Partial Differential Algebraic Equations Research Articles

Physics-Informed Neural Networks for Solving High-Index Differential-Algebraic Equation Systems Based on Radau Methods

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10−6, and the absolute errors for algebraic variables are maintained at 10−5. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

A Posteriori Error Estimation for Parabolic Problems with Dynamic Boundary Conditions

This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf–sup stable discretization schemes for a stationary model problem as a preliminary step. Based on an alternative formulation of the system as a partial differential–algebraic equation, we introduce a posteriori error estimators which allow local refinements as well as a special treatment of the boundary. We prove reliability and efficiency of the estimators and illustrate their performance in several numerical experiments.

Application of Adomian Decomposition Method for Solving Helmholtz Equation

The Adomian decomposition method (ADM) is effective for solving linear and nonlinear ordinary and partial differential equations, integral equations, algebraic equations and integro-differential equations and it leads to an analytical solution in the form of an infinite power series. We will solve the Helmholtz equation in two dimensional by the Adomian decomposition method.
 Key words: Application of Adomian Decomposition Method, Helmholtz Equation, Adomian Decomposition Method. 

Reactive transport model for calcite-rich caprocks in the context of ge-ological carbon storage

Increased concentrations of greenhouse gases in the atmosphere are known to be the primary cause of the increase in global surface temperature and, consequently, climate change [6]. In order to limit global warming, several decarbonisation strategies have been proposed: among them, geological CO2 storage represents very likely the only short- to medium-term option for substantially improving CO2 sinks and reducing net carbon emissions into the atmosphere.
 To achieve secure and lasting storage of CO2 in underground spaces, it is necessary to have both a reservoir and a low-permeability caprock (seal), that maintains its integrity. Deep saline aquifers, depleted oil and gas fields and unminable coal seams are thus the primary targets for the underground storage of supercritical CO2 [2]. The occurrence of natural reservoirs implies that certain lithotypes have a certain sealing capacity, which can prevent leakage of gas to the atmosphere over long geological time periods (i.e. million of years); however, in order to assess the risk of CO2 leakage through caprock above storage sites, the potential caprock alterations due to the contact with CO2 must be considered. In fact, although certain caprocks can be suitable for hydrocarbons, CO2 in contact with the seal may pose additional risks. As for natural hydrocarbon accumulation, shales and evaporites are potential seals also for carbon storage. With particular reference to shales, their mechanical and transport properties are controlled by the behaviour of clay minerals. The sealing efficiency of intact shale caprocks is in fact dominated by the high specific surface clays, which are characterized by high capillary entry pressures, but are also susceptible to electro-chemical interaction. From the engineering perspective, capillary and electrical phenomena also have relevant effects at the Representative Elementary Volume scale, causing mechanical couplings which could affect porosity, clay fabric, hydraulic conductivity, compressibility, and shear strength [3]. Besides these electro-chemical effects, CO2 dissolution and diffusion in water also result in acidification of the in situ brine, causing chemical reactions with some caprock minerals and potentially affecting the mechanical and transport properties [4].
 This work presents a reactive transport model to assess the effects of pore water acidification on caprock materials. The model includes the water mass balance equation for the saturated porous medium and the mass balance equation for all the primary species dissolved in water, according to the theoretical approach presented in [7]. The modelling approach proposed accounts for both the aqueous (homogeneous) reactions of the CO2 dissolved in water (assumed to be in equilibrium) and the dissolution kinetics of calcite in the acidic environment induced by CO2 injection (see [1] for details). Calcite dissolution is finally linked to porosity changes via the reactive surface area of the mineral and the reaction rate. Chemo-hydraulic coupling is addressed by considering porosity changes in the storage term of the balance equations and by introducing a suitable link between hydraulic conductivity and current porosity. The model requires the solution of an initial chemical speciation problem, which has been performed with the software PHREEQ-C, and then the integration of a set of partial differential equations (the mass balance equations of water and of the master primary species dissolved in water) and non-linear algebraic equations (to obtain the concentration of all the chemical species involved), which in this study has been performed via the Finite Element software Comsol Multiphysics ®, according to the approach presented in [8].
 The numerical model has been validated according to the numerical benchmark proposed in [5], developed to reproduce a geo-chemical scenario where mineral dissolution causes permanent alterations in the transport properties of a porous medium. In partic-ular, the benchmark reproduces a flow-through columns experiment, with the porosity and permeability of a calcite-rich interlayer changing under sulfuric acid attack. Figure 1 shows a comparison between the predictions of the current model and the model proposed in [5] in terms of the spatial evolution of porosity, permeability and dissolved calcite, after 400 hours since the injection of sulfuric acid. The model is able to reproduce that, as the acidic front proceeds into the sample from left to right, calcite is progressively dissolved causing porosity changes and the consequent permeability increase. The agreement between the two models is satisfactory, despite some slight differences that may be attributed to the different numerical mesh and the different numerical solver scheme.

