter of the idea of reducing the Taylor series in this study is the use of very small terms in the Taylor series to reduce terms with values reaching towards zero, namely the third term and the fourth term. This small term reduction method is spread into problems of mechanics, oscillations, heat, electricity, magnetism and quantum mechanics. The results of this study show that after using the small term reduction principle, the equations to be solved are like first-order and second-order differential equations. By solving these equations we get a variable physical quantity.

Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses.

For some communicable endemic diseases (e.g., influenza, COVID-19), vaccination is an effective means of preventing the spread of infection and reducing mortality, but must be augmented over time with vaccine booster doses. We consider the problem of optimally allocating a limited supply of vaccines over time between different subgroups of a population and between initial versus booster vaccine doses, allowing for multiple booster doses. We first consider an SIS model with interacting population groups and four different objectives: those of minimizing cumulative infections, deaths, life years lost, or quality-adjusted life years lost due to death. We solve the problem sequentially: for each time period, we approximate the system dynamics using Taylor series expansions, and reduce the problem to a piecewise linear convex optimization problem for which we derive intuitive closed-form solutions. We then extend the analysis to the case of an SEIS model. In both cases vaccines are allocated to groups based on their priority order until the vaccine supply is exhausted. Numerical simulations show that our analytical solutions achieve results that are close to optimal with objective function values significantly better than would be obtained using simple allocation rules such as allocation proportional to population group size. In addition to being accurate and interpretable, the solutions are easy to implement in practice. Interpretable models are particularly important in public health decision making.

Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection–diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection–diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k≥2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h2k) and O(hk+2), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(hk+1). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

Comprehensive design strategy of the NSD–TMD controlled system with closed-form frequency-amplitude solutions

A negative stiffness device (NSD) utilizing pre-compressed helical springs has been integrated with diverse vibration control techniques. Recently, the incorporation of NSD has been proposed to enhance the efficacy of a tuned mass damper (TMD) controlled system. This study introduces two design strategies for the NSD–TMD system, which functions either as a TMD with an adjustable operating bandwidth or as a pure non-linear energy sink. First, the non-linear force of NSD was expressed in a Taylor series approximation form incorporating a negative linear stiffness and a positive cubic stiffness term. Second, the equations of motion were formulated, and closed-form solutions were derived using the complexification–averaging method. The accuracy of the Taylor series approximation was substantiated within a specified range through a comparison of force–displacement curves. Third, the accuracy of the dynamic closed-form solutions was affirmed by comparing the amplitude–frequency curves with the numerical simulations. For a case study, an actual timber pedestrian bridge with decaying natural frequency was selected, and the two design strategies were examined individually. The vibration control performance of the TMD and NSD–TMD was also comparatively analyzed. The assessment focused on the peak force transmissibility value and the effective operating frequency bandwidth. Results highlighted a notable enhancement in operational performance, particularly for primary structures exhibiting diminishing modal frequencies. In a TMD-controlled system, the efficacy of vibration control deteriorates and may even amplify the dynamic response when a modal frequency shift occurs in the primary structure. Conversely, both design strategies of NSD–TMD can achieve lower transmissibility and broaden the operating bandwidth.

Tuning a PI/PID Controller with Direct Synthesis to Obtain a Non-Oscillatory Response of Time-Delayed Systems

The article presents a method for tuning PI/PID regulators controlling inertial time-delayed plants. The method was formulated on the basis of direct synthesis using the FOPDT (First-Order Plus Dead-Time) and the SOPDT (Second-Order Plus Dead-Time) models to represent the dynamics of the object. The performance objective is to achieve the non-oscillatory response of the system in the shortest time possible while set-point tracking. By employing a novel approach to the synthesis of the control systems with a time delay based on a root-locus analysis, the method eliminates errors resulting from the omission of high-order terms in the Taylor series expansion or the Padé approximation in simplified mathematical descriptions of the process. Consequently, the proposed method makes it possible to precisely determine controller settings to achieve the set control objective. The method was tested with a computer simulation, and the results obtained were compared with those of other methods designed to ensure specific system responses in the designed control systems. The comparisons confirm the proposed method’s ability to achieve the desired reference response and demonstrate its effectiveness in overcoming the limitation of permissible time delays present in other similar methods.

