Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.     

Third-order temporal discrete scheme for the non-stationary Navier–Stokes equations

In this article, we suggest a new third-order time discrete scheme for the two-dimensional non-stationary Navier–Stokes equations. After presenting the Galerkin finite element approximation for the spatial discretization, we consider an implicit/explicit time discrete scheme for the problem, which is based on the two-step Adams–Moulton scheme (implicit scheme) for the linear term and the three-step Adams–Bashforth scheme (explicit scheme) for the nonlinear term. In this method, we only need to solve a linearized discrete system at each time step, so the scheme can converge fast and the computational cost can be reduced. Moreover, under some assumptions, we deduce the stability and optimal error estimate for the velocity in L ²-norm.     

A lattice Boltzmann based implicit immersed boundary method for fluid–structure interaction

The immersed boundary method is a practical and effective method for fluid–structure interaction problems. It has been applied to a variety of problems. Most of the time-stepping schemes used in the method are explicit, which suffer a drawback in terms of stability and restriction on the time step. We propose a lattice Boltzmann based implicit immersed boundary method where the immersed boundary force is computed at the unknown configuration of the structure at each time step. The fully nonlinear algebraic system resulting from discretizations is solved by an Inexact Newton–Krylov method in a Jacobian-free manner. The test problem of a flexible filament in a flowing viscous fluid is considered. Numerical results show that the proposed implicit immersed boundary method is much more stable with larger time steps and significantly outperforms the explicit version in terms of computational cost.     

An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme.     

A semi-implicit approach for fast parameter estimation by means of the extended Kalman filter

An assessment is performed of the impact of the time-integration scheme on data assimilation based on the extended Kalman filter approach. The usage of the implicit Euler scheme, which leads to unconditionally stable model updates for the whole range of time steps, results in a less accurate estimation of model parameters, which can be overcome using a parameter estimator based on the explicit Euler scheme. However, the latter limits the maximum time step which may lead to an unacceptable increase in computational load. The alternative can be found using a semi-implicit estimator, where a model is updated using the implicit Euler scheme, whereas the propagation of the error covariance matrix in time is based on the explicit time integration scheme. The improved properties of the proposed algorithm are demonstrated on a one-dimensional problem of permeability estimation in a porous medium for a single phase oil flow.     

Discretization error estimator for transcient dynamic simulations

In this paper, we demonstrate that the concept of error in constitutive relation provides an answer to the problem of error estimation in transient dynamic analysis.The construction of our error measure is based on a reformulation of the transient dynamic problem. From the solution to the discretized model, we build a set of fields, which satisfy the kinematic constraints, the initial conditions and the equilibrium equation exactly. The quality of this numerical solution depends on the extent to which the constitutive relations are satisfied.Our error estimator can be used with explicit as well as implicit time integration schemes. Here, it is first calculated on a simple single-degree-of-freedom linear dynamic problem. Its satisfactory behavior is demonstrated by different tests. Moreover, it is compared with several other indicators from the literature.Next, we explain how this error measure can be applied to problems involving both time and space. Then, preliminary one-dimensional test results for a bar fixed at one end are presented and discussed.Finally, we introduce a new error indicator which turns out to be an indicator of the error on the time integration for the initial reference problem. This indicator enables us to extract from the global error estimation the main contribution, which is relative to the time integration scheme chosen. Then, this quantity is calculated in order to evaluate the error due to the lumped mass assumption for problems solved by the explicit central difference method.     

A comparison of implicit and explicit natural element methods in large strains problems: Application to soft biological tissues modeling

The natural element method (NEM) is one of the members of the large family of meshless methods, with clear advantages over the finite element method (FEM) in problems involving large mesh distortions or complex geometries where the design of the mesh is costly. These problems are found in many applications like, for instance, simulation of biological structures involving soft tissues, such as, articular joints. One additional advantage of NEM is that it can be easily coupled with finite elements and implemented into any FE framework, including well-known commercial packages. NEM as most other spatial approximation approaches can be applied to evolution problems in two types of time (or pseudo-time) integration schemes, namely implicit and explicit. However, the NEM explicit version has neither been implemented nor sufficiently analyzed, so a comparative study of those two types of NEM time integration schemes is still missing. The main aim of this paper is to discuss issues related to NEM accuracy and stability in its explicit version, and problems related to its implementation into an explicit FE commercial code. Finally, a comparative study addressing the main properties, advantages and disadvantages of both types of NE schemes, implicit and explicit, is presented. Several examples of application are discussed including aspects where NEM is competitive with FEM including modeling of human articular joints like the knee. Explicit NEM allows achieving accurate results for high distortions and complex contact conditions although constraints on time step still are a major drawback and comparable to those known in finite elements to keep stability and accuracy despite the less NEM sensitivity to mesh distortion.     

