Differential Equation Solver Research Articles

Gas discharge mechanism of weakly ionized gas sensor using nanomaterials

Gas discharge plays a very important role in the performance of ionized gas sensor. Due to the structure of nanomaterial-based electrode, gas discharge becomes more complexity under atmospheric pressure. In this article, we show a theoretical framework for the calculation of gas discharge of weakly ionized gas sensor using nanomaterials. It consists of the mass conservation equation for each species with the drift-diffusion approximation, the electron energy conservation equation, and the Poisson’s equation. The effects of ionization, excitation, attachment, recombination, electron diffusion, and is formulated by a time-dependent finite-element partial differential equation solver. The simulation results agree well with experimental data under the same conditions and gas discharge characters of ionized gas sensor can be extracted from the model to understand the discharge mechanism. Our approach is not only suitable for explaining the experimental phenomena by clarifying the gas chemical reaction process but also to design the sensor structure.

Effect of the conditional scalar dissipation rate in the conditional moment closure

In the context of modeling turbulent scalar mixing using probability density function (PDF) methods, the treatment of molecular mixing is of paramount importance. The conditional moment closure (CMC) offers a high-fidelity description for molecular mixing in nonpremixed flows. Recent work has demonstrated that first-order CMC can be implemented numerically using the moments of the conditioning variable and first-order joint moments of the scalar of interest. When solving the CMC using, for example, quadrature-based moment methods (QBMM), a functional form must be chosen for the conditional scalar dissipation rate (CSDR) of the conditioning variable. In prior work, the CSDR was chosen to produce a β-PDF for the conditioning variable (mixture fraction) at steady state. This choice has the advantage that the system of moment equations used in QBMM-CMC can be written in closed form. In this work, an alternative choice for the CSDR is investigated, namely, the amplitude mapping closure (AMC). With the AMC, the moment equations can be closed using the quadrature method of moments incorporated into a realizable ordinary differential equation solver. Results are compared with the β-CSDR closure for binary, passive scalar mixing in homogeneous single- and disperse-phase turbulent flows. It is also demonstrated that the moment formulation of CMC provides a straightforward method for modeling the effect of differential diffusion in the context of CMC.

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

Mathematical model for the dynamics of Savanna ecosystem considering fire disturbances

In this article, the stability of equilibrium solutions of a recently formulated mathematical model of Savanna ecosystem is analytically and numerically analyzed. The mathematical model is formulated by generalizing all plant life into three components; trees, tree saplings, and grass under ecologically valid effects of fire, rainfall and competition for space. Fire has a considerable effect on trees by delaying the recruitment of saplings to trees and the recruitment rate is a piecewise linear decreasing function of grass with a sigmoidal shape. This leads to there existing different equilibria in the plant community of a Savanna ecosystem. It is rigorously demonstrated that the local stability of equilibria depends on the slope and value of the recruitment function. Moreover, it is found that the composition of high grass cover and low tree cover or low grass cover and high tree cover are the stable equilibria, while intermediate cover results in unstable equilibria. In analyzing the global stability of solutions, it is found that the limit set is an equilibrium solution. Several numerical simulations are provided to validate the analytical studies of the behavior of the equilibrium solutions. The numerical solutions are generated using a Python ordinary differential equation(ODE) solver. The analytical and numerical solutions presented in this work are very important for further developments in the area of mathematical ecology.

Operator evolution from the similarity renormalization group and the Magnus expansion

The Magnus expansion is an efficient alternative to solving similarity renormalization group (SRG) flow equations with high-order, memory-intensive ordinary differential equation solvers. The numerical simplifications it offers for operator evolution are particularly valuable for in-medium SRG calculations, though challenges remain for difficult problems involving intruder states. Here we test the Magnus approach in an analogous but more accessible situation, which is the free-space SRG treatment of the spurious bound-states arising from a leading-order chiral effective field theory (EFT) potential with very high cutoffs. We show that the Magnus expansion passes these tests and then use the investigations as a springboard to address various aspects of operator evolution that have renewed relevance in the context of the scale and scheme dependence of nuclear processes. These aspects include SRG operator flow with band- versus block-diagonal generators, universality for chiral EFT Hamiltonians and associated operators with different regularization schemes, and the impact of factorization arising from scale separation. Implications for short-range correlations physics and the possibilities for reconciling high- and low-resolution treatments of nuclear structure and reactions are discussed.