Non-Iterative Calculation of Quasi-Dynamic Energy Flow in the Heat and Electricity Integrated Energy Systems

Quasi-dynamic energy flow calculation is an indispensable tool for the heat and electricity integrated energy system (HE-IES) analysis. One solves the nonlinear partial differential algebraic equations to obtain thermal, hydraulic and electric variations. However, mainstream iteration solvers face the challenges of inefficiency and bad robustness. For one thing, the frequent update and factorization of Jacobian matrices utilize high CPU time. For another, the per-step iteration numbers grow exponentially as the system loading level creeps up. This paper presents a novel non-iterative algorithm for the quasi-dynamic energy flow calculation. The kernel of the proposed algorithm is to transform these nonlinear equations into linear recursive ones, by solving which, we obtain explicit closed-form solutions of unknown variables. In each step, the proposed algorithm requires only one matrix factorization and fixed times of arithmetic operations regardless of the loading levels, so that it achieves small and consistent per-step time costs. A semi-discrete scheme is used in PDE solution to avoid dissipative and dispersive errors that are often overlooked in previous literature. To ensure convergence, we also propose to control the temporal step sizes adaptively by estimating the simulation errors. Case studies showed that the proposed method manifested efficient and robust time performance compared with the iterative algorithms, and meanwhile preserved high accuracy.

A second-order bulk–surface splitting for parabolic problems with dynamic boundary conditions

Abstract This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential–algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a backward differentiation formula in time. Within this paper, we focus on the second-order case, resulting in a $3$-step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically.

Simulation of Reactive Transport in Fractured Porous Media

Numerical simulations of reactive transport in fractured porous media require the solution of coupled physical and chemical processes that depend on the fractures. Such coupled processes are described by a system of nonlinear partial differential-algebraic equations, while strong heterogeneities characterise fractures. This paper presents an approach to simulate single-phase flow and non-isothermal reactive transport with mineral dissolution and precipitation in fractured porous media. Our numerical solution strategy is based on two ingredients. First, the model equations consist of coupled partial differential equations for the fluid flow, heat transfer and solute transport and nonlinear algebraic equations representing the chemical reactions. Second, fractures are explicitly represented and treated as lower-dimensional objects. The partial differential equations are discretised using finite-volume methods, and at each time step, we solve a nonlinear system of equations using Newton’s method. With numerical simulations, we illustrate our model’s ability to accurately describe the two-way interaction between coupled multi-physical processes and two- and three-dimensional porous media with intersecting fractures.

Canonical Euler splitting method for parabolic partial functional differential algebraic equations

A novel canonical Euler splitting method is presented for semilinear composite stiff parabolic partial functional differential algebraic equations with initial and Dirichlet boundary conditions. The original partial differential problems are transformed into the semi-discrete problems by spatial discretization, and then the canonical Euler splitting method is employed to solve the resulting semi-discrete problems. Under appropriate assumptions, the stability and convergence theories of this method are established. A series of numerical experiments are given to illustrate the effectiveness of this method and the correctness of theoretical results.Numerical results also demonstrate that the constructed method can significantly improve the calculation speed.

Nonlinear Dynamic Energy Flow Calculation for Integrated Gas and Power System Based on Method of Lines

The rapid development of gas turbine (GT) and power-to-gas (P2G) technology promotes the integration of power systems (PS) and natural gas systems (NGS). To accurately describe the state distribution, this paper first introduces the model of integrated gas and power system (IGPS) governed by nonlinear partial differential-algebraic equations (PDAEs). A third-order method of lines (MoL) is proposed to solve the nonlinear partial differential equations (PDEs) in the natural gas system. The distribution of mathematical equations and variables in the energy flow model is then analyzed. The solution strategy for nonlinear dynamic energy flow calculation (EFC) in integrated gas and power system is then developed, including the equation set formulation, solvability analysis, and initial guess selection method. Finally, the accuracy and efficiency of the proposed method are verified through simulation in two different cases.

Theoretical Analysis of a Nonequilibrium Transport Model of Two-Dimensional Nonisothermal Reactive Chromatography Accounting for Bi-Langmuir Isotherm.

