Integration Of Ordinary Differential Equations Research Articles

Exponentially fitted two-step peer methods for oscillatory problems

This paper concerns the construction of a general class of exponentially fitted two-step implicit peer methods for the numerical integration of Ordinary Differential Equations (ODEs) with oscillatory solution. Exponentially fitted methods are able to exploit a-priori known information about the qualitative behaviour of the solution to efficiently furnish an accurate solution. Moreover, peer methods are very suitable for a parallel implementation, which may be necessary in the discretization of Partial Differential Equations (PDEs) when the number of spatial points increases. Examples of methods with 2 and 3 stages are provided. Numerical experiments are carried out in order to confirm theoretical expectations.

Second derivative two-step collocation methods for ordinary differential equations

We introduce second derivative two-step collocation methods for the numerical integration of ordinary differential equations. In these methods, the solution of the problem in each step depends on the numerical solution in some points in the two previous steps. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. The construction technique, analysis of the order of accuracy and linear stability properties of the methods are described.

Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a random forcing term, we show that a probability measure over the numerical solution of ODEs can be obtained by introducing suitable random time-steps in a classical time integrator. This intrinsic randomization allows for the conservation of geometric properties of the underlying deterministic integrator such as mass conservation, symplecticity or conservation of first integrals. Weak and mean-square convergence analysis are derived. We also analyse the convergence of the Monte Carlo estimator for the proposed random time step method and show that the measure obtained with repeated sampling converges in the mean-square sense independently of the number of samples. Numerical examples including chaotic Hamiltonian systems, chemical reactions and Bayesian inferential problems illustrate the accuracy, robustness and versatility of our probabilistic numerical method.

Locally Exact Integrators for the Duffing Equation

A numerical scheme is said to be locally exact if after linearization (around any point) it becomes exact. In this paper, we begin with a short review on exact and locally exact integrators for ordinary differential equations. Then, we extend our approach on equations represented in the so called linear gradient form, including dissipative systems. Finally, we apply this approach to the Duffing equation with a linear damping and without external forcing. The locally exact modification of the discrete gradient scheme preserves the monotonicity of the Lyapunov function of the discretized equation and is shown to be very accurate.

Загальний розв’язок та адаптація до параметричного оцінювання SIR моделі

The article deals with analytical solution and adaptation to the parameter estimation of the SIR model of the epidemic. By a special replacement of the exponential function by inverse proportionality, the approximate general solution of the SIR model is found. It is spoken in detail about the process of integration of ordinary differential equations of the SIR model. The equality of the sum of the obtained analytical solutions and population size is checked. The obtained solutions are simple and understandable. To parametrically estimate the SIR model, its general solution is adapted to paired linear regressions. The article is of interest for students, graduate students and scientists involved in mathematical epidemiology.

Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.

An explicit marching-on-in-time scheme for solving the time domain Kirchhoff integral equation.

A fully explicit marching-on-in-time (MOT) scheme for solving the time domain Kirchhoff (surface) integral equation to analyze transient acoustic scattering from rigid objects is presented. A higher-order Nyström method and a PE(CE)m-type ordinary differential equation integrator are used for spatial discretization and time marching, respectively. The resulting MOT scheme uses the same time step size as its implicit counterpart (which also uses Nyström method in space) without sacrificing from the accuracy and stability of the solution. Numerical results demonstrate the accuracy, efficiency, and applicability of the proposed explicit MOT solver.

Inverse problem on conservation laws

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. Within this problem, one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing conservation of generalized circulation, generalized momentum and energy.

Some Study Methods for Ordinary Differential Equation Integrability of the Second Order of a Certain Type

The paper deals with treating some study methods of the equation integrability of a certain type that are little studied in the theory of differential equations. It is known that a significant part of the differential equations cannot be integrated. Then, to develop methods for their study is, certainly, of scientific interest. The obtained results, formulated as theorems and statements, are of scientific and practical interest because of their importance for applications in modern science.In the paper we present an alternative method for studying the integrability of both linear and nonlinear differential equations of the second order. An introduction of parameters allowed us to develop a study method for the integrability of ordinary differential equations of the second order. We also formulate the theorems describing some General conditions for the integrability of the second-order linear equation and consider special cases of integrability, which arise out of the above facts.Based on the obtained parameter method, some General conditions for the integrability of the nonlinear differential equation of the second order are given, and the consequences of these General conditions are indicated.We have obtained new results related to the construction and development of methods for studying the differential equation to which some types of differential equations are reduced and laid the foundations for a rigorous and systematic study of the introduced special nonlinear differential equation of the second order.

