Stiff Systems Of Differential Equations Research Articles

VENICE: A multi-scale operator-splitting algorithm for multi-physics simulations

Context. We present VENICE, an operator-splitting algorithm to integrate a numerical model on a hierarchy of timescales. Aims. VENICE allows a wide variety of different physical processes operating on different scales to be coupled on individual and adaptive time-steps. It therewith mediates the development of complex multi-scale and multi-physics simulation environments with a wide variety of independent components. Methods. The coupling between various physical models and scales is dynamic, and realised through (Strang) operators splitting using adaptive time-steps. Results. We demonstrate the functionality and performance of this algorithm using astrophysical models of a stellar cluster, first coupling gravitational dynamics and stellar evolution, then coupling internal gravitational dynamics with dynamics within a galactic background potential, and finally combining these models while also introducing dwarf galaxy-like perturbers. These tests show numerical convergence for decreasing coupling timescales, demonstrate how VENICE can improve the performance of a simulation by shortening coupling timescales when appropriate, and provide a case study of how VENICE can be used to gradually build up and tune a complex multi-physics model. Although the examples provided here couple dedicated numerical models, VENICE can also be used to efficiently solve systems of stiff differential equations.

Atmospheric Chemistry Surrogate Modeling With Sparse Identification of Nonlinear Dynamics

AbstractModeling atmospheric chemistry is computationally expensive and limits the widespread use of chemical transport models. This computational cost arises from solving high‐dimensional systems of stiff differential equations. Previous work has demonstrated the promise of machine learning (ML) to accelerate air quality model simulations but has suffered from numerical instability during long‐term simulations. This may be because previous ML‐based efforts have relied on explicit Euler time integration—which is known to be unstable for stiff systems—and have used neural networks which are prone to overfitting. We hypothesize that parsimonious models combined with modern numerical integration techniques can overcome this limitation. Using a small‐scale mechanism to explore the potential of these methods, we have created a machine‐learned surrogate by (a) reducing dimensionality using singular value decomposition to create an interpretably‐compressed low‐dimensional latent space and (b) using Sparse Identification of Nonlinear Dynamics (SINDy) to create a differential‐equation‐based representation of the underlying dynamics in the compressed latent space with reduced stiffness. The root mean square error of ML model prediction for ozone concentration over 9 days is 37.8% of the root mean square concentration across all simulations in our testing data set. The surrogate model is 10× faster with 12× fewer integration timesteps compared to reference model and is numerically stable in all tested simulations. Overall, we find that SINDy can be used to create fast, stable, and accurate surrogates of a simple photochemical mechanism. In future work, we will explore the application of this method to more detailed mechanisms and their use in large‐scale simulations.

Multi-objective optimization through a novel Bayesian approach for industrial manufacturing of Polyvinyl Acetate

ABSTRACT Manufacturing long-chain branched polymers enhances the quality of products at the cost of production time, translating into high manufacturing costs. In this manuscript, we study the cost-versus-quality trade-off while optimizing the operating conditions for large-scale industrial production of Polyvinyl acetate (PVAc). PVAc polymerization is emulated using a large system of stiff differential equations resulting in time-expensive function evaluation. Thus, we aim to achieve the Pareto optimality using Multi-Objective Bayesian Optimization (MOBO) that introduces q-Expected Hyper-Volume Improvement (qEHVI) as the novel acquisition function. Comparison with Non-dominated Sorting Genetic Algorithm-II (NSGA-II), applied to the high-fidelity model, resulted in a similar Pareto front with only 1% of the high-fidelity calls required by NSGA-II. The results indicate the efficiency of MOBO for PVAc optimization and present a real-world application of a generic method that can be implemented to solve time-expensive multi-objective optimization in manufacturing processes

Computer Simulation of Coke Sediments Burning from the Whole Cylindrical Catalyst Grain

