Stable Numerical Algorithm Research Articles

Higher Order Stable Numerical Algorithm for the Variable Order Time-Fractional Sub-diffusion Equation

Higher Order Stable Numerical Algorithm for the Variable Order Time-Fractional Sub-diffusion Equation

Effects of OpenCL-Based Parallelization Methods on Explicit Numerical Methods to Solve the Heat Equation

In recent years, the need for high-performance computing solutions has increased due to the growing complexity of computational tasks. The use of parallel processing techniques has become essential to address this demand. In this study, an Open Computing Language (OpenCL)-based parallelization algorithm is implemented for the Constant Neighbors (CNe) and CNe with Predictor–Corrector (CpC) numerical methods, which are recently developed explicit and stable numerical algorithms to solve the heat conduction equation. The CPU time and error rate performance of these two methods are compared with the sequential implementation and Euler’s explicit method. The results demonstrate that the parallel version’s CPU time remains nearly constant under the examined circumstances, regardless of the number of spatial mesh points. This leads to a remarkable speed advantage over the sequential version for larger data point counts. Furthermore, the impact of the number of timesteps on the crossover point where the parallel version becomes faster than the sequential one is investigated.

Stable implicit numerical algorithm of time-dependent Ginzburg–Landau theory coupled with thermal effect for vortex behaviors in hybrid superconductor systems

Stable implicit numerical algorithm of time-dependent Ginzburg–Landau theory coupled with thermal effect for vortex behaviors in hybrid superconductor systems

High order stable numerical algorithms for generalized time-fractional deterministic and stochastic telegraph models

High order stable numerical algorithms for generalized time-fractional deterministic and stochastic telegraph models

The impact of dissipative algorithms on assessment of high-speed train running safety on railway bridges

The global expansion of high-speed railway networks has elevated their significance as a primary mode of transportation, particularly concerning train running safety (TRS) on bridges during seismic events. Achieving an accurate simulation model for train-bridge coupled (TBC) systems is often hindered by oversimplified assumptions. Refined TBC modeling adds complexity to the relationship between train wheels and tracks, necessitating the discretization of continuous models. However, this process leads to numerical errors and overshooting phenomena, significantly affecting TRS assessment. To tackle these challenges, an efficient unconditionally stable explicit numerical dissipation algorithm is designed to minimize errors and mitigate overshooting in TRS calculations. Furthermore, three TRS indices, the derailment factor, offload factor, and velocity-related spectral intensity, are investigated using spectral radii. The results show that adjusting the spectral radius can effectively reduce false peaks in these indices. Nonetheless, excessively small values may lead to inaccurate TRS assessments. Comparative analysis reveals that setting the spectral radius to 0.5 achieves relatively precise TRS assessments. This study underscores the pivotal role of dissipative algorithms in the safety assessment of TBC systems, an aspect overlooked in previous research.

A flux-based moving mesh method applied to solving the Poisson–Nernst–Planck equations

The moving mesh method is one of the important adaptive mesh methods which is practically useful when the mesh size and overall resolution are provided and fixed as seen in many engineering computations. We employ a moving mesh technique combined with a finite element method (FEM) to solve a widely-used charged carrier transport model, the Poisson–Nernst–Planck (PNP) equations, the solutions of which often have boundary or internal layers and sharp interfaces when the convection is dominated. Considering that flux is a significant physical quantity in designing stable and accurate numerical algorithm for transport problems such as the PNP system, we start from a relatively general functional and propose a flux-based monitor function and an adaptive step size-controlling algorithm for guiding the mesh movement in the FEM solution of the PNP equations. The numerical results illustrate that the moving mesh method is effectively addressing challenges arising from solution singularities and the convection-dominated effects. It also shows that the moving mesh finite element method we propose exhibits superior performance compared to both the traditional moving mesh finite element method and fixed mesh finite element method in some scenarios. Furthermore, it demonstrates better adherence to the physical properties of energy dissipation inherent in the PNP equation.