Integration of Ordinary Differential Equations on Riemann Surfaces with Unbounded Precision

We consider analytical systems of ODE with a real or complex time. Integration of such ODE is equivalent to an analytical continuation of a solution along some path, which usually belongs to the real axis. The problems that may appear along this path are often caused by singularities of the solution that lie outside the real axis. It is possible to circumvent problematic parts of the path (including singularities) by going on the Riemann surface of the solution (i.e., in the complex domain). A natural way to realize this program is to use the method of Taylor expansions, which does not require a formal complexification of the system (i.e., a change of variables). We use two classical problems, i.e., the Restricted Three-Body problem, and Van der Pol equation, for demonstration of how Taylor expansions can be used for integration of ODE with an arbitrary precision. We obtained some new results in these problems.

Analytical and numerical solutions to the problem on a heat wave initiating for the nonlinear heat equation with a source

The article deals with the nonlinear heat (porous medium) equation with a source. This equation has a large number of applications, as well as non-standard non-linear properties. An important class of solutions to the equation is heat waves (waves of filtration), propagating on a zero background with a finite velocity. This paper continues the cycle of articles devoted to the construction and study of heat waves with a closed front. Here we consider a problem with a given boundary condition with a source in cases of plane, circular and spherical symmetry. The solution is constructed in the form of a double power series, the theorem of existence and uniqueness is proved. We also look for some exact solutions of an equation with a power source, the construction of which reduces to the integration of ordinary differential equations (ODE) with a singularity. To solve the obtained ODE, we propose a computational algorithm based on the boundary element method. The results of numerical calculations are presented and discussed.

A relationship between λ‐symmetries and first integrals for ordinary differential equations

In this paper, we provide some geometric properties of λ‐symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of λ‐symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries. We also investigate the properties of λ‐symmetries in the sense of the deformed Lie derivative and differential operator. We show that λ‐symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases.

Adapted explicit two-step peer methods

Abstract In this paper, we present a general class of exponentially fitted two-step peer methods for the numerical integration of ordinary differential equations. The numerical scheme is constructed in order to exploit a-priori known information about the qualitative behaviour of the solution by adapting peer methods already known in literature. Examples of methods with 2 and 3 stages are provided. The effectiveness of this problem-oriented approach is shown through some numerical tests on well-known problems.

Dynamic adaptive chemistry with mechanisms tabulation and in situ adaptive tabulation (ISAT) for computationally efficient modeling of turbulent combustion

To deal with the major challenges of combustion simulation with detailed chemical kinetics, a new dynamic adaptive chemistry method with mechanisms tabulation (DAC-ST: DAC with single tabulation) and a method in which DAC-ST is combined with the ISAT (In Situ Adaptive Tabulation) approach (DAC-DT: DAC with double tabulation) to accelerate numerical calculation of large chemical mechanisms in turbulent combustion flows are proposed. Compared to the conventional DAC, DAC-ST expedites calculations by tabulating and reusing the reduced mechanisms, and DAC-DT further expedites the calculations by reducing the number of direct ordinary differential equations (ODEs) integrations and tabulating and retrieving the solutions. The correlation of the two mechanisms is evaluated by a pre-specified criterion comprising temperature, pressure, fuel, oxygen, OH, CH2O, and HO2, and can be updated by a user-specified threshold value. The proposed methods were validated in a piloted partially premixed methane/air diffusion flame (Sandia Flame D) with the 53-species GRI-Mech 3.0 and a hydrogen fueled model scramjet combustor (DLR) with 9 species and 27 reactions. The results indicate that the two proposed methods can accurately capture the flame structure and that the relative percentage errors of temperature and species concentration are well-controlled and proportional to the threshold values. Further, compared to the conventional DAC, detailed diagnostics show that DAC-ST and DAC-DT reduce the computational overhead of mechanism reduction by factors of 41 and 96 for Flame D and 165 and 365 for DLR, respectively. They also speed up ODE integration by factors of 3.9 and 7.8 for Flame D and 1.6 and 4.0 for DLR, respectively. The successful validation demonstrates that the two proposed methods can be efficiently used in the simulation of reactive flow for detailed kinetic mechanisms, especially for the case with a large number of cells, e.g., DNS.