Theoretical Analysis of Pre-Steady State Behaviour of Non-Linear Double Intermediate Enzymatic Reaction

This paper explores the mathematical modelling of a non-linear double intermediate enzymatic reaction, focusing on its pre-steady state behaviour. Using homotopy perturbation and Taylor's series methods, an approximate analytical solution is derived for the concentrations of substrate, the first enzyme-substrate complex, and the second enzyme-substrate complex. The model is applicable in scenarios where the initial substrate concentration dramatically exceeds that of the enzyme and when they are comparable. Comparisons between analytical and numerical solutions are made, highlighting the efficacy of the proposed model. Additionally, an improved understanding is achieved by comparing results obtained from Matlab simulations with analytical solutions. Moreover, sensitivity analysis on parameters affecting the concentrations is conducted. This mathematical model enhances comprehension of biochemical reactions in living organisms.

Self-adaptive system state optimization based on nonlinear affine transformation for renewable energy volatility

The short-term power variations of renewable energy sources result in system state fluctuations and deviations from the scheduled operating point. Tracking short-term power variations and maintaining the system's optimality is challenging for traditional optimization methods due to their highly time-consuming nature. This paper proposes a short-timescale self-adaptive optimization strategy based on the nonlinear affine transformation to deal with this problem. Firstly, the multi-period optimization model is established and solved by considering static power-frequency characteristics to obtain an optimal operating point on a longer temporal scale, with the state security margins reserved using chance constraint programming. Next, analogously with the Taylor series, the nonlinear relationship between the system state variable and short-term power fluctuations is revealed through an analytic expression of the nonlinear affine transformation. Then, a self-adaptive optimization algorithm based on the nonlinear affine transformation is proposed to achieve frequency and voltage optimizations on a shorter temporal scale. With less communication, self-adaptive optimization is implemented at the local bus level to achieve more optimal states for short-term renewable power variations rapidly. Finally, simulations demonstrate that the proposed optimization strategy can effectively enhance frequency and voltage qualities, and decrease objective function, thereby improving the operation safety and economy.

Divergence beneath the Brillouin sphere and the phenomenology of prediction error in spherical harmonic series approximations of the gravitational field

The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential, V, will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin’s formula, for the upper bound, EN , on the absolute value of the prediction error, eN , of a SH series model, VN(θ,λ,r) , truncated at some maximum degree, N=nmax . When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in 1/r . Costin’s formula is EN≃BN−b(R/r)N , where R is the radius of the Brillouin sphere. This formula depends on two positive parameters: b, which controls the decay of error amplitude as a function of N when r is fixed, and a scale factor B. We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound, An on the absolute value of the TS coefficients, an , for the same radial line. This formula, An≃Kn−k , depends on degree, n, and two positive parameters, k and K, that are analogous to b and B. We use synthetic planets, for which we can compute the potential, V, and also the radial component of gravitational acceleration, gr=∂V/∂r , to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscript V refer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscript g to the coefficients and predictions errors associated with gr . For polyhedral planets of uniform density we show that bV=kV=7/2 and bg=kg=5/2 almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle, α, between that radial line and the singular radial line. We also derive useful identities connecting KV,BV,Kg , and Bg . These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities are α and R. The phenomenology of ‘series divergence’ and prediction error (when r < R) can be described as a function of the truncation degree, N, or the depth, d, beneath the Brillouin sphere. For a fixed r⩽R , as N increases from very low values, the upper error bound EN shrinks until it reaches its minimum (best) value when N reaches some particular or optimum value, Nopt . When N>Nopt , prediction error grows as N continues to increase. Eventually, when N≫Nopt , prediction errors increase exponentially with rising N. If we fix the value of N and allow R/r to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth, d, beneath the Brillouin sphere. Because bg=bV−1 everywhere, divergence driven prediction error intensifies more rapidly for gr than for V, both in terms of its dependence on N and d. If we fix both N and d, and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as does B) as we approach high-amplitude topography.

On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L-\)function

Let \(E\) be a Galois extension of \(\mathbb{Q}\) of finite degree and let \(\pi \) and \(\pi'\) be two irreducible automorphic unitary cuspidal representations of \(GL_m(\mathbb{A}_E)\) and \(GL_{m'}(\mathbb{A}_E)\), respectively. Let \(\Lambda(s,\pi\times\widetilde{\pi}')\) be a Rankin-Selberg \(L-\)function attached to the product \(\pi\times\widetilde{\pi}'\), where \(\widetilde{\pi}'\) denotes the contragredient representation of \(\pi'\), and let its finite part (excluding Archimedean factors) be \(L(s,\pi\times\widetilde{\pi}')\). The Euler-Stieltjes constants of the Rankin-Selberg \(L-\)function are the coefficients in the Laurent (Taylor) series expansion around \(s=1+it_0\) of the function \(L(s, \pi \times \widetilde{\pi}')\). In this paper, we derive an upper bound for these constants.

