Low-order Equations Research Articles

Mixed finite element algorithm for a nonlinear time fractional wave model

In this article, a mixed element algorithm is presented to look for the numerical solution for a class of nonlinear wave model with the Caputo fractional derivative. By introducing two auxiliary functions and reducing order technique of fractional derivative, the studied model with high-order derivative in time is transformed into a coupled system including three lower order equations. Next, a fully discrete mixed element algorithm is formulated, where the temporal direction is approximated by a second-order scheme. The stability analysis of the proposed mixed scheme is done and optimal error estimates for three functions are derived. Finally, the numerical tests are carried out to verify the theory results.

Online Spark Timing Optimization With Complex High-Fidelity Combustion Phasing, Knock, and Coefficient of Variation of IMEP Models

Abstract The combustion phasing of spark ignition (SI) engines is traditionally regulated with map-based spark timing (SPKT) control. The calibration of these maps is time-consuming for SI engines with a high number of control actuators. This paper proposes three online SPKT optimization algorithms that can utilize control-oriented semiphysics-based combustion models making the SPKT control algorithm more adaptive to different engine designs. These three SPKT optimizers do not require model inversion and derivative information. These methods also preserve the dependence between combustion phasing, knock, and coefficient of variation (COV) of indicated mean effective pressure (IMEP) models to avoid evaluating combustion models multiple times within one iteration. The two-phase and constraint relaxation methods are derived from direct search optimization theories. The recursive least square (RLS) polynomial fitting method can be considered as a virtual extreme seeking (ES) process that converts the original “black” box nonlinear constrained optimization into the solution of three low-order polynomial equations. Although these three online SPKT optimization approaches have unique properties making them preferable with certain types of combustion models, simulation and test results show that all of them can find the optimal SPKT with less than 10 evaluations of the combustion models. This fact makes it possible to implement the proposed model-based SPKT control strategy in future engine control units (ECUs).

Superconvergence analysis of a nonconforming MFEM for nonlinear Schrödinger equation

In this paper, an efficient nonconforming mixed finite element method (MFEM) is studied with element and zero-order Raviart–Thomas element for a generalized nonlinear Schrödinger equation. On the one hand, by introducing as an intermediate variable, we split the equation into two low-order equations and present a semi-discrete scheme for nonconforming MFEM and prove its existence and uniqueness. And the superconvergence results for original variable u in broken -norm and flux in norm are derived by using the proved properties of the above nonconforming MFEs. On the other hand, a linearized Crank–Nicolson fully discrete scheme is constructed and the superclose and superconvergence results with order for above variables are also derived. The keys to our analysis are the following two skills: one is an important novel property for the above MFEs (see Lemma 2.3) and another is a splitting technique for nonlinear terms, while previous literature always only obtain the convergent estimates in a routine way. Finally, two numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

An accurate model for free vibration of porous magneto-electro-thermo-elastic functionally graded cylindrical shells subjected to multi-field coupled loadings

An accurate model for vibration of a porous magneto-electro-thermo-elastic functionally graded (METE-FG) cylindrical shell made of barium titanate (BaTiO3) and cobalt diiron tetraoxide (CoFe2O4) with magneto-electro-thermal loadings is proposed within the framework of Hamiltonian system. Four types of porosity distribution profiles in the thickness direction are considered. By introducing a new total eigenvector, the higher-order governing differential equations are transformed into a set of lower-order equations. The exact solution for free vibration of METE-FG shells can be expanded in terms of specific symplectic eigenfunctions having seven possible explicit forms. Subsequently, analytical frequency equations and vibration mode shapes for METE-FG shells with various boundary conditions are derived simultaneously. A comparison study is presented to demonstrate the accuracy of the proposed model and very good agreement is observed. The effects of material properties and magneto-electro-thermal loadings on free vibration characteristics of METE-FG cylindrical shells are analyzed and discussed in detail.

