System Of Partial Differential Algebraic Equations Research Articles

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.

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.

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.

Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

This paper presents a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.

Mathematics and Numerics for Balance Partial Differential-Algebraic Equations (PDAEs)

We study systems of partial differential-algebraic equations (PDAEs) of first order. Classical solutions of the theory of hyperbolic partial differential equation such as discontinuities (shock and contact discontinuities), rarefactions and diffusive traveling waves are extended for variables living on a surface $$\mathcal {S}$$ , which is defined as solution of a set of algebraic equations. We propose here an alternative formulation to study numerically and theoretically the PDAEs by changing the algebraic conditions into partial differential equations with relaxation source terms (PDREs). The solution of such relaxed systems is proved to tend to the surface $$\mathcal {S}$$ , i.e., to satisfy the algebraic equations for long times. We formulate a unified numerical scheme for systems of PDAEs and PDREs. This scheme is naturally parallelizable and has faster convergence. We do not perform a rigorous analysis about the convergence or accuracy for the method, the evidence of its effectiveness is presented by means of simulations for physical and synthetical problems.

Analytical and approximate solutions of Fractional Partial Differential-Algebraic Equations

Abstract In this paper, we have extended the Fractional Differential Transform method for the numerical solution of the system of fractional partial differential-algebraic equations. The system of partial differential-algebraic equations of fractional order is solved by the Fractional Differential Transform method. The results exhibit that the proposed method is very effective.

Rapid Modeling and Parameter Estimation of Partial Differential Algebraic Equations by a Functional Spreadsheet Paradigm

We present a systematic spreadsheet method for modeling and optimizing general partial differential algebraic equations (PDAE). The method exploits a pure spreadsheet PDAE solver function design that encapsulates the Method of Lines and permits seamless integration with an Excel spreadsheet nonlinear programming solver. Two alternative least-square dynamical minimization schemes are devised and demonstrated on a complex parameterized PDAE system with discontinues properties and coupled time derivatives. Applying the method involves no more than defining a few formulas that closely parallel the original mathematical equations, without any programming skills. It offers a simpler alternative to more complex environments which require nontrivial programming skill and effort.

On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks

We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semidiscretization in space via a mixed finite element method and also during the further dimension reduction by structure- preserving Galerkin projection, which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models, and we discuss certain modifications needed to satisfy some algebraic compatibility conditions which are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our arguments are based on a careful analysis of the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests, and the necessity of the compatibility conditions is demonstrated by examples.

Modeling boyciana–fish–human interaction with partial differential algebraic equations

Under the influence of human population distribution, the boyciana–fish ecological system is considered. First, the system can be described as a nonlinear partial differential algebraic equations system (PDAEs) with Neumann boundary conditions and ratio-dependent functional response. Second, we examine the system’s persistence properties: the loacl stabilities of positive steady states, the absorbtion region and the global stability. And the proposed approach is illustrated by numerical simulation. Finally, by using the realistic data collected in the past fourteen years, the PDAEs parameter optimization model is built to predict the boyciana population.

Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

Steric effects on ion dynamics near charged electrodes

We present a modified Poisson–Nernst–Planck (mPNP) model to include steric effects on ion dynamics near charged electrodes. This contribution appears directly on chemical potential of each ion in solution as a non-electrostatic term. The model consists in a partial-differential-algebraic equation (PDAE) system, which after spatial discretization was solved by the well-established DASSLC code, implemented in EMSO simulator. A careful dimensional analysis was carried out to turn the PDAE system dimensionless with normalized spatial domain. Size correlation (SC) effects for equally sized ions avoid unphysical charge densities near the electrodes (especially at high voltages), and also give some insights on double layer formation. Time evolution of free charge density profiles show a diffusion controlled dynamics with a very fast response (t∼10−8s). The methodology proposed here allows the inclusion of new contributions on PNP model framework for ion dynamics, as well as to solve both stationary and dynamical ion distributions, taking advantage of the EMSO software simulator features.

On the convergence rate of dynamic iteration for coupled problems with multiple subsystems

In multiphysical modeling coupled problems naturally occur. Each subproblem is commonly represented by a system of partial differential-algebraic equations. Applying the method of lines, this results in coupled differential-algebraic equations (DAEs). Dynamic iteration with windowing is a standard technique for the transient simulation of such systems. In contrast to the dynamic iteration of systems of ordinary differential equations, convergence for DAEs cannot be generally guaranteed unless some contraction condition is fulfilled. In the case of convergence, it is a linear one.In this paper, we quantify the convergence rate, i.e., the slope of the contraction, in terms of the window size. We investigate the convergence rate with respect to the coupling structure for DAE and ODE systems and also for two and more subsystems. We find higher rates (for certain coupling structures) than known before (that is, linear in the window size) and give sharp estimates for the rate. Furthermore it is revealed how the rate depends on the number of subsystems.