A constitutive operator splitting method for nonlinear transient analysis

A constitutive operator splitting method for the time integration of the nonlinear dynamic equations of motion which result from the consideration of nonlinear material behavior is studied. In the method, the material constitutive law is split into a constant, history independent relation (implicit portion) and a variable history dependent relation (explicit portion); the resulting constituents are then integrated by implicit and explicit methods, respectively. Analysis of stability and some example solutions indicate that for materials with decreasing stiffness (materials which soften with increasing strain) the method is unconditionally stable. Several examples are considered which compare the accuracy of the method to exact solutions and solutions obtained by explicit central difference time integration; in all cases there is good agreement.     

An expert system for setting time steps in dynamic finite element programs

An expert system, ETUDES—Expert Time integration control Using Deep and Surface Knowledge System, which addresses the determination of the timestep for time integration of linear structural dynamic equations is described. This time-step may also be applicable for a moderately nonlinear simulation of the same structure. The program also determines whether an explicit or implicit method is most efficient for the particular simulation. A production rule programming system written in OPS5 is used for the implementation of this prototype expert system. Issues relating to the expert system architecture for this application, such as knowledge representation and structure, as well as domain knowledge are discussed. The prototype is evaluated by measuring it's performance in various benchmark model problems.     

基于IMEX的织物动态仿真的近似解法

织物在空间运动的刚性特征始终是困扰织物动态仿真的难题．显式方法简单灵活，易于实现，但受稳定因素影响，无法实现具有刚性特征的织物动态模拟；隐式方法稳定性好，却忽略了非线性因素，而且计算复杂，直接影响到仿真的最终结果和实际效率．针对这一问题，提出了基于隐式一显式的近似解法，该方案从系统受力形变的非线性特征出发，将质点受力分为线性和非线性两部分，线性部分采用隐式解法，非线性部分利用显式解法，线性方程组的求解则运用近似解法．实验结果表明，该方法兼具两种方法的优点，既保留了隐式方法的稳定性，又充分利用了显式方法的简易性处理非线性特征，从而从真正意义上解决织物仿真中的刚性问题．     

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.     

Shape design optimization of SPH fluid–structure interactions considering geometrically exact interfaces

Fluid–structure interaction problems are solved by applying a smoothed particle hydrodynamics method to a weakly compressible Navier–Stokes equation as well as an equilibrium equation for geometrically nonlinear structures in updated Lagrangian formulation. The geometrically exact interface, consisting of B-spline basis functions and the corresponding control points, includes the high order geometric information such as tangent, normal, and curvature. The exactness of interface is kept by updating the control points according to the kinematics obtained from response analysis. Under the scheme of explicit time integration and updated Lagrangian formulation, the required shape design velocity should be updated at every single step. The update scheme of design velocity is developed using the sensitivity of physical velocity. The developed sensitivity analysis method is further utilized in gradient-based shape optimization problems and turns out to be very efficient since the interaction pairs of particles determined in the response analysis can be directly utilized.     

Decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data

Based on the mixed finite element method, we consider the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data in this paper. The temporal treatment of the spatial discrete Boussinesq equations is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms. Thanks to the decoupled method, the considered problem is split into two subproblems and these subproblems can be solved in parallel. Under some restriction on the time step, we present the stability and convergence results of numerical solutions, Finally, some numerical experiments are provided to test the performance of the developed numerical scheme and verify the established theoretical findings.     

双曲方程基于分离算子的交替方向差分算法

因为在自然科学领域有着广泛的应用,双曲型方程组的数值求解一直是研究的热点．本文中,为求解一类非线性二阶双曲型方程,将方程中的非线性椭圆微分算子分解为线性部分和非线性部分,对线性部分用隐格式逼近,对非线性部分用显格式逼近,这种方法可以把非线性问题转化成每一时间层只有右端项不同的线性方程组,计算简单且计算格式绝对稳定;交替方向格式可以把多维问题转化成一维问题,x,y两个方向的迭代矩阵均为三对角矩阵,结构相同,易于编程并行计算．最后通过数值实验表明结果符合理论分析．     

On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method

Using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger equation are shown to be conserved for two stabilized version of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank–Nicolson; and the explicit circularly exact Leapfrog scheme. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann stability analysis. A series of numerical experiments with different nonlinear potentials are presented.     