Implicit linear difference equations over a non-Archi-medean ring

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

ВРАЩЕНИЯ ТОРОВ В СТРУКТУРЕ ЖИДКОГО КРИСТАЛЛА

Liquid crystals are a collection of flattened molecules. On the one hand, they have a welldefined structure, on the other hand, this structure is deformable, since its elements can change their position in space. In columnar structures, consisting of disks, plates, and tori, the fluid lines and cords are distinguished. In the case of tori and circular disks, a hexagonal structure of the arrangement of molecules is observed in a combination of the cord. The aim of this work is to obtain the stability of tori positions in an elementary fragment of a liquid crystal and to analyze their rotations in a self-consistent field of the surrounding toroidal molecules. To solve this problem, the method of mathematical modeling was used, based on classical models of molecular dynamics. The calculation is carried out on the basis of the model of cross-atom-atom interactions for molecular tori. The minimal fragment of the cord is selected, which makes it possible to determine the characteristic dynamic state of the central torus in the fragment. To describe the motion of the molecular tori, the equations of motion for their centers of mass and the Euler equations for their angular displacements are used. The minimal fragment of the material contains twenty-nine tori. The equations for displacements of the centers of mass of the tori are initially represented as ordinary differential equations of the second order. However, by introducing fictitious points into the consideration of velocities, they can be reduced to a system of first-order equations with a doubled number of lower-order equations. The resulting system of the first-order differential equations is integrated numerically using a highorder accuracy step-by-step scheme. All calculations are performed with a constant time step. The accuracy of the obtained numerical results is verified in terms of the balance of total energy of the system. Calculations show that the central torus of the presented fragment executes angular oscillations around its main axis with amplitude of more than one revolution. Thus, the performed calculations show that a representative fragment of the liquid crystal structure of molecular tori can be used as a generator of high-frequency mechanical vibrations.

A diffusion synthetic acceleration approach to k-eigenvalue neutron transport using PJFNK

This paper investigates a diffusion synthetic acceleration (DSA) approach to the k-eigenvalue problem of the neutron transport equation. The novelties of this study reside in 1) casting the low-order diffusion equation directly as a nonlinear problem, which is solved with the PJFNK (preconditioned Jacobian-free Newton Krylov) method, circumventing the need for power iteration; 2) showing the appealing features on solving the diffusion equation with DSA compared with nonlinear diffusion acceleration (NDA) with a simplified model of ATR (Advanced Test Reactor); 3) proposing the DSA weak form for the discretization scheme with SAAF (self-adjoint angular flux) formulation that is stable with anisotropic scatterings; 4) showing the impact of the vacuum boundary coefficient of the low-order equation on the convergence of DSA. Both DSA and NDA are implemented in and tested with Rattlesnake code, the INL MOOSE-based radiation transport application for multi-physics modeling and simulations.

Multilevel quasidiffusion method with mixed-order time discretization for multigroup thermal radiative transfer problems

In this paper we present a numerical method for solving multigroup thermal radiative transfer (TRT) problems in 2D Cartesian geometry. It is based on the Quasidiffusion (aka Variable Eddington Factor) method and defined by the multilevel system of multigroup high-order radiative transfer (RT) equations and multigroup and grey low-order equations for moments of the intensity with the exact closures. We apply time integration schemes of different orders of accuracy to approximate the high-order and low-order equations. The first-order scheme is used for the high-order RT equations. The second-order scheme is applied to the low-order equations. This improves the accuracy of the TRT solution while using robust and relatively inexpensive scheme for the high-order RT equations. The solution of the low-order equations is non-monotonic because the hyperbolic low-order Quasidiffusion (QD) equations are discretized by the second-order scheme. To reduce non-monotonicity of the low-order solution we apply a monotonization procedure to the discretized time-dependent low-order equations based on the Limited-Trapezoidal method. Numerical results of the Fleck-Cummings test are presented to demonstrate performance of the developed mixed-order time integration scheme for the multilevel system of high-order and low-order QD equations. We use this TRT test problem to analyze the convergence in time of the mixed-order scheme.