Numerical simulation of foam displacement impacted by kinetic and equilibrium surfactant adsorption

This study explores the effectiveness of foam in reducing gas mobility in porous media impacted by surfactant adsorption in highly heterogeneous porous media. The research focuses on assessing the influence of kinetic and equilibrium adsorption during foam displacement on reservoir sweep efficiency, considering co-injection and SAG (Surfactant Alternating Gas) injection strategies. Including the surfactant on the liquid phase and considering the surfactant adsorption effects fitted by experimental data, the mathematical modeling combines Darcy’s law, fluid phase conservation, surfactant transport equations, and a non-Newtonian foam model in local equilibrium. The simulations utilize FOSSIL (FOam diSplacement SImuLator), an in-house software composed of a stable and conservative numerical algorithm with reduced numerical diffusion. These properties are achieved by combining a hybrid mixed finite element method to solve the Darcy system with a central-upwind finite volume scheme to approximate the transport equations and adopting an implicit adaptive method in time. The results reveal that while omitting adsorption phenomena enhances sweep efficiency due to reduced surfactant loss, differences between equilibrium and kinetic adsorption models are minor (up to 0.3% in production). Consequently, these results suggest using simple adsorption models, such as equilibrium isotherms, rather than more complex models, such as kinetics. Additionally, the SAG approach outperforms co-injection (up to 19% in production), despite the fact that the adsorption is detrimental to both of them, aligning with literature results and previous findings.

An Unconditionally Stable Numerical Algorithm for Two-Dimensional Convection-Diffusion-Reaction Equations

An Unconditionally Stable Numerical Algorithm for Two-Dimensional Convection-Diffusion-Reaction Equations

Численное моделирование цунами оползневого типа

The article proposes mathematical models and stable numerical algorithms for solving problems of deformation and flow of water-saturated porous materials. This class of problems arises in many engineering and applied fields, from field development to biomechanics. Of particular interest are the problems of the collapse of the banks of rivers, lakes and bays of the seas due to their catastrophic impact on possible consequences, for example, blocking the riverbed, the occurrence of a tsunami on the water surface of a reservoir, etc. For the mathematical description of the stress-strain state of saturated porous media, the Biot theory of three-dimensional consolidation is used, supplemented by the kinetics of damage accumulation in the medium and the Drucker-Prager yield conditions. After the formation of the main yield surface (or crack) in the material at the moment of the beginning of the collapse, its rheological properties are presented in the form of a Bingham model with a mathematical description similar to the Navier-Stokes equations with effective viscosity. A characteristic hydrodynamic feature of the problem under consideration is the presence of a free surface. Numerical modeling of the considered problems is carried out by the finite element method. The evolution of the free surface and the occurrence of a tsunami generated by a landslide are investigated.

Stable numerical algorithm with localized radial basis function for solution of fractional convection–diffusion–reaction equation

Designing an efficient high-order numerical discretization is one of the challenges in numerical solution of fractional differential equations. Therefore in this paper to overcome the ill-conditioning that often destroys the convergence rate of global RBF methods, we apply a local meshfree method known as radial basis function-generated finite difference (RBF-FD) method equipped with a greedy algorithm to design stable stencil weights and approximate spatial derivatives for parabolic fractional partial differential equations (FPDEs) of convection–diffusion–reaction type. In RBF-FD method to select stable stencils that make weights with desirable properties for differentiation matrix, we apply a greedy algorithm, then convergence and stability of proposed scheme are investigated theoretically and numerically to show efficiency of developed technique. Finally, numerical results are provided by some FPDEs in regular and irregular shaped domains to illustrate the approximation quality and convergence of our approach in comparison with the results presented in literatures.

Applying recent efficient numerical methods for long-term simulations of heat transfer in walls to optimize thermal insulation

Since transient simulations are quite resource intensive, the winter heat loss through building walls is often estimated by simple steady-state calculation based on e.g. the Degree-days method, which is often rather inaccurate. In this work, transient simulations are performed using the very recent leapfrog-hopscotch and modified Dufort-Frankel techniques, which are the most effective explicit and stable numerical algorithms to cope with heat transport issues, according to prior investigations. The optimum insulation thickness, energy savings, and payback time are determined using an economic model that takes into account the orientation of the outside walls, solar radiation, the cost of insulating material, the present cost of energy consumption, and the cost over the 25-year lifetime of a building in Miskolc City and a case is studied in the cold season. Three materials (Expanded Polystyrene (EPS), glass-wool, and rock-wool) and several thicknesses are examined. As a result, it was found that for the north-oriented wall, the optimum insulation thickness is (17, 22, and 12 cm) and the life cycle energy savings are (142.5, 153, and 132.12 kWh/m2) for the EPS, glass-wool, and rock-wool, respectively, while the payback time is (3.73, 3.14, and 4.33 years). According to the analysis of the life cycle energy savings, the optimal insulation properties can be achieved using 22 cm thick glass wool, which only slightly depends on the orientation of the wall.