The article is devoted to the development of the mathematical model of oxidative regeneration of the cylindrical catalyst grain. The model is constructed using a diffusion approach to modeling catalytic processes. The model is based on the equations of material and thermal balance. Mass transfer in the catalyst grain is carried out due to diffusion and the Stefan flow resulting from a decrease in the reaction volume during sorption processes. Chemical transformations of substances are taken into account as a source term in the equation. The thermal balance of the catalyst grain is described by a thermal conductivity equation, with an inhomogeneous term responsible for heating the grain during exothermic chemical reactions. The effective coefficients of heat capacity and thermal conductivity of the catalyst grain, which are determined taking into account the porosity of the grain depending on temperature, were used to calculate the thermal balance of the catalyst grain. The dependencies are approximated using the method of least squares based on experimental data. Different boundary conditions for the developed model allow calculating the main characteristics of the oxidative regeneration process for a whole catalyst grain under different conditions. The mathematical model of oxidative regeneration of a cylindrical catalyst grain is described by a stiff system of differential equations. Splitting by physical processes is applied to avoid computational difficulties. The calculation of flows is carried out sequentially: first, chemical problems are solved using the Radau method, then the diffusion and thermal conductivity equations are solved by the finite volume method. The result of the algorithm implemented in C++ is a picture of the distribution of substances and temperature along the cylindrical grain of the catalyst.

Simulation study on thermal characteristics of primary coolant system of lead-bismuth cooled reactor

According to the structural characteristics of the lead-bismuth (LBE) cooled reactor and the flow and heat transfer characteristics of the coolant, a real-time simulation model of the primary coolant system (PCS) is developed. The Reference Fluid Properties (REFPROP) is called to calculate the properties of supercritical carbon dioxide (S–CO2), and the International Mathematics Statistics Library (IMSL) is used to call D_IVPAG function to solve the system of stiff differential equations in fully implicit scheme. Using the idea of matrix modularity, the transient code is developed in FORTRAN language. The transient characteristics of power step, primary and secondary flow rate and temperature step are studied respectively. The results show that the transient code can well reflect the dynamic characteristics of the LBE cooled reactor, and the code and algorithm meet the real-time requirements.

Feasibility study for the creation and implementation of an instrumental complex for determining and predicting the technical characteristics of locomotives

A modern tool for research and rapid prototyping of new developments are various methods of digital system modeling. When creating or upgrading any electric transport systems and electric rolling stock, the correct choice of all elements of the traction power supply system and electric rolling stock is of great importance. To solve this problem, it is necessary to take into account the movement of a set of moving loads in the form of electric transport in power supply systems of various levels and, at the same time, to study not only the mechanical processes of electric transport movement, but also the electromagnetic processes of interaction of electric transport with the traction power supply system and the electric power system, as well as their mutual interaction. electromagnetic influence on each other. The construction of a software and hardware complex for real-time simulation of the joint operation of electric power systems, traction power supply systems and electric trains corresponds to the modern moment. The implementation of such a system is possible using modern high-performance computing systems or stationary supercomputers with integrated hardware that increase the speed of calculating stiff systems of differential equations that describe large electro-magnetic-mechanical systems. Such computing software and hardware systems are quite expensive. The article describes the composition of such a complex and the feasibility study for its development and implementation.

ОСОБЕННОСТИ ВТОРИЧНОГО ЗАГРЯЗНЕНИЯ ВОЗДУХА ПРИ ИЗМЕНЕНИИ ПЛОТНОСТИ ПОТОКА ТРАНСПОРТА

The pollution of the urban atmosphere in the area of the direct action of the motorway is considered. A certified environmental impact assessment methodology was used as the master. With its help, the masses of substances – that are primary pollutants – emitted per unit of time are calculated depending on the intensity of the traffic flow. Further, to calculate their concentrations, the dispersion is calculated. In order to determine the concentration of the secondary pollutant, ozone, a mathematical model is proposed. It is a stiff system of differential equations that describes the kinetics of the process of chemical transformation of primary pollutants. To solve it, the Rosenbrock method was applied. The influence of the traffic flow density on the ozone content near the highway is being studied

Solutions of Stiff Systems of Ordinary Differential Equations Using Residual Power Series Method

The stiff differential equations occur in almost every field of science. These systems encounter in mathematical biology, chemical reactions and diffusion process, electrical circuits, meteorology, mechanics, and vibrations. Analyzing and predicting such systems with conventional numerical techniques require more time and memory; still accurate solution is completely uneconomical and uncertain. Most of the numerical techniques have stability issues while dealing with stiff systems. To overcome these limitations, residual power series method (RPSM) is proposed for stiff systems of differential equations (DEs). RPSM is applied to various linear and nonlinear stiff systems, and closed‐form solutions are achieved. This indicates the effectiveness of proposed scheme for stiff family of DEs. Since this method leads to better results with less computational cost, it can be extended for more complex systems which arise in different areas of engineering and sciences.