Euler-like discrete models of the logistic differential equation

To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the standard linear stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation.Among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other. We find that these schemes are superior to explicit schemes when they are stable and the blow-up time has not passed: for these λh-values they are dynamically faithful. When these schemes ‘turn unstable’, however, they have much less desirable properties than explicit or fully implicit schemes: they become simultaneously superstable and unstable. This is explained by the fact that these schemes are not self-adjoint: the linearly implicit self-adjoint scheme is dynamically faithful in an Euler-typical range of step sizes and gives correct stability for all step sizes.     

An Implicit Eulerian–Lagrangian WENO3 Scheme for Nonlinear Conservation Laws

We present a new, formally third order, implicit Weighted Essentially Non-Oscillatory (iWENO3) finite volume scheme for solving systems of nonlinear conservation laws. We then generalize it to define an implicit Eulerian–Lagrangian WENO (iEL-WENO) scheme. Implicitness comes from the use of an implicit Runge–Kutta (RK) time integrator. A specially chosen two-stage RK method allows us to drastically simplify the computation of the intermediate RK fluxes, leading to a computationally tractable scheme. The iEL-WENO3 scheme has two main steps. The first accounts for particles being transported within a grid element in a Lagrangian sense along the particle paths. Since this particle velocity is unknown (in a nonlinear problem), a fixed trace velocity v is used. The second step of the scheme accounts for the inaccuracy of the trace velocity v by computing the flux of particles crossing the incorrect tracelines. The CFL condition is relaxed when v is chosen to approximate the characteristic velocity. A new Roe solver for the Euler system is developed to account for the Lagrangian tracings, which could be useful even for explicit EL-WENO schemes. Numerical results show that iEL-WENO3 is both less numerically diffusive and can take on the order of about 2–3 times longer time steps than standard WENO3 for challenging nonlinear problems. An extension is made to the advection–diffusion equation. When advection dominates, the scheme retains its third order accuracy.     

A Chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrödinger equations

The collision of solitary waves is an important problem in both physics and applied mathematics. In this paper, we study the solution of coupled nonlinear Schrödinger equations based on pseudospectral collocation method with domain decomposition algorithm for approximating the spatial variable. The problem is converted to a system of nonlinear ordinary differential equations which will be integrated in time by explicit Runge–Kutta method of order four. The multidomain scheme has much better stability properties than the single domain. Thus this permits using much larger step size for the time integration which fulfills stability restrictions. The proposed scheme reduces the effects of round-of-error for the Chebyshev collocation and also uses less memory without sacrificing the accuracy. The numerical experiments are presented which show the multidomain pseudospectral method has excellent long-time numerical behavior and preserves energy conservation property.     

无条件稳定的显式降维非线性有限元

针对传统降维非线性有限元计算速度与精确度难以兼顾的问题,提出了一种无条件稳定的显式迭代算法。基于泰勒展开式得到速度、加速度的三阶精度差分表达式从而获得新的有限元显式迭代方程,并分析其单自由度系统下的传递矩阵谱半径。改进迭代方程使谱半径始终小于1从而满足无条件稳定的要求。实验表明,改进后的显式迭代算法在等效阻尼比的精度上优于中心差分法和隐式迭代法;在降维非线性有限元模型计算中的计算耗时优于隐式迭代方法,提高了降维非线性有限元的迭代计算速度。模型在降维后维度数值较高时,仍能维持良好的计算耗时和帧率,保证了模型的精确度。     

Implicit-explicit schemes for nonlinear consolidation

A general analytical procedure capable of performing linear and nonlinear consolidation analysis of saturated porous media is proposed. A brief review of the coupled field equations is included and the constitutive assumptions are stated explicitly. Time integration of the resulting nonlinear semidiscrete finite element equations is performed by using an implicit/explicit predictor/multicorrector scheme developed by Hughes and co-workers. It is shown that the algorithm can be simply and concisely implemented. The technique allows for a convenient selection of implicit and explicit elements, and for a convenient implicit-explicit split of the various operators appearing in the equations. The procedure proves to be extremely effective in dealing with consolidation problems. Numerical results which demonstrate the versatility and accuracy of the proposed procedures are presented.