A Multilevel in Space and Energy Solver for 3-D Multigroup Diffusion and Coarse-Mesh Finite Difference Eigenvalue Problems

The Multilevel in Space and Energy Diffusion (MSED) method accelerates the iterative convergence of multigroup diffusion eigenvalue problems by performing work on lower-order equations with only one group and/or coarser spatial grids. It consists of two primary components: (1) a grey (one-group) diffusion eigenvalue problem that is solved via Wielandt-shifted power iteration (PI) and (2) a multigrid-in-space linear solver. In previous work, the efficiency of MSED was verified using Fourier analysis and numerical results from a one-dimensional multigroup diffusion code. Since that work, MSED has been implemented as a solver for the coarse-mesh finite difference (CMFD) system in the three-dimensional Michigan Parallel Characteristics Transport (MPACT) code. In this paper, the results from the implementation of MSED in MPACT are presented, and the changes needed to make MSED more suitable for MPACT are described. For problems without feedback, the results in this paper show that MSED can reduce the CMFD run time by an order of magnitude and the overall run time by a factor of 2 to 3 compared to the default CMFD solver in MPACT [PI with the generalized minimal residual (GMRES) method]. For problems with feedback, the convergence of the outer Picard iteration scheme is worsened by the well-converged CMFD solutions produced by the standard MSED method. To overcome this unintuitive deficiency, MSED may be run with looser convergence criteria (a modified version of the MSED method called MSED-L) to circumvent the issue until the multiphysics iteration in MPACT is improved. Results show that MSED-L can reduce the CMFD run time in MPACT by an order of magnitude, without negatively impacting the outer Picard iteration scheme.

A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

Second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage.Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal.In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. We show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

Multigrid treatment of implicit continuum diffusion

Implicit treatment of diffusive terms of various differential orders common in continuum mechanics modeling, such as computational fluid dynamics, is investigated with spectral and multigrid algorithms in non-periodic 2D domains. In doubly periodic time dependent problems these terms are handled efficiently by spectral methods, but in non-periodic systems solved with distributed memory parallel computing and 2D domain decomposition, this efficiency is lost for a large number of processors. We built and present here a multigrid algorithm for these types of problems that outperforms a spectral solution employing the highly optimized FFTW library. This solver is suitable for high performance computing and may be able to efficiently treat implicit diffusion of arbitrary order by introducing auxiliary equations of lower order. We test these solvers for fourth and sixth order diffusion with harmonic test functions as well as turbulent 2D magnetohydrodynamic simulations. It is also shown that an anisotropic operator without mixed-derivative terms improves model accuracy and speed, and we examine the impact that the various diffusion operators have on the energy, the enstrophy, and the qualitative aspect of a simulation.

A hierarchy of dynamic equations for solid isotropic micropolar circular cylinders

This work considers the derivation procedure and evaluation of dynamic equations for isotropic micropolar circular cylinders by adopting a power series expansion method in the radial coordinate. Variationally consistent equations of motion together with pertinent sets of boundary conditions are expressed in a systematic fashion up to arbitrary order. The numerical results cover eigenfrequencies, mode shapes and field distributions over cross sections for axisymmetric and flexural motion adopting different sets of end boundary conditions for equations of different truncation orders of the present method. The results illustrate that the present approach may render benchmark solutions provided that higher order equations are used, and act as accurate approximate engineering solution for lower order equations.

Parametric conditions and exact solution for the Duffing-Van der Pol class of equations

This paper presents a methodology to find the exact solution and respective parametric conditions to the Duffing-Van der Pol class of equations. The supposed method in this paper is different from the Prelle and Singer method and the Lie symmetry method. The main idea of the supposed method is implemented in finding the first integrals of the original equation and leading this equation to a solved equation of lower order to which the exact solution can be obtained. As results the parametric conditions and the exact solutions in parametric forms are indicated. The algorithm for determining integral constants and the investigation of solution characteristics are considered.

Two-dimensional equations for thin-films of ionic conductors

A theoretical model is developed for predicting both conduction and diffusion in thin-film ionic conductors or cables. With the linearized Poisson-Nernst-Planck (PNP) theory, the two-dimensional (2D) equations for thin ionic conductor films are obtained from the three-dimensional (3D) equations by power series expansions in the film thickness coordinate, retaining the lower-order equations. The thin-film equations for ionic conductors are combined with similar equations for one thin dielectric film to derive the 2D equations of thin sandwich films composed of a dielectric layer and two ionic conductor layers. A sandwich film in the literature, as an ionic cable, is analyzed as an example of the equations obtained in this paper. The numerical results show the effect of diffusion in addition to the conduction treated in the literature. The obtained theoretical model including both conduction and diffusion phenomena can be used to investigate the performance of ionic-conductor devices with any frequency.