The current study investigates a nonequilibrium and nonlinear two-dimensional lumped kinetic transport model of nonisothermal reactive liquid chromatography, considering the Bi-Langmuir adsorption isotherm, heterogeneous reaction rates, radial and axial concentration variations, and the adsorption and reaction enthalpies. The mathematical models of packed bed chromatographic processes are expressed by a highly nonlinear system of coupled partial differential algebraic equations connecting the phenomena of convection, diffusion, and reaction, for mass and energy balance, the differential algebraic equations for mass balance in the solid phase, and the algebraical expressions for the adsorption isotherms and for the reaction rates. The nonlinearity of the reaction term and the adsorption isotherm preclude the derivation of an analytical solution for the model equations. For this reason, a semidiscrete, high-resolution, finite-volume technique is extended and employed in this study to obtain the numerical solution. Several consistency checks are performed to evaluate the model predictions and analyze the precision of the proposed numerical scheme. A number of heterogeneously catalyzed stoichiometric reactions are numerically simulated to examine reactor performance under the influence of temperature and Bi-Langmuir adsorption dynamics, the level of coupling between mass and energy fronts, and to study the effects of various critical parameters. The numerical results obtained are beneficial for optimal predictive control and process optimization during production and the development of methods for systematic design and fault detection of nonisothermal liquid chromatographic reactors, and hence constitute the first step to provide deeper insight into the overall evaluation of integrated reaction and separation processes.

Reduced-Order Model of Lithium-Iron Phosphate Battery Dynamics: A POD-Galerkin Approach

Lithium iron phosphate batteries with plateau in the open circuit voltage, hysteresis, and path dependence dynamics due to phase transition during intercalation/de-intercalation are challenging to model and even more challenging to control. A core-shell electrochemical modeling approach is able to capture the phase transition behavior at the cost of using a fine-grained spatial grid to transform the governing Partial Differential Algebraic Equations into Ordinary Differential Algebraic Equations, resulting in a computationally expensive system intractable for the design of real-time battery management system algorithms. This letter presents a reduced-order modeling paradigm to transform the high-dimensional model into a low-dimensional yet accurate control oriented electrochemical model. The Proper Orthogonal Decomposition-Galerkin method is used to reduce the state variable vector from 169 to a meager 9 with negligible loss in fidelity. The reduced-order model is validated against both experimental data and the high-dimensional model for discharging-charging load profiles of different C-rates and real driving cycles. Promising results with one-third the computational burden and a voltage RMS error of less than 0.6% are achieved.

Model-Based Analysis of a Highly Efficient CO2 Separation Process Using Flexible Metal–Organic Frameworks with Isotherm Hysteresis

To realize large-scale CO2 separation to prevent global warming, energy and space saving separation technologies are required. Flexible metal–organic frameworks (flexible MOFs) with gate-opening properties have been found to be promising adsorbents, but the unconventional sigmoidal shapes and hysteresis of the isotherms have hindered quantitative evaluation of their applications in separation processes. This paper is the first to evaluate ELM-11, a flexible MOF that exhibits sigmoidal isotherms with hysteresis, in a process model and quantify its advantages over a conventional adsorbent. Based on the experimental uptake data, isotherm models for adsorption and desorption were developed, and their parameters were estimated. The resulting isotherm models are incorporated into a rigorous dynamic model of partial differential algebraic equations (PDAEs) to simulate a vacuum pressure swing adsorption (VPSA) process to evaluate the process performance. Numerical challenges to solve the PDAEs, including sigmoidal and hysteresis isotherm models, were resolved with the proposed numerical approaches. Sensitivity analysis for feed pressure and temperature was performed to identify the optimal operating strategy. A comparison with a conventional adsorbent, zeolite 13X, showed that high selectivity and sigmoidal isotherms of ELM-11 give higher productivity and CO2 product purity exceeding 99% without rinse and purge operations as well as lower power consumption for the compressor and vacuum pump.

Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions

Abstract In this paper, we introduce a stochastic partial differential equation model for the spatial dynamic of tumor–immune interactions. We perform numerical simulations in order to investigate the propagation of traveling waves in model system under the influence of random space-time fluctuations. One of methods is to solve a stochastic partial differential equation system for tumor–immune cell densities. The second method is to solve a stochastic partial differential algebraic equation system in order to assess the wave behavior of the solution in comparison with the deterministic approach. Finally, we discuss the implications of the model results.

Pyomo.DOE: An open‐source package for model‐based design of experiments in Python

AbstractPredictive mathematical models are a cornerstone of science and engineering. Yet selecting, calibrating, and validating said science‐based models often remains an art in practice. Model‐based design of experiments (MBDoE) provides a systematic framework to maximize information gain from experiments while minimizing time and resource costs. But MBDoE remains limited to niche application areas, in part because practitioners must integrate expertise in statistics, computational optimization, and modeling. To help reduce this barrier, we introduce Pyomo.DOE, an open‐source package for MBDoE. Pyomo.DOE uses a nonlinear sensitivity analysis code k_aug to quickly approximate the Fisher information matrix and leverages a new stochastic programming abstraction. We demonstrate Pyomo.DOE with the first application of MBDoE to fixed‐bed breakthrough experiments, which highlights the power of Pyomo.DOE to quantify the value of experimental modifications a priori for large‐scale partial differential‐algebraic equation (PDAE) models. We also provide a mathematical primer on MBDoE targeted at general chemical engineers.

