Implicit Midpoint Research Articles

Conformal structure‐preserving schemes for damped‐driven stochastic nonlinear Schrödinger equation with multiplicative noise

AbstractSince many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped‐driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped‐driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal‐symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure‐preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.

Arbitrary High-Order Unconditionally Stable Methods for Reaction-Diffusion Equations with inhomogeneous Boundary Condition via Deferred Correction

Abstract In this paper, we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. To avoid possible order reduction, the IBVP is first transformed into an IBVP with homogeneous boundary conditions (IBVPHBC) via a lifting of inhomogeneous Dirichlet, Neumann or mixed Dirichlet–Neumann boundary conditions. The IBVPHBC is discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order 2 ⁢ p + 2 2p+2 of accuracy at the stage p = 0 , 1 , 2 , … p=0,1,2,\ldots of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage 𝑝 of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVPHBC. This fully discretized scheme is unconditionally stable with order 2 ⁢ p + 2 2p+2 of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular, while an increment of one order is proven for a quasi-uniform family of meshes. Numerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio, a Fisher equation, a linear reaction-diffusion equation addressing order reduction and two linear IBVPs in two dimensions are performed and demonstrate the unconditional convergence of the method. The orders 2, 4, 6, 8 and 10 of accuracy in time are achieved. Except for some linear problems, the accuracy of DC methods is better than that of BDF methods of same order.

Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics. The isospectral property thereby often corresponds to mathematically or physically important conservation laws. The most prominent feature of these systems, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can in the Lie-Poisson case be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonally implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincaré reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.

Solving Lorenz System by Using Lower Order Symmetrized Runge-Kutta Methods

Runge-Kutta is a widely used numerical method for solving the non-linear Lorenz system. This study focuses on solving the Lorenz equations with the classical parameter values by using the lower order symmetrized Runge-Kutta methods, Implicit Midpoint Rule (IMR), and Implicit Trapezoidal Rule (ITR). We show the construction of the symmetrical method and present the numerical experiments based on the two methods without symmetrization, with one- and two-step active symmetrization in a constant step size setting. For our numerical experiments, we use MATLAB software to solve and plot the graphical solutions of the Lorenz system. We compare the oscillatory behaviour of the solutions and it appears that IMR and two-step active IMR turn out to be chaotic while the rest turn out to be non-chaotic. We also compare the accuracy and efficiency of the methods and the result shows that IMR performs better than the symmetrizers, while two-step active ITR performs better than ITR and one-step active ITR. Based on the results, we conclude that different implicit numerical methods with different steps of active symmetrization can significantly impact the solutions of the non-linear Lorenz system. Since most study on solving the Lorenz system is based on explicit time schemes, we hope this study can motivate other researchers to analyze the Lorenz equations further by using Runge-Kutta methods based on implicit time schemes.

A charge-preserving compact splitting method for solving the coupled stochastic nonlinear Schrödinger equations

In this paper, a splitting method for solving a system of coupled stochastic nonlinear Schrödinger (CSNLS) equations is proposed. Based on its Hamiltonian structure, the CSNLS equations split into two subproblems such that one of them is linear. The solution of the nonlinear subproblem is computed exactly, and the linear subproblem is discritized through a compact method in the spatial direction and the symplectic implicit midpoint rule with respect to the time variable. Stability, discrete stochastic multi-symplectic conservation law, discrete charge and energy conservation laws for the proposed method are investigated. Numerical results are reported for various amplitudes of noise. We present the propagation and collision of solitons which are different from the deterministic case. Finally, the mean-square order of convergence of the proposed algorithm in the temporal and spatial directions for the CSNLS equations is investigated in a numerical example.

Error analysis for full discretizations of quasilinear wave-type equations with two variants of the implicit midpoint rule

AbstractWe study the full discretization of a general class of first- and second-order quasilinear wave-type problems with the implicit midpoint rule and a linearized variant thereof. Based on a proof by induction, we prove wellposedness and a rigorous error estimate for both schemes, combining energy techniques, inverse estimates and a linearized fixed-point iteration for the analysis of the nonlinear scheme. To confirm the relevance of the general framework, we derive novel error estimates for the full discretization of two prominent examples from nonlinear physics: the Westervelt equation and the Maxwell equations with Kerr nonlinearity.

A conservative implicit-PIC scheme for the hybrid kinetic-ion fluid-electron plasma model on curvilinear meshes

The hybrid kinetic-ion fluid-electron plasma model is widely used to study challenging multi-scale problems in space and laboratory plasma physics. Here, a novel conservative scheme for this model employing implicit particle-in-cell techniques [1] is extended to arbitrary coordinate systems via curvilinear maps from logical to physical space. The scheme features a fully non-linear electromagnetic formulation with a multi-rate time advance – including sub-cycling and orbit-averaging for the kinetic ions. By careful choice of compatible particle-based kinetic-ion and mesh-based fluid-electron discretizations in curvilinear coordinates, as well as particle-mesh interpolations and implicit midpoint time advance, the scheme is proven to conserve total energy for arbitrary curvilinear meshes. In the electrostatic limit, the method is also proven to conserve total momentum for arbitrary curvilinear meshes. Although momentum is not conserved for arbitrary curvilinear meshes in the electromagnetic case, it is for an important subset of Cartesian tensor-packed meshes. The scheme and its novel conservation properties are demonstrated for several challenging numerical problems using different curvilinear meshes, including a merging flux-rope simulation for a space weather application, and a helical m=1 mode simulation for magnetic fusion energy application.

