Time-dependent Nonlinear Equations Research Articles

Possible properties of TeV spectra in PKS 2155–304

We present a one-zone homogeneous lepton-hadronic model and obtain steady-state spectra by solving the time-dependent equations to study a plausible origin of hard TeV spectra in PKS 2155–304. In this model, we assume a steady electron and proton injection rate in the source and solve the non-linear time-dependent kinematic equations that self-consistently consist of proton-photon interaction, synchrotron radiation of electron/positron pairs and proton, inverse Compton scattering, and synchrotron self-absorption. We employ this model to reproduce the multi-wavelength spectrum of PKS 2155–304, then find that the possible bump located at E ∼ 1 TeV which may originate from the synchrotron radiation of secondary electrons produced by Bethe-Heitler pair production, resulting in the hard TeV spectrum.

The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov–Maxwell equations

We discuss the development, analysis, implementation, and numerical assessment of a spectral method for the numerical simulation of the three-dimensional Vlasov–Maxwell equations. The method is based on a spectral expansion of the velocity space with the asymmetrically weighted Hermite functions. The resulting system of time-dependent nonlinear equations is discretized by the discontinuous Galerkin (DG) method in space and by the method of lines for the time integration using explicit Runge–Kutta integrators. The resulting code, called Spectral Plasma Solver (SPS-DG), is successfully applied to standard plasma physics benchmarks to demonstrate its accuracy, robustness, and parallel scalability.

L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations

The task of repeatedly solving parametrized partial differential equations (pPDEs) in optimization, control, or interactive applications makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold on which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However this decomposition, usually implemented with aid from the empirical interpolation method (EIM) for nonlinear and/or parametric-nonaffine PDEs, can be challenging to implement, or results in severely degraded online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding efficiency degradation exhibited in traditional applications of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced approximation space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values whose snapshots span the reduced approximation space. Together, these two ingredients ensure that the proposed L1-ROC scheme is both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.

Brief numerical analysis of (3+1) Ginzburg-Landau equations

In this contribution, we show the implementation of the Link-variable method for solve the complete set of acopled non-linear time dependent Ginzburg-Landau differential equations in a three-dimensional homogeneous and isotropic mesoscopic superconducting system. In this case, the sample is immersed in an external magnetic d at zero applied current. The effects of demagnetization are taken in count and we show the order parameter and its phase in zero field cooling and field cooling process. This numerical analysis shows good results when this solution is applied to a superconducting cubic sample.

Modelling of lateral forces generated by pedestrians walking across footbridges

The mechanism which results in the synchronization of people walking across footbridges with the bridges fundamental horizontal frequency is studied. The lateral vibration of a bridge subjected to walking pedestrians is modelled by considering the bridge to be a slender beam. The effect of bridge motion on the footfall forces of walking pedestrians is not considered. The contribution of pedestrians varies with their position on the bridge. It is assumed that the walking gait frequency of a pedestrian follows a Gaussian distribution and pedestrians can unconsciously adjust the phase of their walking cycle closer to that of the bridge, based on the theory of coupled oscillators. A time-dependent nonlinear dynamic equation is derived using the modal expansion approach. The model is then applied to the north span of the Millennium Bridge in London, to produce its excessive lateral vibrations. Agreement between the results and the existing observations supports the rationality and reliability of the method. Several parameters, in particular the walking gait frequency of pedestrians, crowd sensitivity to bridge motion, bridge frequency, bridge damping and different load conditions are investigated. The numerical simulation shows that these parameters have different degrees of impact on the critical number of pedestrians triggering excessive vibrations.