Estimating Physics Models and Quantifying Their Uncertainty Using Optimization With a Bayesian Objective Function

This paper reports a verification study for a method that fits functions to sets of data from several experiments simultaneously. The method finds a maximum a posteriori probability estimate of a function subject to constraints (e.g., convexity in the study), uncertainty about the estimate, and a quantitative characterization of how data from each experiment constrains that uncertainty. While this work focuses on a model of the equation of state (EOS) of gasses produced by detonating a high explosive, the method can be applied to a wide range of physics processes with either parametric or semiparametric models. As a verification exercise, a reference EOS is used and artificial experimental data sets are created using numerical integration of ordinary differential equations and pseudo-random noise. The method yields an estimate of the EOS that is close to the reference and identifies how each experiment most constrains the result.

Nonlinear orbital uncertainty propagation with differential algebra and Gaussian mixture model

Nonlinear uncertainty propagation is of critical importance in many application fields of astrodynamics. In this article, a framework combining the differential algebra technique and the Gaussian mixture model method is presented to accurately propagate the state uncertainty of a nonlinear system. A high-order Taylor expansion of the final state with respect to the initial deviations is firstly computed with the differential algebra technique. Then the initial uncertainty is split to a Gaussian mixture model. With the high-order state transition polynomial, each Gaussian mixture element is propagated to the final time, forming the final Gaussian mixture model. Through this framework, the final Gaussian mixture model can include the effects of high-order terms during propagation and capture the non-Gaussianity of the uncertainty, which enables a precise propagation of probability density. Moreover, the manual derivation and integration of the high-order variational equations is avoided, which makes the method versatile. The method can handle both the application of nonlinear analytical maps on any domain of interest and the propagation of initial uncertainties through the numerical integration of ordinary differential equation. The performance of the resulting tool is assessed on some typical orbital dynamic models, including the analytical Keplerian motion, the numerical J2 perturbed motion, and a nonlinear relative motion.

L'origine des méthodes multipas pour l'intégration numérique des équations différentielles ordinaires

L'origine des méthodes multipas pour l'intégration numérique des équations différentielles ordinaires

Towards developing an adaptive time stepping for compressible unsteady flows

PurposeThe purpose of this paper is to solve unsteady compressible Navier–Stokes equations without the commonly used dual-time loop. The authors would like to use an adaptive time-stepping (ATS)-based local error control instead of CFL-based time-stepping technique. Also, an all-speed flow algorithm is implemented with simple low dissipation AUSM convective scheme, which can be computed without preconditioning which in general destroys the time accuracy.Design/methodology/approachIn transient flow computations, the time-step is generally determined from the CFL condition. In this paper, the authors demonstrate the usefulness of ATS based on local time-stepping previously used extensively in ordinary differential equations (ODE) integration. This method is implemented in an implicit framework to ensure the numerical domain of dependence always contains the physical domain of dependence.FindingsIn this paper, the authors limit their focus to capture the unsteady physics for three cases: Sod’s shock-tube problem, Stokes’ second problem and a circular cylinder. The use of ATS with local truncation error control enables the solver to use the maximum allowable time-step, for the prescribed tolerance of error. The algorithm is also capable of converging very rapidly to the steady state (if there is any) after the initial transient phase. The authors present here only the first-order time-stepping scheme. An algorithmic comparison is made between the proposed adaptive time-stepping method and the commonly used dual time-stepping approach that indicates the former will be more efficient.Originality/valueThe original method of ATS based on local error control is used extensively in ODE integration, whereas, this method is not so popular in the computational fluid dynamics (CFD) community. In this paper, the authors investigate its use in the unsteady CFD computations. The authors hope that it would provide CFD researchers with an algorithm based on an adaptive time-stepping approach for unsteady calculations.

