Implicit Midpoint Scheme Research Articles

Two-dimensional growth of incompressible and compressible soft biological tissues

Soft tissues have the ability of growth as a response to some especial changes of their environment. The aim of this work is to develop a non-linear finite element formulation for two-dimensional growth of soft tissues. The formulation is derived in Total Lagrangian (TL) framework. Moreover, the growth variable is considered as a single-valued scalar parameter known as growth multiplier. To deal with evolution equation resulting from the growth parameter, an implicit midpoint scheme is employed. Furthermore, a method is presented for compressible hyperelastic material models, and the formulas can be applied to incompressible and compressible, isotropic and anisotropic tissues. Finally, to investigate the performance of the proposed model, some examples are provided and compared to those available in the literature.

Arbitrary high order A-stable and B-convergent numerical methods for ODEs via deferred correction

This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme is A-stable, satisfies a B-convergence property, and that the correction on a scheme DC2j of order 2j of accuracy leads to a scheme DC2j + 2 of order 2j + 2. The order of accuracy is guaranteed by a deferred correction condition. Numerical experiments with standard stiff and non-stiff ODEs are performed with the DC2, ..., DC10 schemes. The results show a high accuracy of the method. The theoretical orders of accuracy are achieved together with a satisfactory stability.

A framework of the finite element solution of the Landau-Lifshitz-Gilbert equation on tetrahedral meshes

A framework for the numerical solution of the Landau-Lifshitz-Gilbert equation is developed in this paper. The numerical framework is based on the finite element method on tetrahedral meshes for the spatial discretization and the implicit midpoint scheme for the temporal discretization. The computational complexity for calculating the demagnetization field is effectively reduced by using a PDE approach, in which a gradient recovery technique is used for preserving the numerical accuracy. The numerical convergence of the proposed method is studied in detail for the μMAG standard problem #3, from which a limit is predicted for the desired side length. The capability of the proposed method on handling problems defined on complex domains is successfully demonstrated by several examples, in which the computational domains are thin films with irregular defects.

Strange Dynamic Behavior Induced by Time Delay for a Rijke Tube System with Periodic Excitation

Thermoacoustic instability problems widely exist in many real-world applications such as gas turbines and rocket motors. A Rijke tube is a typical thermoacoustic system, and it is difficult to analyze such a system due to the nonlinearity and time delay. In this paper, a set of nonlinear ordinary differential equations with time delay which represent a Rijke tube system will be studied. The state space of such a tube system consists of velocity and pressure, and the periodic motion can be discretized based on an implicit midpoint scheme. Through Newton–Raphson method, the node points on the periodic motion will be solved, and the analytical solution of such a periodic motion for Rijke tube system can be recovered using a set of Fourier representations. According to the theory of discrete maps, the stability of the periodic motion will be obtained. With such a proposed technique, the stable and unstable branches of period-1 motions with one mode and multiple modes will be presented. The Neimark–Sacker bifurcation will be observed, and the periodic and quasi-periodic motions related to Neimark–Sacker bifurcation will be discussed. The strange dynamic behavior for stable period-1 motion at the neighborhood of Neimark–Sacker bifurcation will be observed, and the motion can only be attracted to the stable period-1 limit cycle when the initial conditions are given exactly from the analytic solution of such a stable period-1 motion. Otherwise, it will be attracted to another coexisting quasi-periodic motion.

The numerical influence of additional parameters of inertia representations for quaternion-based rigid body dynamics

Different inertia representations can lead to different formulations of the differential-algebraic equations for the quaternion-based rigid body dynamics. In this paper, the inertia representations are classified into $\alpha $-type and $\gamma $-type, according to the additional parameters in the kinetic energy. These two types of representations and the corresponding parameters $\alpha $ and $\gamma $ are theoretically equivalent if the constraint $\boldsymbol{q}^{T}\boldsymbol{q} = 1$ is satisfied exactly. Nevertheless, the error estimation demonstrates that they can present entirely different numerical features in simulation and suggests that the parameter $\gamma $ can be used to optimize the numerical performance of the integrations in simulation. To further verify the numerical difference between the inertia representations of $\alpha $-type and $\gamma $-type, the corresponding modified Hamilton’s equations are discretized by the IMS (implicit midpoint scheme), EMS (energy–momentum preserving scheme) and Gauss–Lobatto SPARK methods. Numerical performance for the examples of the spinning symmetrical top is shown to result from the comprehensive effect of the discretization schemes including the distribution of discretized points and the convergence order, the inertia representations and their combinations. Numerical results further suggest that the integrations of $\gamma $-type are superior to those of $\alpha $-type and the optimized values of $\gamma $ can be used to achieve better numerical accuracy, convergence speed and stability.

Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics

The numerical integration of particle trajectories in curved spacetimes is fundamental for obtaining realistic models of the particle dynamics around massive compact objects such as black holes and neutron stars. Generalized algorithms capable of handling generic metrics are required for studies of both standard spacetimes (Schwarzschild and Kerr metrics) and non-standard spacetimes (e.g. Schwarzschild metric plus non-classical perturbations or multiple black hole metrics). The most commonly employed explicit numerical schemes (e.g. Runge-Kutta) are incapable of producing highly accurate results at critical points, e.g. in the regions close to the event horizon where gravity causes extreme curvature of the spacetime, at an acceptable computational cost. Here, we describe a generalized algorithm for the numerical integration of time-like (massive particles) and null (photons) geodesics in any given 3+1 split spacetime. We introduce a new, exactly energy-conserving implicit integration scheme based on the preservation of the underlying Hamiltonian, and we compare its properties with a standard fourth-order Runge-Kutta explicit scheme and an implicit midpoint scheme. We test the numerical performance of the three schemes against analytic solutions of particle and photon orbits in Schwarzschild and Kerr spacetimes. We also prove the versatility of our framework in handling more exotic metrics such as Morris-Thorne wormholes and quantum-perturbed Schwarzschild black holes. The generalized approach is also discussed in the perspective of future extensions to more complex particle dynamics, e.g. the addition of the Lorentz force acting on charged particles, which allows for test particle diagnostics in GRMHD simulations.

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

A Multi-Symplectic Compact Method for the Two-Component Camassa–Holm Equation with Singular Solutions**Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570, the Open Foundation of State Key Laboratory of High Performance Computing of China, the Research Fund of the National University of Defense Technology under Grant No JC15-02-02, and the Fund from HPCL.

The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.

Stability of Implicit Midpoint Algorithm Applied to Structures with Nonlinear Damping

Stability of implicit midpoint algorithm applied to solving the equation of motion with nonlinear damping. The analytical and numerical study exhibit the unconditional stability of implicit midpoint scheme for nonlinear damping systems. The work of the present paper provides theoretic base for further application of this implicit midpoint approach.

Birkhoffian Symplectic Scheme for a Quantum System

In this paper, a classical system of ordinary differential equations is built to describe a kind of n-dimensional quantum systems. The absorption spectrum and the density of the states for the system are defined from the points of quantum view and classical view. From the Birkhoffian form of the equations, a Birkhoffian symplectic scheme is derived for solving n-dimensional equations by using the generating function method. Besides the Birkhoffian structure-preserving, the new scheme is proven to preserve the discrete local energy conservation law of the system with zero vector f. Some numerical experiments for a 3-dimensional example show that the new scheme can simulate the general Birkhoffian system better than the implicit midpoint scheme, which is well known to be symplectic scheme for Hamiltonian system.

Satellite attitude dynamics and estimation with the implicit midpoint method

We describe the application of the implicit midpoint integrator to the problem of attitude dynamics for low-altitude satellites without the use of quaternions. Initially, we consider the satellite to rotate without external torques applied to it. We compare the numerical solution with the exact solution in terms of Jacobi’s elliptic functions. Then, we include the gravity-gradient torque, where the implicit midpoint integrator proves to be a fast, simple and accurate method. Higher-order versions of the implicit midpoint scheme are compared to Gauss–Legendre Runge–Kutta methods in terms of accuracy and processing time. Finally, we investigate the performance of a parameter-adaptive Kalman filter based on the implicit midpoint integrator for the determination of the principal moments of inertia through observations.

Discretization methods for optimal control problems with state constraints

We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise state constraints. Since this problem may have no classical solutions, it is also formulated in relaxed form. The classical control problem is then discretized by using the implicit midpoint scheme, while the controls are approximated by (not necessarily continuous) piecewise linear classical controls. We first study the behavior in the limit of properties of discrete optimality, and of discrete admissibility and extremality. We then apply a penalized gradient projection method to each discrete classical problem, and also a corresponding progressively refining combined discretization-optimization method to the continuous classical problem, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal in some sense for the corresponding discrete or continuous, classical or relaxed, problem. For nonconvex problems whose solutions are nonclassical, we show that we can apply the above methods to the problem formulated in Gamkrelidze relaxed form. Finally, numerical examples are given.