Convergence rates of Gaussian ODE filters

A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss–Markov process {varvec{X}}, which is then iteratively conditioned on information about {dot{x}}. This article establishes worst-case local convergence rates of order q+1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of q=1 and an integrated Brownian motion prior, and analyses how inaccurate information on {dot{x}} coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to q in {2,3,ldots }.

Nonlinear Propagation in Optical Fibers With Gain Saturation and Gain Dispersion

There have been many developments in nonlinear propagation models in the past few decades. Especially, a form of such model has been developed to allow a standard ordinary differential equation (ODE) solver to be directly used for its solution. But such a model currently does not consider gain saturation and wavelength-dependent gain, which are very important in high-pulse-energy lasers. In this work, the directly-ODE-integrable nonlinear propagation equation is extended to include gain saturation and gain dispersion, which is then used to study maximum pulse energy limited by amplified spontaneous emission and minimum pulse width limited by gain narrowing in ultrafast fiber lasers in order to demonstrate these new capabilities.

Development of a Massively Parallelized Fluid-Based Plasma Simulation Code With a Finite-Volume Method on an Unstructured Grid

A massively parallelized unstructured-grid plasma simulation code based on a multifluid plasma model has been developed and validated. The code uses a collocated cell-centered finite-volume method and is designed to simulate intermediate low-pressure and atmospheric-pressure (AP) plasma sources with complex machine geometries. One of the novel features of this code is that it is implemented in a highly flexible computational platform named ultrafast massively parallel processing (ultraMPP), which allows straightforward addition and integration of different partial differential equation (PDE) solvers in a self-consistent manner. As to the numerical methods, the Scharfetter–Gummel scheme is adopted to handle the drift-diffusion flux of electrons, whereas the Harten–Lax-van Leer (HLL)-type approximate Riemann solver is used to handle the convection terms of ion momentum equations. For discretization of the diffusion terms in an unstructured grid, the Taylor expansion is used to deal with the effects of nonorthogonality of the cells, and the cell-center gradient is calculated using a least-squares method. The simulation code with a local-field approximation (LFA) was validated for two cases of AP plasmas as well as a case of an argon capacitively coupled plasma generated in a Gaseous Electronic Conference (GEC) reference cell with local mean energy approximation (LMEA) at intermediate low pressure. The simulation results were found to be in good agreement with the previously published experimental and simulation data.

Mathematical modeling of canonical and non-canonical NF-κB pathways in TNF stimulation

Background and objectiveNF-κB can be activated by the canonical and non-canonical pathways. These two pathways interplay via the TRAF1|NIK complex after stimulation by TNF. However existing mathematical models of two pathways are inadequate. In this context, an improved mathematical model is constructed to simulate these two pathways and their coupling stimulated by TNF. MethodsA schematic description of two NF-κB pathways and their relation after TNF stimulation is constructed at first. Then twenty-eight ordinary differential equations are utilized to build the mathematical model. Model equations are solved via the ordinary differential equation solver (ode23). ResultsThe proposed model firstly reconstructs the changes in concentrations of NF-κB pathway related biochemical factors with time, and further investigates the underlying mechanism of interaction between two pathways through the TRAF1|NIK complex after stimulation. ConclusionsThe model is validated through good agreement between simulation results and published experimental observations. This study helps to well understand the canonical and non-canonical pathways and their interaction. It also provides a potential tool to investigate how the dysregulated pathways act in pathological conditions.

Robust numerical solution for the two parameter solute solvent transport model

Prediction of solute and solvent transport in cells is central to developing and testing cryopreservation protocols. As we show here, however, the models used can be difficult to accurately numerically integrate in some key cases, and thus are a challenge to implement when determining the time dependent cell state during cryoprotectant equilibration and cooling. Exact solution techniques exist for overcoming this problem, but their implementation is also challenging: inversion of a nonlinear function is required that negates much of the utility of the approach. This communication describes a simple approach for more robust numerical integration that can be implemented using any numerical differential equation solver, and can facilitate arbitrarily accurate solutions to transport models without the complication of inversion formulae or complicated numerical integration schemes. Further, a simple relevant example of red blood cell equilibration with 40% glycerol is presented with comments on extending the approach to other settings.