An investigation of earth and sea-return impedances of power electrical cables

This paper investigates the numerical performance of the Wedepohl series and of the Gauss-Kronrod quadrature to calculate ground- and sea-return impedances of power cables. Many computational routines to calculate ground-return impedances of cables used in EMTP-type programs have been proposed without technical justification of their application range. In this paper, we explore the convergence and accuracy of the Wedepohl series based on the number of terms in the series and as a ratio test. As a result, an accurate and stable numerical algorithm to compute ground- and sea-return impedances is proposed. The proposed method is capable of accounting for very low resistivity values and large depths.

Radon transform with Gaussian beam: Theoretical and numerical reconstruction scheme

The Radon transform and its various types have been studied since its introduction by Johann Radon in 1917. Since the Radon transform is an integral transform that maps a given function to its line integral, it has been studied in the field of computerized tomography, which deals with electromagnetic waves that primarily travel along straight lines, such as X-rays. However, in many laser optics applications, it is assumed that the laser beam is shaped like a Gaussian bell rather than a straight line. Therefore, in tomographic modalities using optical beams, such as optical projection tomography, images reconstructed with the inversion algorithms for the standard Radon transform are subject to a loss of quality. To address this issue, one needs to consider theoretical inversion methods for Radon transforms with Gaussian beam kernels and associated numerical reconstruction methods. In this study, we consider a Radon transform with a Gaussian beam kernel (also known as the point spread function) and show the uniqueness of the inversion of this transform. Furthermore, we provide an accurate and stable numerical reconstruction algorithm using the point spread function-sequential quadratic Hamiltonian scheme. Numerical experiments with disk and Shepp–Logan phantoms demonstrate that the proposed framework provides superior reconstructions compared to the traditional filtered back-projection algorithm.

Assessment of Correction Methods Applied to BEMT for Predicting Performance of Horizontal-Axis Wind Turbines

Blade Element Momentum Theory (BEMT) is the most used method to design horizontal-axis wind turbines worldwide. This is because BEMT has a low computational cost and easy numerical implementation. Additionally, it is demonstrated in the literature that the prediction of output power using BEMT agrees well with experimental data. Another important feature of the BEMT is its applicability to small, medium, and large-sized turbines. However, BEMT models are usually implemented and adjusted for a specific power range turbine, and they are not applied in a more general form. Thus, this article presents an analysis of additional correction methods for tip and root losses, high values of the axial induction factor, and high angle of attack to better represent horizontal-axis turbines in terms of numerical stability. The approach has the intention of combining several complementary correction methods strategically inserted in the BEMT in order to compile an algorithm that is more general, stable, and workable for any turbine size. The main contribution of this work is to propose a stable BEMT numerical algorithm through the assessment of the combination of the correction methods available in the literature, i.e., classical and modern ones. The algorithm ensures applicability for small, medium, and large-sized wind turbines, as well as being fast and easy to implement in any computer and extendable even to turbines with a diffuser. This approach is validated by comparing the results with experimental data from four turbines of different power ranges (1.9 kW to 7.3 MW). The results show the best approximations for performance power curves against the measured values of all turbines. Moreover, it is effective, less complex, and quick in analyzing the performance of those turbines. Furthermore, the need for high-performance computers to analyze the performance of horizontal-axis turbines is avoided.

Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation

We collected 20 explicit and stable numerical algorithms for the one-dimensional transient diffusion equation and analytically examined their consistency and convergence properties. Most of the methods used have been constructed recently and their truncation errors are given in this paper for the first time. The truncation errors contain the ratio of the time and space steps; thus, the algorithms are conditionally consistent. We performed six numerical tests to compare their performance and try to explain the observed accuracies based on the truncation errors. In one of the experiments, the diffusion coefficient is supposed to change strongly in time, where a nontrivial analytical solution containing the Kummer function was successfully reproduced.

