Semi-Lagrangian multistep exponential integrators for index 2 differential–algebraic systems

Implicit-explicit (IMEX) multistep methods are very useful for the time discretization of convection diffusion PDE problems such as the Burgers equations and the incompressible Navier–Stokes equations. In the latter as well as in PDE models of plasma physics and of electromechanical systems, semi-discretization in space gives rise to differential–algebraic (DAE) system of equations often of index higher than 1. In this paper we propose a new class of exponential integrators for index 2 DAEs arising from the semi-discretization of PDEs with a dominating and typically nonlinear convection term. This class of problems includes the incompressible Navier–Stokes equations. The integration methods are based on the backward differentiation formulae (BDF) and they can be applied without modifications in the semi-Lagrangian integration of convection diffusion problems. The approach gives improved performance at low viscosity regimes.     

Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods

Integration factor methods are a class of “exactly linear part” time discretization methods. In [Q. Nie, Y.-T. Zhang, R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006) 521–537], a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. In this paper, we solve this problem by applying the Krylov subspace approximations to the matrix exponential operator. Then we apply this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction–diffusion equations. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the method in resolving the stiffness of the DG spatial operator for reaction–diffusion PDEs. Application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown.     

Symplectic and multisymplectic numerical methods for Maxwell’s equations

In this paper, we compare the behaviour of one symplectic and three multisymplectic methods for Maxwell’s equations in a simple medium. This is a system of PDEs with symplectic and multisymplectic structures. We give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments. We also show that these numerical methods preserve the divergence. Furthermore, we extend the discussion on dispersion for (multi)symplectic methods applied to PDEs with one spatial dimension, to include anisotropy when applying (multi)symplectic methods to Maxwell’s equations in two spatial dimensions. Lastly, we demonstrate how varying the Courant–Friedrichs–Lewy (CFL) number can cause the (multi)symplectic methods in our comparison to behave differently, which can be explained by the study of backward error analysis for PDEs.     

Exact Solutions to a Combined sinh-cosh-Gordon Equation

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived.     

风扇/压气机数值稳定性模型

本文给出了一种用于预测轴流压气机旋转失速发生的数值稳定性模型,并描述了模型的理论基础以及求解模型方程的方法。模型的建立基于线性稳定性理论的基础之上,通过特征值的虚部来判断系统的稳定性。一种全局性方法被用于求解离散系统的所有特征值。离散的方法包括二阶精度的有限差分方法以及切比雪夫谱配置的方法。计算结果表明,用这两种方法均可以得到准确的特征值,但是收敛的速度有所不同。另外,将文中的分析与实验进行了比较,结果表明实验中的失速点可以用这种模型进行合理的预测。     

New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection–reaction equation. By using a Fourier–Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.     

Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method

We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine–Gordon, Korteweg–de Vries, nonlinear Schrödinger, (linear) time-dependent Schrödinger, and Maxwell equations. In the dissipative case the examples are: the Allen–Cahn, Cahn–Hilliard, Ginzburg–Landau, and heat equations.     

New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.     

Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models

In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients’ fourth-order partial differential equations (FOPDEs) that arise in Euler–Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical solution is quite a tedious and challenging problem, which is our main concern in this paper. The current scheme is hybrid in nature in which the second-order finite difference is used for temporal discretization, while spatial derivatives and solutions are approximated via the Haar wavelet. Next, the integration and Haar matrices are used to convert partial differential equations (PDEs) to the system of linear equations, which can be handled easily. Besides this, we derive the theoretical result for stability via the Lax–Richtmyer criterion and verify it computationally. Moreover, we address the computational convergence rate, which is near order two. Several test problems are given to measure the accuracy of the suggested scheme. Computations validate that the present scheme works well for such problems. The calculated results are also compared with the earlier work and the exact solutions. The comparison shows that the outcomes are in good agreement with both the exact solutions and the available results in the literature.     

Stability switching at transcritical bifurcations of solitary waves in generalized nonlinear Schrödinger equations

Linear stability of solitary waves near transcritical bifurcations is analyzed for the generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions. Bifurcation of linear-stability eigenvalues associated with this transcritical bifurcation is analytically calculated. Based on this eigenvalue bifurcation, it is shown that both solution branches undergo stability switching at the transcritical bifurcation point. In addition, the two solution branches have opposite linear stability. These analytical results are compared with the numerical results, and good agreement is obtained.     

Energy Decay for the Damped Wave Equation Under a Pressure Condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This result holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.     

Exact dynamic and static stiffness matrices of shear deformable thin-walled beam-columns

Total potential energy of non-symmetric thin-walled beam-columns in the general form is presented by introducing the displacement field based on semitangential rotations and deriving transformation equations between displacement and force parameters defined at the arbitrary axis and the centroid-shear center axis, respectively. Next, governing equations and force-deformation relations are derived from the total potential energy for a shear-deformable, uniform beam element and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and exact dynamic stiffness matrices are determined using force-deformatin relationships. In addition, the modified numerical method to eliminate multiple zero eigenvalues and to evaluate the exact static stiffness matrix is developed for spatial stability analysis. Finally, in order to demonstrate the validity and the accuracy of this study, the spatially coupled natural frequencies and buckling loads are evaluated and compared with analytical solutions or results analyzed by thin-walled beam elements and ABAQUS's shell elements.     