Study of stiff differential equations of mathematical description of izomerization of pentane-hexane cut process

This paper provides a brief overview of the existing definitions of a stiff system of differential equations. Further, on the example of the accepted scheme of chemical transformations of the catalytic isomerization process of the pentane-hexane fraction, the stiffness of the system of differential equations was studied. In the course of the work, a method for studying the direct kinetic problem for stiffness is presented. In the Matlab software, the results of solving a system of differential equations by five methods (solvers) were compared. The given method can be tried for solving other problems of chemical kinetics.

Robust optimization of stiff delayed systems: application to a fluid catalytic cracking unit

We apply a robust steady state optimization method for stiff delay differential equations to the economic optimization of a fluidized catalytic cracking unit. Stiff systems of differential equations appear in this case due to the different time scales in the gas and fluid phase. Delays result from the catalyst hold-ups in the standpipes connecting riser and regenerator. We show that the proposed robust optimization method can cope with stiffness and delays. Moreover, the proposed method is capable of simultaneously optimizing the process parameters and tuning controller parameters.

Determination of single-phase transformer saturation characteristic by using Nelder–Mead optimization method

Correctly modelling the nonlinear saturation characteristics of the transformer iron core is of great importance in steady-state calculations (harmonics and power quality) and calculations of the electromagnetic transients (inrush current, ferroresonance, temporary overvoltages). Thus, this paper presents an original estimation algorithm for determining the transformer nonlinear saturation characteristic from the measured inrush and steady-state current waveforms in entire windows of occurrences. The proposed algorithm is based on the minimization of a specifically defined cost function by applying the Nelder–Mead method, which was used to create a balance between the transformer steady-state current and the inrush current at zero residual flux. The algorithm uses the BDF numerical method in the background to solve the stiff differential equation system. By applying the appropriate indexes of quality estimation, it was demonstrated that, out of a total of twelve proposed analytical functions, the best optimization results were achieved with three proposed functions (the Frolich extended, three-term polynomial and special hyperbolic function). Detailed performances and accuracy suggested that the estimation algorithm has been verified by a very close agreement between the simulation results and measured transformer current waveforms in the steady-state and different inrush current scenarios.

ХИМИЧЕСКИЕ ПРЕВРАЩЕНИЯ ВОССТАНОВЛЕННЫХ СОЕДИНЕНИЙ УГЛЕРОДА И ОКСИДОВ АЗОТА В СТЕСНЕННЫХ АЭРОДИНАМИЧЕСКИХ УСЛОВИЯХ ГОРОДСКОЙ АТМОСФЕРЫ

The kinetics of chemical transformations of the most significant pollutants in the cramped aerodynamic conditions of the urban atmosphere is considered, i.e. in the gaps between rows of wide buildings with significant height. High concentrations created in those areas lead to sufficiently high values of reaction rates. The process is described by a stiff system of differential equations, the solution of which is performed by Rosenbrock method. The effect of reduced carbon compounds and nitrogen oxides in a wide range of their concentrations on the rate of formation of a highly toxic secondary pollutant is investigated. In the course of numerical experiments, conditions are determined that correspond to its maximum value. In that case, the transfer of substances from the reaction zone does not have time to occur.

Explicit Stabilized Integrators for Stiff Optimal Control Problems

Explicit stabilized methods are an efficient alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimension. In this paper we derive explicit stabilized integrators of orders one and two for the optimal control of stiff systems. We analyze their favorable stability properties based on the continuous optimality conditions. Furthermore, we study their order of convergence taking advantage of the symplecticity of the corresponding partitioned Runge--Kutta method involved for the adjoint equations. Numerical experiments including the optimal control of a nonlinear diffusion-advection PDE illustrate the efficiency of the new approach.

Contributions of VT and VV exchanges to the CO2 relaxation in the full state-to-state modeling

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a computationally complex problem requiring the solution of a huge system of stiff differential equations. In the present study, we use parallel numerical scheme, based on the extended backward differential formula with adaptive timestep strategy. Using this scheme, we study the contribution of various types of energy exchanges to the solution of kinetics equations. The comparison of the results obtained with the use of the ”cut” and full state-to-state approaches is also shown in the paper.