An accurate and parallel method with post-processing boundedness control for solving the anisotropic phase-field dendritic crystal growth model

A fast, accurate, and stable numerical algorithm is proposed to solve the anisotropic phase-field dendritic crystal growth model. The algorithm combines the first-order direction splitting method and the linear stabilization technique, which preserve the energy stability while ensuring fast computing. The related stability analysis is shown. To obtain high accurate solution, the solution of first-order direction splitting method is extrapolated to be fourth-order, and the fourth-order difference method is applied to spatial discretization. To further improve the stability and prevent the blow-up of numerical solution, two post-processing methods, the bound preserving least-distance modification and the cut-off approach, are developed to control the boundedness of numerical solution, and their theoretical proofs are given. The proposed novel algorithm can be performed in parallel in each time step, significantly improving computing efficiency. Several numerical examples demonstrate the effectiveness of the proposed algorithm, including convergence and stability tests, the effectiveness of two post-processing techniques, and a series of 2D and 3D dendritic crystal growth simulations.

An accurate and stable numerical method for option hedge parameters

We propose an unconditionally stable numerical algorithm, which uses the Feynman-Kac formula of the Black-Scholes equation to obtain accurate option prices and hedge parameters. We discretize the asset and time using uniform grid points. We approximate the option values by piecewise quadratic polynomials for each time step and integrate them analytically over the log-normal distribution. The piecewise quadratic approximation gives the third-order convergence in the asset direction, and the analytic integration reduces truncation error in the time direction. The estimation errors are propagated backward in time following the convection and diffusion characteristics of the Black-Scholes equation, which assures the unconditional stability of our method. The vectorized code implementation reduces the time complexity. The convergence test shows that our approach outperforms the Crank-Nicolson scheme of the finite difference method in both time and asset directions, and the stability test verifies that our method is stable as the Crank-Nicolson. Furthermore, we show that our algorithm reduces the price errors and hedge parameter errors by more than 50% from the benchmark.

An improved morgenstern-price method using gaussian quadrature

The Morgenstern-Price (MP) method is an effective solution to compute the factor of safety (FOS) of a slope against sliding that has been used extensively for decades. In the MP method, the soil mass above the slip surface needs to be tediously divided into vertical slices to compute the force and moment integrals. The proposed research improves the computational efficiency of the MP method by employing the Gaussian quadrature for the integrals, where the soil mass division into a significant number of vertical slices is improved by aligning a few Gauss points along the slip surface. Consequently, the computational efficiency can be dramatically improved, resulting in a reduction in calculation efforts. Moreover, this research further refines the iteration scheme to satisfy the equilibrium conditions for the FOS using a robustly stable numerical algorithm with two initial scaling factors. The critical slip surface was determined by the simulated annealing algorithm. Finally, the proposed method was thoroughly validated via a series of benchmark examples. This research could improve the traditional MP method by enhancing its robustness and efficiency and provide an informative technique for force integrals that may be motivated by other problems in geotechnical engineering.

Stable Numerical Differentiation Algorithms Based on the Fourier Transform in Frequency Domain

A class of stable numerical differential algorithms is constructed based on the Fourier transform. The instability of the numerical differentiation problem is overcome by modifying the integral “kernel” in the frequency domain. The convergence of the approximate derivatives is ensured based on some reasonable assumptions of the modified “kernel” function. The a-posteriori choice strategy of the regularization parameter is considered. Moreover, the convergence analysis and error estimate of the approximate derivatives are also given.

A general framework to derive linear, decoupled and energy-stable schemes for reversible-irreversible thermodynamically consistent models

This paper presents a general numerical platform for designing accurate, efficient, and stable numerical algorithms for incompressible hydrodynamic models that obey thermodynamical laws. The obtained numerical schemes are automatically linear in time. It decouples the hydrodynamic variable and other state variables such that only small-size linear problems need to be solved at each time marching step. Furthermore, if the classical velocity projection method is utilized, the velocity field and pressure field can be decoupled. In the end, only a few elliptic-type equations shall be solved in each time step. This strategy is made possible through a sequence of model reformulations by exploring the models' thermodynamic structures. The generalized Onsager principle directly guides these reformulation procedures. In the reformulated but equivalent models, the reversible and irreversible components can be identified, guiding the numerical platform to decouple the reversible and irreversible dynamics. This eventually leads to decoupled numerical algorithms, given that the coupling terms only involve irreversible dynamics. To further demonstrate the numerical platform's power, we apply it to several specific incompressible hydrodynamic models. The energy stability of the proposed numerical schemes is shown in detail. The second-order accuracy in time is verified numerically through time step refinement tests. Several benchmark numerical examples are presented to further illustrate the proposed numerical framework's accuracy, stability, and efficiency.