Simultaneous approximation of a function and its derivatives by Sobolev polynomials: Applications in satellite geodesy and precise orbit determination for LEO CubeSats

The objectives of this paper are twofold: (1) to present a new method of approximation such that the function and some of its derivatives are simultaneously approximated; and (2) to investigate the potential applications of this new method in the field of satellite geodesy by deriving a numerical solver of ordinary differential equations. To fulfill both the objectives, Sobolev polynomials are used. Explicit formulas for these polynomials are presented. A quadrature rule is derived based on the explicit forms of these polynomials. Solution of first order ordinary differential equations is presented based on the numerical integration method by the new quadrature rule. To present the applications of the results, three problems are investigated in the field of satellite geodesy. In the first application, orbit propagated by the numerical solution of orbital equations is compared with the Keplerian motion. It is shown that the new method for solving ordinary differential equations can be used alongside any other method, including the adaptive Runge-Kutta and Adams-Bashforth-Moulton integration methods. The comparison of this new method with the adaptive Runge-Kutta, Adams-Bashforth-Moulton, and the powerful and newly established adaptive Gaussian numerical integration methods reveals that it is at least 580, 666667, and 72 times more accurate than the mentioned methods, respectively, in any given absolute tolerance and time increment. The method is fast, stable, consistent, and capable of handling high accuracy, even with large time increment and low absolute tolerance, which is shown in this application. In the second application, the Gravity Recovery and Climate Experiment (GRACE) satellites' orbits are propagated using both the Ensemble Kalman Filter (EnKF) with stochastic updates and smoothing for the position and the Sobolev numerical integration. The orbits are then compared with the observed positions of the satellites. It is shown that the Sobolev polynomials work better in this problem, being more than 2 times more accurate. In the third application, the precision orbit determination problem for the Low Earth Orbit (LEO) CubeSats is investigated in the reduced dynamic form. A case study is presented for a pico-nano CubeSat in China. It is shown that the reduced dynamic orbit is approximately 32 percent more accurate than either of the kinematic or static Precise Point Positioning (PPP),representing a maximum error of 3.4 cm, with respect to the observation of the ground tracking stations.

Scalability of high-performance PDE solvers

Performance tests and analyses are critical to effective high-performance computing software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this article, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for partial differential equations (PDEs) that govern a wide range of physical applications. We consider a sequence of PDE-motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.

Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic.

Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model COVID-19 pandemic by fractional order SIDARTHE model which did not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

A robust and accurate solver for some nonlinear partial differential equations and tow applications

In this paper we introduce a robust and accurate solver in order to solve various classes of nonlinear partial differential equations. This solver gives the closed formula for the solutions. Within this respect, a solver is developed to accurately resolve and represent the complete wave structure of the nonlinear partial differential equations. To verify this solver, the two applications are introduced. The obtained solutions may be applicable for some new observations in physics, fluid mechanics, biology, engineering, economy, astrophysics, meteorological topics, gravitational collapse and laser technology. The theoretical study and obtained solutions reported illustrating that the proposed solver efficient and adequate. The shape for the obtained solutions are illustrated via 2D and 3D plots for suitable values of the free parameters.

OpenCAL system extension and application to the three-dimensional Richards equation for unsaturated flow

OpenCAL is a scientific software library specifically developed for the simulation of 2D and 3D complex dynamical systems on parallel computational devices. It is written in C/C++ and relies on OpenMP/OpenCL and MPI for parallel execution on multi-/many-core devices and clusters of computers, respectively. The library provides the Extended Cellular Automata paradigm as a high-level formalism for modeling complex systems on structured computational grids. As a consequence, it can be used as a parallel explicit solver of ordinary and partial differential equations through classical methods, including finite-difference and finite-volume. Here the latest version of the library is described, introducing the MPI infrastructure over the 3D OpenCL and 2D/3D OpenMP components. The implementation of a three-dimensional unsaturated flow model based on a direct discrete formulation of the Richards’ equation is also shown, corresponding to a finite-difference scheme. Computational performances have been assessed on both a scientific workstation equipped with a dual Intel Xeon socket and three Nvidia GPUs, and a 16 nodes cluster with a fast interconnection network. The OpenCAL embedded quantization optimization is also discussed and exploited to drastically reduce computing time.