On inefficiency of the shape memory alloys in dynamically loaded sandwich plates with structural damping: New 3D zigzag-viscoelasticity theory and asymmetric transformations

Abstract The important claim that the presence of the structural damping considerably lessens or even eliminates the effectiveness of the embedded shape memory elements in the dissipation of the vibration/impact energies of the structures, is stated and proven here for the first time. The finite-element-based governing equations of thick composite sandwich plates with homogeneous/FG soft viscoelastic cores and embedded SMA wires are derived based on a new hyperbolic SMA-zigzag sandwich plate theory that considers the transverse flexibility. The tension-compression asymmetry of the SMA behavior and Zener-viscoelasticity-based zigzag nonlinear corrections are implemented to guarantee the transverse stress continuity at the interfaces between layers and updating the relation between the describing parameters of the displacement field, and the stiffness and even mass matrices of the elements within each time step. For the first time, the phase-based rather than the mixture-based stresses are considered in tracing the phase transformations. The resulting time-dependent nonlinear integro-differential equations are solved by a special algorithm. Results confirm, for the first time, that the stress-strain hysteresis loops of the SMAs may shrink or even disappear in presence of the structural damping; so that, the belief that the phase-transformation-based damping and the structural damping directly add to each other, is quite wrong.

Nonlinear thermomechanical vibration mitigation analysis in rotating fractional-order viscoelastic bidirectional FG annular disks under nonuniform shocks

The current article is concerned with the investigation of the formation of the time-dependent three-dimensional distributions and redistributions of the stress and displacement fields of the rotating annular and circular plates/disks under the nonuniform distributions of the dynamic thermomechanical loads or shocks. Redistribution of the stress and displacement fields happens due to the Gerasimov-Caputo-type relaxation kernel of the fractional viscoelastic model of the material in addition to the dynamic nature of the loading. All the mechanical and thermal material properties of the plate or disk may be tailored in both radial and transverse directions. The 3D thermoviscoelasticity theory is employed to develop the governing equations of motion (3D vibration) and heat transfer. Different thermal and mechanical boundary conditions are implemented. The resulting nonlinear time-dependent fractional-order thermoviscoelastic integro-differential equations are solved by proposing and implementing a special procedure that uses numerical evaluation of the singular-kernel Caputo-integral and second-order forward, backward, and central discretization of the spatial and time domains. Eventually, the influences of the bidirectional distribution of the material properties, 2D temperature distribution, fractional order and parameters of the viscoelasticity model, and thermomechanical boundary conditions are investigated on the distributions of the displacement and stress components rigorously and accompanied by 3D demonstrations.

A local character based method for solving linear systems of radiation diffusion problems

The radiation diffusion problem is a kind of time-dependent nonlinear equations. For solving the radiation diffusion equations, many linear systems are obtained in the nonlinear iterations at each time step. The cost of linear equations dominates the numerical simulation of radiation diffusion applications, such as inertial confinement fusion, etc. Usually, iterative methods are used to solve the linear systems in a real application. Moreover, the solution of the previous nonlinear iteration or the solution of the previous time step is typically used as the initial guess for solving the current linear equations. Because of the strong local character in ICF, with the advancing of nonlinear iteration and time step, the solution of the linear system changes dramatically in some local domain, and changes mildly or even has no change in the rest domain.In this paper, a local character-based method is proposed to solve the linear systems of radiation diffusion problems. The proposed method consists of three steps: firstly, a local domain (algebraic domain) is constructed; secondly, the subsystem on the local domain is solved; and lastly, the whole system will be solved. Two methods are given to construct the local domain. One is based on the spatial gradient, and the other is based on the residual. Numerical tests for a two-dimensional heat conduction model problem, and two real application models, the multi-group radiation diffusion equations and the three temperature energy equations, are conducted. The test results show that the solution time for solving the linear system can be reduced dramatically by using the local character-based method.

Multipleweak solutions for a kind of time-dependent equation involving singularity

The existence of at least three weak solutions for a kind of nonlinear time-dependent equation is studied. In fact, we consider the case that the source function has singularity at origin. To this aim, the variational methods and the well-known critical points theorem are main tools.

Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls

• We consider the temporal stability of multiple solutions for flows in a porous channel with moving walls for the first time. • We construct second order finite difference schemes for eigenproblems associated with the linear stability analysis . • We verify the linear stability results for the first time by solving the perturbed nonlinear time dependent equation. In this paper, a viscous, incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is considered. We present multiple symmetric steady-state solutions of this flow problem at several different expanding ratios, and use the linear stability theory to analyse the temporal stability for these solutions under symmetric, antisymmetric and general perturbations. We construct second order finite difference schemes for the eigenvalue problems with boundary conditions associated with those perturbations, and observe that most of these solutions which are stable under symmetric perturbations are unstable under antisymmetric perturbations. Furthermore, we verify the linear stability analysis results by directly solving the original perturbed nonlinear time dependent problem, and find that both stability results are consistent.

Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations

In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S2 in R3. The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.

Simulation of planar mechanisms with revolute clearance joints using the multipatch based isogeometric analysis

The dynamic analysis of planar mechanisms with multiple revolute clearance joints is performed using the isogeometric analysis for the first time. A general multiple patches based revolute clearance joint model, which can be integrated into other parts or structures without changing the procedure of contact treatment, is proposed. A stable and optimized contact detection algorithm that accompanies the joint model is also introduced. Benefit from the high order NURBS basis functions, the geometrically exact covariant formulations of contact geometry, contact force and its Jacobian matrix are implemented. The implicit generalized-α method is employed to solve the time dependent nonlinear system equations. The effect of joint clearance on rigid and flexible mechanisms is studied through numerical examples. The accuracy and efficiency of the proposed approach are verified through a comparison with other conventional methods. The proposed approach exhibits great stability in long-term simulation regardless of the number of clearance joints, the size of clearance, the flexibility of the link and bearing. Results show that isogeometric analysis can provide a unified framework for joint clearance analysis of rigid and flexible systems. In the field of analyzing the combined effect of joint clearance and bodies’ large flexibility, the proposed method has significant advantages.

Meshless Singular Boundary Method for Nonlinear Sine-Gordon Equation

A meshless method based on the singular boundary method is developed for the numerical solution of the time-dependent nonlinear sine-Gordon equation with Neumann boundary condition. In this method, by using a time discrete scheme to approximate the time derivatives, the time-dependent nonlinear problem is transformed into a sequence of time-independent linear boundary value problems. Then, the singular boundary method is used to establish the system of discrete algebraic equations. The present method is meshless, integration-free, and easy to implement. Numerical examples involving line and ring solitons are given to show the performance and efficiency of the proposed method. The numerical results are found to be in good agreement with the analytical solutions and the numerical results that exist in literature.

Is the GeV-TeV emission of PKS 0447-439 from the proton synchrotron radiation?

We study the multi-wavelength emission features of PKS 0447-439 in the frame of the one-zone homogeneous lepto-hadronic model. In this model, we assumed that the steady power-laws with exponential cut-offs distributions of protons and electrons are injected into the source. The non-linear time-dependent kinematic equations, describing the evolution of protons, electrons and photons, are defined; these equations self-consistently involve synchrotron radiation of protons, photon-photon interaction, synchrotron radiation of electron/positron pairs, inverse Compton scattering and synchrotron self-absorption. The model is applied to reproduce the multi-wavelength spectrum of PKS 0447-439. Our results indicate that the spectral energy distribution (SED) of PKS 0447-439 can be reproduced well by the model. In particular, the GeV-TeV emission is produced by the synchrotron radiation of relativistic protons. The physically plausible solutions require the magnetic strength $10~\text{G}\lesssim B \lesssim 100~\text{G}$ . We found that the observed spectrum of PKS 0447-439 can be reproduced well by the model whether $z = 0.16$ or $z = 0.2$ , and the acceptable upper limit of redshift is $z=0.343$ .

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms.