Performance optimization and energy minimization of cloud data center using optimal switching and load distribution model

Cloud computing is an effective computing methodology used in all stages of business. Most of the Cloud Data Centers (CDC) operates on the basis of peak load and huge scales. Hence, it necessitates saving the energy in CDC. This study introduces an energy-efficient strategy based on the fat tree. Here, Taylor-based Manta-Ray Foraging Optimization (Taylor-MRFO) is developed by combining the Taylor series with Manta Ray Foraging Optimization (MRFO) to distribute the load in a CDC. In load distribution, the cloud data switching to the preferred mode is done by the Actor critic neural network (ACNN). Furthermore, the developed Taylor-MRFO+ACNN provided a better outcome than the conventional approaches with the least energy consumption of 0.4930, least load of 0.3631, and least fitness of 0.4343. For setup-1, when the population size is 15, the load value obtained by the proposed method is 23.43 %, 10.19 %, 7.18 %, 5.31 %, 4.43 %, and 2.58 % higher when compared to the existing approaches namely, Artificial Bee colony(ABC), Efficient Load Optimization and Resource Minimization (ELORM), Adaptive Parameter- Ant Colony Optimization (AP-ACO), Multi-Objective Memetic Algorithm-Adaptive Plant Intelligent Behavior Optimization (MOMA-APIBO), Cooling Control Algorithm (CCA), and Minimum Total Power (MinPR).

Analysis of parabolic trough solar collector thermal efficiency with application of a graphene oxide nanosheet‐based nanofluid

AbstractNowadays, clean energy production and reconciliation with nature is one of the best solutions known to rectify the global warming issue. Solar energy, as a clean and green one, has recently been drawing scientists' attention to itself more than ever. Applying parabolic trough solar collectors (PTSCs) is one of the state‐of‐the‐art ways to extract energy from the sun. As one of the most interesting research topics, the thermal efficiency enhancement of PTSCs is studied. The current study demonstrates how the thermal efficiency of PTSCs is enhanced through the application of graphene oxide (GO) nanosheets, as an inexpensive nanomaterial with superb thermal conductivity and unique structure, in thermal oil, namely, Behran oil, as heat transfer fluid, within the effective volume fraction range of 0.05. The above‐mentioned analysis is based on the theoretical calculations performed through a mathematical model based on the Taylor series approximation. All the calculations are performed with respect to a PTSC unit of a solar–thermal power plant situated in Shiraz, Iran, as the research case study. In addition, the Nan et al. model is used to analytically predict how the increase in the volume fraction of GO nanosheets raises the thermal conductivity and consequently increases the convection heat transfer coefficient of the nanofluid, leading to the thermal efficiency enhancement of PTSCs. The results reveal a maximum thermal efficiency of 71.10% for the volume fraction of 0.05. In comparison to the thermal efficiency of the real‐world case‐study mentioned above with pure thermal oil, that is, 67.71%, the thermal efficiency enhancement by about 5.5%, is achieved. Moreover, it is discussed how the thermal efficiency achieved from GO nanosheets surpasses the ones obtained from the application of particle shape nanomaterials, or carbon nanotubes in the same conditions.

Formulation and convergence analysis of an efficient higher order iterative scheme

&lt;p&gt;This contribution presents a highly efficient three-step iterative scheme. The proposed scheme is different in itself by achieving seventh-order convergence. The scheme is very useful for equations of nonlinear nature having multiple roots. The Taylor series expansion is employed to rigorously analyze the convergence of the presented scheme. That the scheme is effective and robust can be fit through a variety of examples from different fields. Numerical experimentation demonstrates the scheme’s rapid and reliable convergence to the true root and comparing its performance against existing techniques in the literature. Additionally, basins of attraction are visualized to offer a clear, comparative view of how different methods perform with varying initial guesses. The results show that this new scheme consistently compete well over other methods. This makes it a powerful tool for solving complex equations.&lt;/p&gt;

Geometrically nonlinear transient analyses of rotating structures through high-fidelity models

This work presents geometrically nonlinear transient analyses of various rotating blades. The structures are discretized through refined beams or multi-dimensional finite element models, which are generated using the Carrera Unified Formulation (CUF). The CUF offers a procedure to develop low- and high-fidelity one-dimensional, two-dimensional, and three-dimensional finite element models hierarchically and automatically. Various beam models were developed using different kinematics models based on Taylor or Lagrange expansion functions. Multi-dimensional models were obtained by merging beam and solid elements, exploiting the unique feature of Lagrange polynomials to have only pure displacements as unknowns. This property allows beam and solid elements to be coupled at the node level without requiring complicated mathematical formulations. By utilizing the Finite Element Method in conjunction with the CUF, the governing equations are written by including all rotation effects, namely the Coriolis term, spin-softening, and geometrical stiffening. In a total Lagrangian scenario, the Hilbert-Hughes-Taylor-α method and the iterative Newton-Raphson scheme are employed to solve the equations of motion. The proposed methodology has been applied to evaluate different blade configurations, comparing the solution obtained using linear, linearized, and nonlinear approaches. The results have been verified and validated by comparing them with existing solutions present in the literature.