Notes on the convergence order of gradient schemes for time fractional differential equations

We apply the GDM (Gradient Discretization Method), developed recently, as discretization in space to time-fractional diffusion and diffusion-wave equations with a fractional derivative of Caputo type in any space dimension.In the case of time-fractional diffusion equations, we establish an implicit scheme, and we prove an L∞(L2)-error estimate. A similar result in a discrete L∞(H01)–norm is also stated.To construct the numerical scheme for the time-fractional diffusion-wave equation, we write the equation in the form of a system of two low-order equations. We state an a prior estimate result that helps us to derive error estimates in discrete semi-norms of L∞(H1) and H1(L2). The convergence is unconditional. Another gradient scheme is also suggested. We state its convergence results, which improve the convergence order proved recently for a SUSHI scheme.These results hold then for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes.

A numerical method for the solution of fifth order boundary value problem in ordinary differential equations

In this article we have proposed a technique for solving the fifth order boundary value problem as a coupled pair of boundary value problems. We have considered fifth order boundary value problem in ordinary differential equation for the development of the numerical technique. There are many techniques for the numerical solution of the problem considered in this article. Thus we considered the application of the finite difference method for the numerical solution of the problem. In this article we transformed fifth order differential problem into system of differential equations of lower order namely one and four. We discretized the system of differential equations into considered domain of the problem. Thus we got a system of algebraic equations. For the numerical solution of the problem, we have the system of algebraic equations. The solution of the algebraic equations is an approximate solution of the problem considered. Moreover we get numerical approximation of first and second derivative as a byproduct of the proposed method. We have shown that proposed method is convergent and order of accuracy of the proposed method is at lease quadratic. The numerical results obtained in computational experiment on the test problems approve the efficiency and accuracy of the method.

Spectral parameter power series for arbitrary order linear differential equations

Let L be the n‐th order linear differential operator Ly=ϕ0y(n)+ϕ1y(n−1)+⋯+ϕny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.

Two-dimensional noisy autoregressive estimation with application to joint frequency and direction of arrival estimation

The present study addresses the problem of two-dimensional autoregressive estimation in the presence of additive white noise. The estimation method is based on combining the low-order and high-order Yule-Walker equations. The noise-compensated YW equations are solved using an iterative algorithm. The proposed method is also applied to joint frequency and direction of arrival estimation in uniform linear arrays. Using simulation study, the performance of the proposed algorithm is evaluated and compared with other methods.

The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak

The two-dimensional (2D) kinetic ballooning theory is developed for the ion temperature gradient mode in an up-down symmetric equilibrium (illustrated via concentric circular magnetic surfaces). The ballooning transform converts the basic 2D linear gyro-kinetic equation into two equations: (1) the lowest order equation (ballooning equation) is an integral equation essentially the same as that reported by Dong et al., [Phys. Fluids B 4, 1867 (1992)] but has an undetermined Floquet phase variable, (2) the higher order equation for the rapid phase envelope is an ordinary differential equation in the same form as the 2D ballooning theory in a fluid model [Xie et al., Phys. Plasmas 23, 042514 (2016)]. The system is numerically solved by an iterative approach to obtain the (phase independent) eigen-value. The new results are compared to the two earlier theories. We find a strongly modified up-down asymmetric mode structure, and non-trivial modifications to the eigen-value.

Accurate simulation of thin dielectric sheet by higher order surface integral equation

ABSTRACTA higher order approach for the accurate simulation of a thin dielectric sheet is proposed in this paper. Due to the small thickness of thin dielectric sheet, an equivalent surface integral equation containing tangential and normal fields can be derived from the conventional volume integral equation. Higher order bases are then applied to approximate both components. Although the programming complexity increases slightly compared with the traditional low-order thin dielectric sheet surface integral equation method, the number of unknowns required in the same problem can be drastically reduced in the proposed higher order approach. At the same time, a higher accuracy can be achieved. Numerical results are presented to show the high efficiency and stability of the proposed method.