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

Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.

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.

By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs) and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equationy′′′=0is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.

The numerical solution of a ordinary differential equation (ODE) basically describes any physical phenomenon of interest. Traditionally, the numerical solution of ODE is solved by a discretization process using the classical finite difference approximation. Recently, many studies have used deep neural networks (DNN) to approximate ODE solutions without discretization. However, without the optimization process, DNN cannot run independently. In this work, we proposed an alternative method that combines DNN with a genetic algorithm (GA) as an optimization method. GA initiates a population of DNN trainable parameters, making the DNN training process more streamlined on a single-generation GA. We focused on DNN-GA hybrid solution for linear and nonlinear ordinary differential equations (ODE‘s). Our method is achieved by solving several test problems and demonstrating a comparison with stochastic gradient descent (SGD) optimizers. In most ODE problems, using the same number of iterations and architecture, DNN-GA produces a better loss function than using DNN-SGD.

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Geometric Numerical Integration of Differential Equations'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with symplectic or multisymplectic integration; volume-preserving integration; symmetry-preserving integration; integrators that preserve first integrals, Lyapunov functions, or dissipation; exponential integrators; integrators for highly oscillatory systems; Lie-group integrators, etc. Papers on geometric integration of both ODEs and PDEs will be considered, as well as application to molecular-scale integration, celestial mechanics, particle accelerators, fluid flows, population models, epidemiological models and/or any other areas of science. We believe that this issue is timely, and hope that it will stimulate further development of this new and exciting field.

This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.

Ordinary differential equation (ODE) solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity and, implicitly, in stochastic optimization. In this work, we propose and study the stochastic Euler dynamics—a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics, which we derive through a first-order approximation to the infinitesimal generator of the stochastic Euler dynamics. In the context of linear ODEs, these deterministic Euler dynamics describe the dynamics of the expectation of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next, we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster–Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di- stability) in problem discretizations. However, computational cate- gories are not precise, and qualitative descriptors such as of moderate size cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive New- ton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate bound- ary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood.

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.

Ordinary differential equations (ODEs) are often used to model the behavior of physical phenomena and textbooks today especially demonstrate this fact. Since only a very small collection of ODEs can be solved analytically, there is often no alternative than to use computer software to gain some insight into the behavior of solutions (and sometimes even if solution formulas are available--the formulas are often complicated!). A classic work on the numerical solution of ODEs was authored by (Numerical Solution of Ordinary Differential Equations, Chapman and Hall/CRC,1994). There are some questions about the behavior of solutions of ODEs that are not quite appropriate for numerical solvers. In this paper we present examples which illustrate some of these features. However, there is no disputing the fact that the output of numerical solvers is often useful for portraying and understanding the behavior of solutions of ODEs and their utility in modeling physical phenomena, as our final example shows.

The focus of this paper is to use fluid dynamics and object-oriented approach in particle subsystems for interactive real-time visualization in 3D gaming. By exploiting all the areas of fluid dynamics and exercising it in the particle subsystem of gaming engine, we create an environment where we can map an artificial object to a real-world entity to such an extent that it can add realism to the virtual gaming environment. To do so, particle integration and particle rendering play an important part. Also, graphics accelerators and graphics processor units add a mammoth advantage calculating the particle coordinates followed by rendering. The current graphic processor though fast, fails to give the exact trajectory of particles when collision, sliding, outburst, explosion or stabbing by one character to another character occurs. They lack in tracing the exact path followed by the particles when certain forces are applied on a heavenly body or on liquidus body. The particle that emerges is random and has no correlation to reality as they follow random path. All these must be considered to enhance the user interaction in 3D gaming to enrich the real-time virtual environment gaming experience. Many particle subsystems tools are available that can create extraordinary particle effects in gaming, but they all lack in giving directions to the particle. Thus in this paper, we create a particle subsystem that is not only stable but also follows the law of physics so that each particle in the gaming environment can be advanced with random time steps. The principal motivation behind the paper is to examine the flow of particles when they are embedded with the laws of fluid dynamics and to calculate the rendering complexity which makes it impossible to implement it on the large scale on conventional graphic processing unit without using quantum technology.

The Random-Time Binomial Model

This work is devoted to the implementation and testing of the Adams method for solving ordinary differential equations in the Sage computer algebra system. The Sage computer algebra system has, to some extent, trivial means for numerical integration of ordinary differential equations, but at the same time, it is worth noting that this environment is convenient and practical for conducting computer experiments related to symbolic numerical calculations in it. The article presents the FDM package developed on the basis of the RUDN, which contains the developments of recent years, performed by M. D. Malykh and his students, for numerical integration of differential equations. In this package, attention is paid to the visualization of the calculation results, including the construction of various kinds of auxiliary diagrams, such as Richardson diagrams, as well as graphs of dependence, for example, the value of a function or step from a moment in time. The implementation of the Adams method will be considered from this package. In this article, this implementation of the Adams method will be tested on various examples of input data, and the method will also be compared with the Jacobi system. Exact and approximate values will be found and compared, and an estimate for the error will be obtained.

New off-grid methods, that are not limited by the Dahlquist stability theorem, are introduced for the numerical integration of first- and second-order systems of differential equations. The methods are characterized by having all derivative evaluations performed at locations off the grid of final solution values. All on-grid solution values and off-grid derivative evaluations are used over m back steps to calculate the solution value at the next on-grid location. There are n off-grid derivative evaluations associated with each of these intervals located at fractional positions γ1h, ..., γnh relative to the on-grid locations. Off-grid geometric parameters are found which give highly stable integrators of maximum possible order, O(hm+mn+n-1). These integrators also have the property that their coefficients decrease geometrically with remoteness from the current interval of integration; and although the second mean value theorem is not applicable in obtaining an exact error expression for these methods, the error coefficients are generally several orders of magnitude less than comparable on-grid Adams or Cowell error coefficients. Comparisons are made with other integration techniques for satellite orbital calculations and other systems or ordinary differential equations.

Combining a Lie algebraic approach that is due to Wei and Norman [J. Math. Phys. 4, 475 (1963)] and the ideas suggested by Drach [Compt. Rend. 168, 337 (1919)], we have constructed several classes of systems of linear ordinary differential equations that are integrable by quadratures. Their integrability is ensured by integrability of the corresponding stationary cubic Schrödinger, KdV, and Harry–Dym equations. Next, we obtain a hierarchy of integrable reductions of the Dirac equation of an electron moving in the external field. Their integrability is shown to be in correspondence with integrability of the stationary mKdV hierarchy.