An exponential compact difference scheme for solving 2D steady magnetohydrodynamic (MHD) duct flow problems

In this article, an exponential high-order compact (EHOC) difference scheme on the nine-point stencil is developed for the solution of the coupled equations representing the steady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight channel of rectangular section. A key property of the EHOC scheme is that it has excellent stability and higher accuracy so that the high gradients near the boundary layer areas can be effectively resolved without refining the mesh. Numerical experiments are carried out to validate the performance of the currently proposed scheme. Computation results of the MHD flow in the 2D square-channel problems with different wall conductivities are presented for Hartmann numbers ranging from 10 to 10⁶. The numerical solutions obtained with the newly developed EHOC scheme are also compared with analytic solutions and numerical results by other available methods in the literature.     

The connection of geometrical optics with the propagation of gaussian beams and the theory of optical resonators

The propagation of light near the axis of astigmatic optical systems may be described by the geometrical-optics approximation with the aid of ray-matrices. The application of the theory of diffraction to the propagation of light in such systems leads to integrals containing essentially the elements of the ray-matrices as parameters. The ABCD-law is derived by evaluating these integrals for gaussian beams. Integral equations applicable to astigmatic optical resonators, having nearly vanishing diffraction losses, are set up. They are only valid under certain conditions, which are comprehensively discussed. The eigensolutions and the eigenvalues of these integral equations are given. The spot-sizes at the resonator mirrors are derived from the eigensolutions, and the eigenvalues lead to the resonance condition. Spot-sizes and resonance condition appear as functions of the elements of the characteristic resonator matrices. The methods described here are applied to the propagation of gaussian beams through gas-lenses and to a resonator containing an internal gas-lens.     

Computation of invariant tori by Newton–Krylov methods in large-scale dissipative systems

A method to compute invariant tori in high-dimensional systems, obtained as discretizations of PDEs, by continuation and Newton–Krylov methods is described. Invariant tori are found as fixed points of a generalized Poincaré map so that the dimension of the system of equations to be solved is that of the original system. Due to the dissipative nature of the problems studied, the convergence of the linear solvers is extremely fast. The computation of periodic orbits inside the Arnold’s tongues is also considered. Thermal convection of a binary mixture of fluids, in a rectangular cavity, has been used to test the method.     

Efficient computation of the spectrum of viscoelastic flows

The understanding of viscoelastic flows in many situations requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations. Linear stability analysis leads to a generalized eigenvalue problem (GEVP), whose numerical analysis may be challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to non-physical eigenvalues at infinity. For viscoelastic flows, the difficulties increase due to the presence of continuous spectrum, related to the constitutive equations.The Couette flow of upper convected Maxwell (UCM) liquids has been used as a case study of the stability of viscoelastic flows. The spectrum consists of two discrete eigenvalues and a continuous segment with real part equal to ?1/We (We is the Weissenberg number). Most of the approximations in the literature were obtained using spectral expansions. The eigenvalues close to the continuous part of the spectrum show very slow convergence.In this work, the linear stability of Couette flow of a UCM liquid is studied using a finite element method. A new procedure to eliminate the eigenvalues at infinity from the GEVP is proposed. The procedure takes advantage of the structure of the matrices involved and avoids the computational overhead of the usual mapping techniques. The GEVP is transformed into a non-degenerate GEVP of dimension five times smaller. The computed eigenfunctions related to the continuous spectrum are in good agreement with the analytic solutions obtained by Graham [M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341].     

An Acceleration Method for Stationary Iterative Solution to Linear System of Equations

An acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided.     

A second order self-consistent IMEX method for radiation hydrodynamics

We present a second order self-consistent implicit/explicit (methods that use the combination of implicit and explicit discretizations are often referred to as IMEX (implicit/explicit) methods ,  and ) time integration technique for solving radiation hydrodynamics problems. The operators of the radiation hydrodynamics are splitted as such that the hydrodynamics equations are solved explicitly making use of the capability of well-understood explicit schemes. On the other hand, the radiation diffusion part is solved implicitly. The idea of the self-consistent IMEX method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations. In our self-consistent IMEX method, we solve the hydrodynamics equations inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-free Newton Krylov (JFNK) method ,  and . This is done to avoid order reductions in time convergence due to the operator splitting. We present results from several test calculations in order to validate the numerical order of our scheme. For each test, we have established second order time convergence.     

Dynamical Stability of Non-Constant Equilibria for the Compressible Navier–Stokes Equations in Eulerian Coordinates

In this paper we establish global existence and uniqueness of strong solutions to the non-isothermal compressible Navier–Stokes equations in bounded domains. The initial data have to be near equilibria that may be non-constant due to considering large external forces. We are able to show exponential stability of equilibria in the phase space and, above all, to study the problem in Eulerian coordinates. The latter seems to be a novelty, since in works by other authors, global strong L _p -solutions have been investigated only in Lagrangian coordinates; Eulerian coordinates are even declared as impossible to deal with. The proof is based on a careful derivation and study of the associated linear problem.     

Efficient low-storage Runge–Kutta schemes with optimized stability regions

A variety of numerical calculations, especially when considering wave propagation, are based on the method-of-lines, where time-dependent partial differential equations (PDEs) are first discretized in space. For the remaining time-integration, low-storage Runge–Kutta schemes are particularly popular due to their efficiency and their reduced memory requirements. In this work, we present a numerical approach to generate new low-storage Runge–Kutta (LSRK) schemes with optimized stability regions for advection-dominated problems. Adapted to the spectral shape of a given physical problem, those methods are found to yield significant performance improvements over previously known LSRK schemes. As a concrete example, we present time-domain calculations of Maxwell’s equations in fully three-dimensional systems, discretized by a discontinuous Galerkin approach.