Linear and Weak Nonlinear Double Diffusive Convection in a Viscoelastic Fluid Saturated Anisotropic Porous Medium with Internal Heat Source

This paper deals linear and weak nonlinear stability analysis of double-diffusive convection in an anisotropic porous layer with internal heat source saturated by viscoelastic fluid. For linear stability analysis we use normal mode technique and obtained the expression for oscillatory thermal Rayleigh number which is used to plot neutral stability curve for oscillatory case. For nonlinear analysis truncated representation of Fourier series upto two terms is used. The system of time dependent nonlinear equation is solved numerically and plot the curve for heat transfer and mass transfer with respect to time for different parameters. Effect of thermal anisotropy parameter, mechanical anisotropy parameter, relaxation parameter, retardation parameter, internal heat source parameter, solute Rayleigh number, diffusivity ratio, Darcy-Prandtl number on the onset of convection, heat and mass transfers have been discussed. We also draw the stream lines, isotherms, isohalines at different times.

Space–time adaptive ADER-DG schemes for dissipative flows: Compressible Navier–Stokes and resistive MHD equations

This paper presents an arbitrary high-order accurate ADER Discontinuous Galerkin (DG) method on space–time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent partial differential equations for compressible dissipative flows : the compressible Navier–Stokes equations and the equations of viscous and resistive magnetohydrodynamics in two and three space-dimensions.The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell finite volume limiting procedure suitable for discontinuous Galerkin methods (Dumbser et al., 2014, Zanotti et al., 2015 [40,41]). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ‘Gibbs phenomenon’. In the present work, a nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a novel a posteriori detection criterion, i.e. the MOOD approach. The main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Furthermore, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Subsequently, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages.We apply the whole approach for the first time to the equations of compressible gas dynamics and magnetohydrodynamics in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector. The distinguished high-resolution properties of the presented numerical scheme standout against a wide number of non-trivial test cases both for the compressible Navier–Stokes and the viscous and resistive magnetohydrodynamics equations. The present results show clearly that the shock-capturing capability of the news schemes is significantly enhanced within a cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS).

Explicit Exponential Finite Difference Scheme for 1D Navier-Stokes Equation with Time Dependent Pressure Gradient

In this paper, numerical technique for solving the one-dimensional (1D) unsteady, incompressible Navier-Stokes equation (NSE) is presented. The governing time dependent non-linear partial equation is reduced to non-linear partial differential equation named as viscous Burgers’ equation by introducing Orlowski and Sobczyk transformation (OST). An explicit exponential finite difference scheme (Expo FDS) has been used for solving reduced 1D NSE. The accuracy of the method has been illustrated by taking two numerical examples. Results are compared with the analytical solutions and those obtained based on the numerical results of reduced 1D NSE as Burgers’ equation. The accuracy and numerical feature of convergence of the Expo FDS is presented by estimating their error norms. Excellent numerical results indicate that the proposed numerical technique is efficient admissible with efficient accuracy for the numerical solutions of the NSE.GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 79-90

Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang’s moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they could be taken as one step of the root searching iteration methods. We improve Huang’s MMPDE by adding one Lagrange speed term. The proposed moving mesh PDE could draw the mesh to equidistribution quickly and stably. The numerical algorithm for the coupled system of the original PDE and the moving mesh equation is proposed and the computational experiments are given to illustrate the validity of the new method.

Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon

In this paper, we establish the unconditional stability and optimal error estimates of a linearized backward Euler---Galerkin finite element method (FEM) for the time-dependent nonlinear thermistor equations in a two-dimensional nonconvex polygon. Due to the nonlinearity of the equations and the non-smoothness of the solution in a nonconvex polygon, the analysis is not straightforward, while most previous efforts for problems in nonconvex polygons mainly focused on linear models. Our theoretical analysis is based on an error splitting proposed in [30, 31] together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated (time-discrete) elliptic system in a nonconvex polygon. With the proved regularity, we establish the stability in $$l^\infty (L^\infty )$$lź(Lź) and the convergence in $$l^\infty (L^2)$$lź(L2) for the fully discrete finite element solution without any restriction on the time-step size. The approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons. Numerical results confirm our theoretical analysis and show clearly that no time-step condition is needed.