Suboptimal control of a 2D molten carbonate fuel cell PDAE model

Numerical results for suboptimal boundary control of an integro partial differential algebraic equation system of dimension 28 are presented. The application is a molten carbonate fuel cell power plant. The technically and economically important fast tracking of the new stationary cell voltage during a load change is optimized. The nonstandard optimal control problem s.t. degenerated PDE is discretized by the method of lines yielding a very large DAE constrained standard optimal control problem. An index analysis is performed to identify consistent initial conditions.

Parameter Identification in One‐Dimensional Partial Differential Algebraic Equations

AbstractIn this paper we discuss a couple of situations, where algebraic equations are to be attached to a system of one‐dimensional partial differential equations. Besides of models leading directly to algebraic equations because of the underlying practical background, for example in case of stationary equations, there are many others where the specific mathematical structure requires a certain reformulation leading to time‐independent equations. To be able to apply our approach to a large class of real‐life problems, we have to take into account flux formulations, constraints, switching points, different integration areas with transition conditions, and coupled ordinary differential algebraic equations (DAEs), for example. The system of partial differential algebraic equations (PDAEs) is discretized by the method of lines leading to a large system of differential algebraic equations which can be solved by any available implicit integration method. Standard difference formulas are applied to discretize first and second partial derivatives, and upwind formulae are used for transport equations. Proceeding from given experimental data, i.e., observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, which depends on the solution of the system of PDAEs. Parameters to be identified can be part of the differential equations, initial, transition, or boundary conditions, coupled DAEs, constraints, fitting criterion, etc. Also the switching points can become optimization variables. The resulting least squares problem is solved by an adapted sequential quadratic programming (SQP) algorithm which retains typical features of a classical Gauss‐Newton method by retaining robustness and fast convergence speed of SQP methods. The mathematical structure of the identification problems is outlined in detail, and we present a number of case studies to illustrate the different model classes which can be treated by our approach. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimization under uncertainty of a composite fabrication process using a deterministic one-stage approach

Process design under uncertainty has received considerable attention in recent years, and has led to the development of several modeling and solution approaches. These approaches are broadly categorized under stochastic formulations (model parameters with probability distributions), multiperiod formulations (where uncertain parameters are discretized into a number of deterministic realizations), and parametric programming formulations. This paper presents an application of the one-stage optimization problem (OSOP), a multiperiod method, to find optimal cure temperature cycle design under uncertainty for polymer-matrix composites fabrication using the pultrusion process. The process is governed by a highly non-linear system of partial differential-algebraic equations. The OSOP method is also systematically compared with a sampling-based approach in terms of computational efficiency and solution quality. Most work done so far using such deterministic methods has focused on problems where the performance objective function (often cost) and process constraints are analytic/algebraic in nature. In contrast, in materials processing simulations, evaluation of the objective function and the process constraints are based on the solution of differential-algebraic equations (DAE).

Efficient Solution and Sensitivity Analysis of Partial Differential-Algebraic Equation Systems: Application to Corrosion Pit Initiation

A mathematical model based on transport and reaction equations describing initiation of pitting corrosion at a MnS sulfide inclusion in NaCl electrolyte was used as a test problem for development of efficient numerical procedures for corrosion simulation and sensitivity analysis. The equations associated with the corrosion model were formulated by using a reaction invariant approach, which resulted in an index-2 partial differential-algebraic equation (PDAE) system. The original PDAE system was regularized to yield a lower index PDAE system, for which we were able to formulate an effective preconditioner for use in the DASPK 3.0 DAE solver. This approach was found to improve the efficiency of the Krylov iterative solver as well as to provide the parameter sensitivities with little additional effort in comparison with another method reported recently. In agreement with previous findings, the results obtained with this approach indicate that the model is most sensitive to the parameters associated with the heterogeneous dissolution of the sulfide inclusion.