The special class of second integrals of the KdV equation

The special class of second integrals, distinguished for its infinite dimension, emerges naturally when generalizing the second integrals of ordinary differential equations to partial differential equations. The conserved quantities of the KdV equation are a special class of second integrals. We proved its uniqueness under the assumption on the cofactor operator. The special class is so peculiar that to find it is almost an algorithm. Thus we managed to generalize the special class of the conserved quantities of the KdV equation to a new 2-parameter special class. Among the special classes, the special class of nonlocal second integrals plays an extra role. As an example, the special class that generates the multi-soliton solutions of the KdV equation is presented.

A Suite of Mechanistic Epidemiological Decision Support Tools

ObjectiveWe will demonstrate tools that allow mechanistic contraints on disease progression and epidemic spread to play off against interventions, mitigation, and control measures. The fundamental mechanisms of disease progression and epidemic spread provide important constraints on interpreting changing epidemic cases counts with time and geography in the context of on-going interventions, mitigations, and controls. Models such as these that account for the effects of human actions can also allow evaluation of the importance of categories of epidemic and disease controls.IntroductionWe present the EpiEarly, EpiGrid, and EpiCast tools for mechanistically-based biological decision support. The range of tools covers coarse-, medium-, and fine-grained models. The coarse-grained, aggregated time-series only data tool (EpiEarly) provides a statistic quantifying epidemic growth potential and associated uncertainties. The medium grained, geographically-resolved model (EpiGrid) is based on differential equation type simulations of disease and epidemic progression in the presence of various human interventions geared toward understanding the role of infection control, early vs. late diagnosis, vaccination, etc. in outbreak control. A fine-grained hybrid-agent epidemic model (EpiCast) with diurnal agent travel and contagion allows the analysis of the importance of contact-networks, travel, and detailed intervention strategies for the control of outbreaks and epidemics.MethodsWe use three types of methods for simulation and analysis. They are: (1) Bayesian and regression methods allowing estimation of the basic reproductive number from case-count data; (2) ordinary-differential equation integration with modifications to account for discreteness of disease spread when case counts are small (we include space- and time-dependent effects); and (3) methods that hybridize agent-based travel, mixing, and disease progression with nested-mass action contagion (i.e. not fully agent-based). From the perspective of decision support, the crucial feature of mechanistic infectious epidemiological models is a way to capture the human interventions that determine epidemic outcome. Categorizing types of mitigation into those that change the force of infection, and those that branch disease progression allows a common framework that can be extended from medium-grained models through fine-grained. Our canonical example is our EpiGrid tool which allows for the modulation of the force of infection (i.e. contagion) with time (and potentially space), the vaccination of a susceptible population in a geographically-targeted manner, movement controls, and branching our disease progression model to account for early- vs. late-intervention during host disease progression.ResultsWe will present analysis of diseases that exemplify the various aspects of analysis in support of outbreak and epidemic control. Human and animal diseases relevant to this demonstration include rinderpest, avian influenza, and measles. We will begin with EpiEarly’s estimate of epidemic potential using aggregated time-dependent case-count data. The key observation for EpiEarly is that under a wide range of situations a disease's reproductive number should be generalized to a distribution of possibilities to account for inherent randomness and other factors (including the variability of a disease contact network). We will then continue with a demonstration of EpiGrid’s capabilities for understanding and modelling the role of interventions including contagion control (the force of infection), treatment (changing disease progression and infectiousness depending on treatment), vaccination, culling, and movement controls. We will briefly touch on the capabilities of EpiCast for more detailed analysis of specific intervention strategies.ConclusionsWe will demonstrate examples where modeling either contributed or plausibly would contribute to informing epidemic and outbreak control constrained by the possibilities of the underlying epidemic and disease dynamics.