A Modification to the Novel Toy Model of the Riemann Zeta Function Roots Equation

It is true that my previous series of papers about the Riemann Hypothesis has raised lots of discussions or the concerns among the mathematics society. Indeed, some professionals may think that the Riemann Zeta function at s = 1 or ∑_(n=1)^∞▒1/n is actually divergent and may tend to an infinity, how it can be acted as a denominator in a proof or something like 1/∞ ? This author’s idea is there may be no need to evaluate such kind of summation as one may consider the sum just as an (imaginary) constant (but NOT the complex number). Or we may only consider such summation as a kind of improper integrals with a black-boxed answer. In other words, the true final value of the summation ∑_(n=1)^∞▒1/n is NOT an concerning serious issue. The function ∑_(n=1)^∞▒1/n^s is in practice a smooth function and equals to log N or the logarithmic function for s = 1 or have the infinite differentiable properties except n = 0, that is why the commercial software Maple Soft can compute the Zeta function’s associated Taylor series successfully. In fact, from the above series together with the harmonic properties, one may calculate the artifical Zeta function’s root model equation such as the 0.5 +/- β*cot(ln(x))/(αx+1)n (will be discussed in my next paper of the series). Indeed, this is NOT the best model. Actually, the true Riemann Zeta non-trivial Root model equation is: {z = x + yI | z∈0.5±(y_1⩽y⩽y_2 )I} where y1= (±cot(k))/ln(k) , y2 = (±tan(k))/ln(k) with 0&lt;k&lt;2π. Certainly, this author cannot avoid mistakes. Some of the defects in the previous paper such as one in the linguistic proof of the RH will be stated and amended. Lastly, this author wants to remind that the present series of paper is first to establish a model equation for zeta function’s roots and the Plank’s like constant. This author then applies the telescopic and logarithmic methods for a 3 cases proof of the RH. Also, we have a Matlab code for the evaluation of RH complex contour integral. Finally, the RH problem will be transformed into a linguistic one and hence we may have solved the Riemann Hypothesis issue completely through a 4 cases of truth table collaborations with the logical inference [7] &amp; [15] to the causality (structural) organization of the sentences [16] or just named as “Logical &amp; Organized Context”.

A new analytical method for designing centralised PI controllers for unstable systems using a direct synthesis approach

This paper introduces a new analytical technique using a direct synthesis strategy to develop centralised proportional–integral (PI) controllers for multivariable processes. The current method design controller is focused on attaining the desired closed-loop response for multi input multi output (MIMO) processes that involve multiple time delays. The conventional multivariable PI controller is obtained by approximating the ideal multivariable controller by the Maclaurin series expansion. This is accomplished by choosing the desired closed-loop response order as the system order plus two. Subsequent analysis investigates the proposed method’s efficacy in designing multivariable PI controllers. Servo and regulatory problems studies were conducted to find the effectiveness of the proposed method and compared it with other methods recently reported in the literature, as well as assessment of performance indices like integral absolute error (IAE) and integral square error (ISE). The proposed controller's robustness is assessed by plotting the inverse maximum singular value against frequency, both input and output multiplicative uncertainties.

An Integral-like Numerical Approach for Solving Burgers’ Equation

The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the &amp;lt;i&amp;gt;ӏ&amp;lt;/i&amp;gt;&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, &amp;lt;i&amp;gt;ӏ&amp;lt;/i&amp;gt;&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, and &amp;lt;i&amp;gt;ӏ&amp;lt;/i&amp;gt;&amp;lt;sub&amp;gt;∞&amp;lt;/sub&amp;gt; error norms down to the order of 10&amp;lt;sup&amp;gt;−4&amp;lt;/sup&amp;gt;. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is &amp;lt;i&amp;gt;ν&amp;lt;/i&amp;gt;∆&amp;lt;i&amp;gt;t&amp;lt;/i&amp;gt;/∆&amp;lt;i&amp;gt;x&amp;lt;/i&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; &amp;gt; 0.02, including the viscosity coefficient &amp;lt;i&amp;gt;ν&amp;lt;/i&amp;gt; due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆&amp;lt;i&amp;gt;t&amp;lt;/i&amp;gt; even when using a small grid spacing ∆&amp;lt;i&amp;gt;x&amp;lt;/i&amp;gt;, emphasizing its suitability for practical applications such as numerical weather prediction.