Adaptive solutions of [formula omitted]-approximations to radiative heat transfer in glass

This paper presents numerical models of radiative heat transfer in glass manufacturing that can be performed on normal workstations, yet are sufficiently accurate for many practical applications. Since many of the glass production processes are so complex that a complete simulation is still unthinkable at present, there is a great interest for such models in order to optimise final glass products. We use simplified approximations of spherical harmonics to obtain approximate solutions of high accuracy in optically thick regimes. The arising systems of partial differential-algebraic equations of mixed parabolic–elliptic type are numerically solved by a self-adaptive discretization method based on an error-controlled finite element method in space and a one-step method of Rosenbrock-type with variable step sizes in time. The method itself judges the quality of the numerical solutions and determines adaptive strategies to keep the discretization error below a user-prescribed tolerance. Multilevel techniques based on reliable and efficient a posteriori error estimators and time embedding are used to improve the spatial discretization by local refinement and to steer the step size selection routine. We present numerical results for a typical step in glass manufacturing, the cooling of a glass cube. Our approximate solutions are validated with solutions to the full radiative transport equation and compared to Rosseland approximations widely used in the engineering practice. The results show that simplified approximations of spherical harmonics are efficient and sufficiently accurate. They are a significant improvement of the classical diffusion models.

Index analysis and numerical solution of a large scale nonlinear PDAE system describing the dynamical behaviour of Molten Carbonate Fuel Cells

This paper deals with the efficient simulation of the dynamical behaviour of Molten Carbonate Fuel Cells (MCFCs). MCFCs allow an efficient and environmentally friendly energy production via electrochemical reactions. Their dynamics can be described by large scale systems of up to currently 22 nonlinear partial differential algebraic equations (PDAE). The paper also serves as a basis for later parameter identification and optimal control purposes. Therefore, the numerical simulations are particularly based on hierarchically embedded systems of PDAE, first of all in one space dimension. The PDAE are of mixed parabolic-hyperbolic type and are completed by nonlinear initial and boundary conditions of mixed type. For a series of embedded models in one space dimension, the vertical method of lines (MOL) is used throughout this paper. For the semi-discretization in space appropriate difference schemes are applied depending on the type of equations. The resulting system of ordinary differential algebraic equations (DAE) in time is then solved by a standard RADAU5 method. In order to justify the numerical procedure, a detailed index analysis of the PDAE systems with respect to time index, spatial index and MOL index is carried through. Because of the nonlinearity of the PDAE system, the existing theory has to be generalized. Moreover, MOL is especially suited for near optimal real time control on the basis of a sensitivity analysis of the semi-discretized DAE system, since a theoretically safeguarded sensitivity analysis does not exist so far for PDAE constrained optimal control problems of the above type. Numerical results complete the paper and show their correspondence with the expected dynamical behaviour of MCFCs.

Transient and steady-state models for simulated moving bed processes: numerical solutions

The objective of this work is to compare different strategies for the numerical solution of mathematical models of simulated moving bed (SMB) processes to predict the transient and steady-state behaviour. The mathematical model is based on the true moving bed (TMB) description of SMB processes. The model assumes axial dispersion flow for the liquid phase and plug flow for the solid phase. Intraparticle mass transfer was described in terms of a simple linear driving force (LDF) approximation. The mathematical models for transient situations (TM_1 and TM_2) can be written in terms of system of partial differential-algebraic equations (PDAEs) or partial differential equations (PDEs), respectively, when the algebraic equation representing the adsorption equilibrium isotherm is inserted into the mass conservation/mass transfer kinetics equations or written as separate equation. Similarly, for steady-state the mathematical models (SSM_3 and SSM_4) are represented by a system of differential-algebraic equations (DAEs) or ordinary differential equations (ODEs). Different public domain solvers were used to numerically solve the four mathematical models described above. For the transient model TM_1 the DASSL solver was used to solve the system of PDAEs and for TM_2 the PDECOL package was used to solve the system of PDEs. For the steady-state model, the solvers COLDAE and COLNEW were used to solve the DAEs and ODEs systems, corresponding to the models SSM_3 and SSM_4, respectively. The glucose/fructose separation was used as a test case in a SMB with a 12 column configuration considering both linear and non-linear isotherms. The study of the effect of the operating conditions on the TMB performance and separation regions was also analysed. For the calculation of steady-state concentration profiles and process performance COLDAE solver is recommended because it requires less computing time and allows working with the adsorption equilibrium isotherm separately permitting easy changes on the algebraic relationship.

Index Analysis of a Nonlinear PDAE System Describing a Molten CarbonateFuel Cell

AbstractWithout combustion, molten carbonate fuel cells (MCFC) convert chemical energy contained in fuel and oxidizer to electric energy via electro‐chemical reaction. Performance and service life of MCFCs depends on its operating temperature. So control of the operation temperature within a specified range and reducing temperature fluctuations is highly desirable. In a very first step towards this goal prior to numerical simulations an index analysis of the partial differential‐algebraic equation system (PDAE) is given. Some numerical solutions can be found in [4].