Optimization of CO2 Vibrational Kinetics Modeling in the Full State-to-State Approach

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a challenging time-consuming computational task that involves solving a huge system of stiff differential equations and requires optimized methods to solve it. In the present study, we propose and analyse optimizations for the Extended Backward Differential Formula (EBDF) scheme. Using adaptive timesteps instead of fixed ones reduces the number of steps in the algorithm many thousands of times, although with an increase in step complexity. The use of parallel computations to calculate relaxation terms allows one to further reduce the computation time. Numerical experiments on the modeling of spatially homogeneous carbon dioxide vibrational relaxation were performed for optimized computational schemes of different orders. Based on them, the most optimal algorithm of calculations was recommended: a parallel EBDF scheme of fourth-order with an adaptive timestep. This method takes less computational time and memory costs and has the high stability.

Investigation of rigid dynamic systems on the example of modelling a tape drive mechanism

The features of modeling the dynamics of mechanical systems on the example of the operation of the tape drive mechanism related to real technological processes are stated. An approach to solving stiff systems of differential equations by the numerical-analytical method is noted. The approach is based on solving systems of higher-order differential equations using elementary functions using procedures for precision search for the roots of the characteristic polynomial of the system. A mathematical model of the tape drive mechanism of a VCR is given as an example of a precision electromechanical object.

Optimization of CO2 vibrational kinetics modeling in the full state-to-state approach

«Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» — научно-теоретический рецензируемый журнал, публикующий исследования в области математики, механики и астрономии, а также математического моделирования.

On stiff, fuzzy IRD-14 day average transmission model of COVID-19 pandemic disease

COVID-19, a new pandemic disease is becoming one of the major threats for surviving. Many new models are arrived to study the disease mathematically. Here we are introducing a new model in which instead of studying a day by day changes we are studying the average of 14 day transmission because its life or the patients incubation period is about an average of 14 days. Also, since this is pandemic, and being not aware of susceptible population among the world's population, we considered the model without S-susceptible population. i.e., IRD-Infectious, Recovered, Death-model. In this new model, we are also introducing a new method of calculating new number called <i>N</i>₀-average transmission number. This is used to study the average spread of infection instead of basic reproduction number <i>R</i>₀. The motto of this paper is not to predict the daily cases but to control the current spread of disease and deaths by identifying the threshold number, exceeding which will increase the spread of infection and number of deaths due to this pandemic. Also if the 14 day average IRD-populations are maintained under this threshold number, will definitely control this pandemic disease globally. Stability analysis and test for stiff system of differential equations are studied. Our main aim is to present the medical world, a threshold population of infected, recovered and death cases for every average of 14 days to quickly overcome this pandemic disease COVID-19.

Reduction of stiffness‐induced round‐off errors in chemical reaction systems

AbstractThe timescales of chemical reactions range from nanoseconds to minutes. Hence, chemical reaction systems result in stiff systems of differential equations. Usually, implicit integration schemes are used in order to solve these stiff systems of differential equations. Thus, Newton's method is used in order to solve a nonlinear equation system in each time step. Thereby the evaluation of the chemical source term requests subtraction of very large numbers, and round‐off errors by cancellation occur. This can cause severe convergence problems within Newton's method, resulting in step size reductions. The system of differential equations is replaced by a less stiff modified system in order to reduce the round‐off errors and the computing time. Thereby the approximation error between the given system of differential equations and the modified system of differential equations is smaller than the given tolerance.

Effects of Langmuir Turbulence on Upper Ocean Carbonate Chemistry

AbstractEffects of wave‐driven Langmuir turbulence on the air‐sea flux of carbon dioxide (CO2) are examined using large eddy simulations featuring actively reacting carbonate chemistry in the ocean mixed layer at small scales. Four strengths of Langmuir turbulence are examined with three types of carbonate chemistry: time‐dependent chemistry, instantaneous equilibrium chemistry, and no reactions. The time‐dependent model is obtained by reducing a detailed eight‐species chemical mechanism using computational singular perturbation analysis, resulting in a quasi steady state approximation for hydrogen ion (H+); that is, fixed pH. The reduced mechanism is then integrated in two half‐time steps before and after the advection solve using a Runge‐Kutta‐Chebyshev scheme that is robust for stiff systems of differential equations. The simulations show that as the strength of Langmuir turbulence increases, CO2 fluxes are enhanced by rapid overturning of the near‐surface layer, which rivals the removal rate of CO2 by time‐dependent reactions. Equilibrium chemistry and nonreactive models are found to bring more and less carbon, respectively, into the ocean as compared to the more realistic time‐dependent model. These results have implications for Earth system models that either neglect Langmuir turbulence or use equilibrium, instead of time‐dependent, chemical mechanisms.