An acceleration scheme for deep learning-based BSDE solver using weak expansions

This paper gives an acceleration scheme for deep backward stochastic differential equation (BSDE) solver, a deep learning method for solving BSDEs introduced in Weinan et al. [Weinan, E, J Han and A Jentzen (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5(4), 349–380]. The solutions of nonlinear partial differential equations are quickly estimated using technique of weak approximation even if the dimension is high. In particular, the loss function and the relative error for the target solution become sufficiently small through a smaller number of iteration steps in the new deep BSDE solver.

Phononic transport and simulations of annealing processes in nanometric complex structures

Modeling thermal transport at the nanoscale is a difficult task, especially when external time-varying heating sources and the complexity of the studied systems make computational schemes that rely on accurate particlelike simulations nonaffordable. Alternative strategies based on corrections of the Fourier law could satisfy the trade-off between accuracy and computational efficiency, since they can be implemented in partial differential equation solvers. This continuum approach could also allow for the coupling between thermal transport and other evolving fields related to the generalized temperature field. Here we demonstrate that corrections due to the finite phonon mean free paths can be suitably included in annealing process simulations of three-dimensional nanosystems. Quantitative predictions can be obtained and readily compared with the experimental characterization of the processed samples.

Systematic Investigation of a Large Two-Stroke Engine Crankshaft Dynamics Model

The crankshaft dynamics model is of vital importance to a multitude of aspects on engine diagnostics; however, systematic investigations of such models performance (especially for large two-stroke diesel engines that are widely used in the power generation and shipping industries) have not been reported in the literature. This study aims to cover this gap by systematically investigating the parameters that affect the performance of a two-stroke diesel engine crankshaft dynamics model, such as the numerical scheme as well as the engine components inertia and friction. Specifically, the following alternatives are analysed: (a) two optimal performing numerical schemes, in particular, a stiff ordinary differential equations (ODE) solver and a fast solver based on a piecewise Linear Time-Invariant (LTI) scheme method, (b) the linear and the non-linear inertia-speed approaches, and (c) three engine friction submodels of varying complexity. All the potential combinations of the alternatives are investigated, and the crankshaft dynamics model performance is evaluated by employing Key Performance Indicators (KPIs), which consider the results accuracy compared to the measured data, the computational time, and the energy balance error. The results demonstrate that the best performing combination includes the stiff ODE solver, the constant inertia-speed approach and the most simplistic engine friction submodel. However, the LTI numerical scheme is recommended for applications that require fast response due to the significant savings in computational time with an acceptable compromise in the model results accuracy.

Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

Two‐phase natural convection analysis and hybrid nanoparticle migration around micromixer blades

AbstractIn this paper, numerical investigations are presented for hybrid nanoparticle migration and free convection heat transfer of two kinds of nanofluids in a micromixer at the fixed propeller condition. The inner blades and outer crust of the micromixer are kept at constant hot and cold temperatures, respectively. Two kinds of hybrid nanofluids, TiO2‐CuO water and ethylene glycol‐(MoS2‐SiO2), are considered. The governing equations including velocity, pressure, temperature formulation, and nanoparticle concentrations are solved by a partial differential equation solver based on the Galerkin finite element method. The results are discussed based on the governing parameters, such as nanoparticle volume fraction, thermal and solutal Rayleigh numbers. The average Nusselt number was found to increase with the increasing nanoparticle volume fractions. Also, increasing the thermal Rayleigh number enhanced heat transfer while the solutal Rayleigh number has an insignificant effect on it. More importantly, increasing the thermal Rayleigh number assisted avoiding the agglomeration of nanoparticles around the blades and ensured more uniform nanoparticle distribution.