Discretization Methods for Nonconvex Optimal Control Problems with State Constraints

ABSTRACT We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise state constraints. Because no convexity assumptions are made, the problem may have no classical solutions, and it is reformulated in relaxed form. The relaxed control problem is then discretized by using the implicit midpoint scheme, while the controls are approximated by piecewise constant relaxed controls. We first study the behavior in the limit of properties of discrete relaxed optimality, and of discrete relaxed admissibility and extremality. We then apply a penalized conditional descent method to each discrete relaxed problem, and also a corresponding discrete method to the continuous relaxed problem that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal for the discrete or the continuous problem. Finally, numerical examples are given.

Symplectic integrators for discrete nonlinear Schrödinger systems

Symplectic methods for integrating canonical and non-canonical Hamiltonian systems are examined. A general form for higher order symplectic schemes is developed for non-canonical Hamiltonian systems using generating functions and is directly applied to the Ablowitz–Ladik discrete nonlinear Schrödinger system. The implicit midpoint scheme, which is symplectic for canonical systems, is applied to a standard Hamiltonian discretization. The symplectic integrators are compared with an explicit Runge–Kutta scheme of the same order. The relative performance of the integrators as the dimension of the system is varied is discussed.

Nonlinear Resonance Artifacts in Molecular Dynamics Simulations

The intriguing phenomenon of resonance, a pronounced integrator-induced corruption of a system's dynamics, is examined for simple molecular systems subject to the classical equations of motion. This source of timestep limitation is not well appreciated in general, and certainly analyses of resonance patterns have been few in connection to biomolecular dynamics. Yet resonances are present in the commonly used Verlet integrator, in symplectic implicit schemes, and also limit the scope of current multiple-timestep methods that are formulated as symplectic and reversible. The only general remedy to date has been to reduce the timestep. For this purpose, we derive method-dependent timestep thresholds (e.g., Tables 1 and 2) that serve as useful guidelines in practice for biomolecular simulations. We also devise closely related symplectic implicit schemes for which the limitation on the discretization stepsize is much less severe. Specifically, we design methods to remove third-order, or both the third- and fourth-order, resonances. These severe low-order resonances can lead to instability or very large energies. Our tests on two simple molecular problems (Morse and Lennard–Jones potentials), as well as a 22-atom molecule, N-acetylalanyl-N′-methylamide, confirm this prediction; our methods can delay resonances so that they occur only at larger timesteps (EW method) or are essentially removed (LIM2 method). Although stable for large timesteps by this approach, trajectories show large energy fluctuations, perhaps due to the coupling with other factors that induce instability in complex nonlinear systems. Thus, the methods developed here may be more useful for conformational sampling of biomolecular structures. The analysis presented here for the blocked alanine model emphasizes that one-dimensional analysis of resonances can be applied to a more complex, multimode system to analyze resonance behavior, but that resonance due to frequency coupling is more complex to pinpoint. More generally, instability, apparently due to numerically induced resonances, has been observed in the application of the implicit midpoint scheme to vibrating structures and could be expected also in the simulation of nonlinear wave phenomena; in such applications it is adequate not to resolve the highest frequency modes, so the proposed methods could be very useful.

Resonance in the dynamics of chemical systems simulated by the implicit midpoint scheme

The numerical behavior of the symplectic, implicit midpoint method with a wide range of integration timesteps is examined through an application to a diatomic molecule governed by a Morse potential. Our oscillator with a 12.6 fs period exhibits notable, integrator induced, timestep- ( Δt) dependent resonances and we predict approximate values of Δt where they will occur. The particular case of a third-order resonance ( Δt ≈ 7 fs here) leads to instability, and higher-order resonances ( n = 4, 5) to large energetic fluctuations and/or corrupted phase diagrams. Significantly, for Δt > 10 fs the energy errors remain bound.

Implicit discretization schemes for Langevin dynamics

We explore here several numerical schemes for Langevin dynamics in the general implicit discretization framework of the Langevin/implicit-Euler scheme, LI. Specifically, six schemes are constructed through different discretization combinations of acceleration, velocity, and position. Among them, the explicit BBK method (LE in our notation) and LI are recovered, and the other four (all implicit) are named LIM1, LIM2, MID1, and MID2. The last two correspond, respectively, to the well-known implicit-midpoint scheme and the trapezoidal rule. LI and LIM1 are first-order accurate and have intrinsic numerical damping. LIM2, MID1, and MID2 appear to have large-timestep stability as LI but overcome numerical damping. However, numerical results reveal limitations on other grounds. From simulations on a model butane, we find that the non-damping methods give similar results when the timestep is small; however, as the timestep increases, LIM2 exhibits a pronounced rise in the potential energy and produces wider distr...