Optimal Operation of an Industrial Natural Gas Fired Natural Draft Heater

In this work, a custom dynamic mathematical model of an industrial vertical-cylindrical type natural gas fired natural draft heater is developed using gPROMS® ProcessBuilder®. The integrated model comprises sub-models for each of the distinct sections of the fired heater which are connected by mass and energy flows. The temperature profiles of the tubular coils and the process fluid, a heat transfer fluid (HTF) are modelled using the distributed parameter system (DPS) in the axial direction (1D). The flue gas temperature in each section is modelled using the lumped parameter approach. Published empirical methods and correlations are used for estimating some unknown model parameters. The resulting model is a system of partial differential-algebraic equations (PDAEs) and serves as a basis for conducting an optimisation study to aid decision-making and to identify the best operating conditions within the specified constraints that minimise the daily operational costs. Through process simulation studies, the model predictions are adjusted to closely approximate collected actual plant data. It is demonstrated through the optimisation study that significant reduction in fuel gas consumption can be achieved compared to the current operating consumption levels. The developed models can be extended for use by other hydrocarbon processing plant operators with slight modifications, by specifying geometric parameters, HTF thermophysical properties, fuel gas composition and properties, among others.

Advantages, disadvantages and methods of applying mathematical models to evaluate water quality in reservoirs: a systematic review

Human activities are affecting reservoir water quality; consequently, methods are necessary to verify those impacts. Mathematical modeling improves the understanding of the anthropic impact on water quality, changes in limnological data, and helps formulate management strategies. However, it is necessary to consider the (dis)advantages as well as the methods used for water-quality assessment in reservoirs. This study conducted a systematic review in four databases: (i) PubMed/Medline; (ii) Scopus; (iii) Web of Science; and (iv) Wiley Online Library. We combined Boolean operators and words aiming to identify papers linked to the scope. Rayyan software allowed the initial screening of the found papers. Peer-reviewed papers and the use of mathematical models to assess reservoir water quality were the inclusion criteria. Exclusion criteria included articles in languages other than English, grey literature, and inaccessible articles. Our research found 169 articles, of which 39 were selected and only 13 were included in the review. Mathematical modeling has many benefits related to real-world problems, but the main disadvantages are process simplification, specific rules of the model, and lack of information or data monitoring. Kinetic equations, regression models, Monte Carlo analysis, finite segment models, modeling tools, zero-order rate equations, partial differential algebraic equations, and predictive analysis are the methods observed in mathematical modeling. This review provides information for unfamiliar managers who intend to use mathematical models to assess the water quality of reservoirs.
 Keywords: limnologic tool, model inventory, water management.

Analytical solutions for fractional partial delay differential-algebraic equations with Dirichlet boundary conditions defined on a finite domain

In this paper, we investigate the solution of multi-term time-space fractional partial delay differential-algebraic equations (MTS-FPDDAEs) with Dirichlet boundary conditions defined on a finite domain. We use Laplace transform method to give the solutions of multi-term time fractional delay differential-algebraic equations (MTS-FDDAEs). Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the MTS-FPDDAEs into the MTS-FDDAEs. By applying our obtained solutions to the resulting MTS-FDDAEs, the desired analytical solutions of the MTS-FPDDAEs are obtained. Finally, we give the solutions of some special cases.

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions

AbstractThis paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.

Singular perturbation results for linear partial differential–algebraic equations of hyperbolic type

We consider constrained partial differential equations of hyperbolic type with a small parameter ε>0, which turn parabolic in the limit case, i.e., for ε=0. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter ε. For the analysis, we consider the system equations as partial differential–algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order estimates are proven and their optimality is shown numerically.

Dynamic Modeling and Nonlinear Model Predictive Control of a Moving Bed Chemical Looping Combustion Reactor

A partial differential-algebraic equation (PDAE) model of a dynamic moving bed chemical looping combustion reduction reactor is developed in the IDAES process modeling framework for nonlinear model predictive control (NMPC). Numerical stability of the PDAE discretization and the index-1 property of the DAE are validated. An NMPC case study is performed, which manipulates inlet flow rates to control the system to a steady-state setpoint over approximately 6,000 s of simulated time, while respecting a lower bound on outlet oxygen carrier conversion. The dynamic optimization problems are solved for each NMPC cycle with an average of 39 CPUs, indicating that this model has the potential for use in a real-time scenario.