Real time evolution with neural-network quantum states

A promising application of neural-network quantum states is to describe the time dynamics of many-body quantum systems. To realize this idea, we employ neural-network quantum states to approximate the implicit midpoint rule method, which preserves the symplectic form of Hamiltonian dynamics. We ensure that our complex-valued neural networks are holomorphic functions, and exploit this property to efficiently compute gradients. Application to the transverse-field Ising model on a one- and two-dimensional lattice exhibits an accuracy comparable to the stochastic configuration method proposed in [Carleo and Troyer, Science 355, 602-606 (2017)], but does not require computing the (pseudo-)inverse of a matrix.

Numerically Efficient Discrete-Time Dielectric Elastomer Actuators Models for Optimal Control

Dielectric elastomers can be used to develop innovative mechatronic actuators (called DEA). In this paper, we consider the optimal control problem (OCP) of transitioning between an actuator's operating states. Following a model-based approach, a set of discrete-time state equations is derived for the DEA, which satisfy accuracy demands and, at the same time, can be solved efficiently by numerical methods. Due to numerical stiffness, the implicit midpoint rule is chosen as an integration method. Then, we discuss different classes of input signals to be considered in the OCP. In particular, zero-order-hold and first-order-hold parametrizations are compared within the OCP formulation. While we show equivalence of two OCP formulations from the theoretical point of view, numerical investigations show benefits of our proposed OCP model regarding the quality of computed solutions and the robustness w.r.t. initial guesses. Altogether, we derive a numerically treatable DEA optimal control set-up, which can be used for various control tasks.

Accuracy and Efficiency of Symmetrized Implicit Midpoint Rule for Solving the Water Tank System Problems

The accuracy and efficiency of water tank system problems can be determined by comparing the Symmetrized Implicit Midpoint Rule (IMR) with the IMR. Static and dynamic analyses are part of a mathematical model that uses energy conservation to generate a nonlinear Ordinary Differential Equation. Static analysis provides optimal working points, while dynamic analysis outputs an overview of the system behaviour. The procedure mentioned is tested on two water tank designs, namely, cylindrical and rectangular tanks with two different parameters. The Symmetrized IMR is used in this study. Results show that the two-step active Symmetrized IMR applied on the proposed mathematical model is precise and efficient and can be used for the design of appropriate controls. The cylindrical water tank model takes the fastest time in emptying the water tank. The approach of the various water tank models shows an increase in accuracy and efficiency in the range of parameters used for practical model applications. The results of the numerical method show that the two-step Symmetrized IMR provides more precise stability, accuracy and efficiency for the fixed step size measurements compared with other numerical methods.

Dynamic phase-field fracture with a first-order discontinuous Galerkin method for elastic waves

We present a new numerical approach for wave induced dynamic fracture. The method is based on a discontinuous Galerkin approximation of the first-order hyperbolic system for elastic waves and a phase-field approximation of brittle fracture driven by the maximum tension. The algorithm is staggered in time and combines an implicit midpoint rule for the wave propagation followed by an implicit Euler step for the phase-field evolution. At fracture, the material is degraded, and the waves are reflected at the diffusive interfaces. Two and three-dimensional examples demonstrate the advantages of the proposed method for the computation of crack growth and spalling initiated by reflected and superposed waves.

An implicit split-operator algorithm for the nonlinear time-dependent Schrödinger equation

The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schrödinger equations. However, when applied to certain nonlinear time-dependent Schrödinger equations, this algorithm loses time reversibility and second-order accuracy, which makes it very inefficient. Here, we propose to overcome the limitations of the explicit split-operator algorithm by abandoning its explicit nature. We describe a family of high-order implicit split-operator algorithms that are norm-conserving, time-reversible, and very efficient. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. Although they are only applicable to separable Hamiltonians, the implicit split-operator algorithms are, in this setting, more efficient than the recently proposed integrators based on the implicit midpointmethod.

Wrinkle and curl distortion of leaves using plant dynamic

An algorithm was proposed to simulate the withering deformation of plant leaves by wrinkle and curl due to dehydration, based on cell dynamics and time-varying external force. First, a leaf boundary expansion algorithm was proposed to locate the feature points on the tip of the vein and construct the primary vein using a discrete geodesic path. Second, a novel mass-spring system by cell dynamics and a non-uniform mass distribution was defined to accelerate the movement of the boundary cells. Third, the cell swelling force was defined and adjusted to generate wrinkle deformation along with dehydration. Fourth, the time-varying external force on the feature points was defined to generate the curl deformation by adjusting the initial value of the external force and multiple iterative parameters. The implicit midpoint method was used to solve the equation of motion. The experimental results showed that our algorithm could simulate the wrinkle and curl deformation caused by dehydration and withering of leaves with high authenticity.

Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations

Coherent or exact equations of motion for a post-Newtonian Lagrangian formalism are the Euler-Lagrange equations without any terms truncated. They naturally conserve energy {and} angular momentum. Doubling the phase-space variables of positions and momenta in the coherent equations, we establish extended phase-space symplectic-like integrators with the midpoint permutations. The velocities should be solved iteratively from the algebraic equations of the momenta defined by the Lagrangian during the course of numerical integrations. It is shown numerically that a fourth-order extended phase-space symplectic-like method exhibits good long-term stable error behavior in energy {and} angular momentum, as a fourth-order implicit symplectic method with a symmetric composition of three second-order implicit midpoint rules or a fourth-order Gauss-Runge-Kutta implicit symplectic scheme does. For given time step and integration time, the former method is superior to the latter integrators in computational efficiency. {The extended phase-space method is used to study the effects of the parameters and initial conditions on the orbital dynamics of the coherent Euler-Lagrange equations for a post-Newtonian circular restricted three-body problem. It is also applied to trace the effects of the initial spin angles, initial separation and initial orbital eccentricity on the dynamics of the coherent post-Newtonian Euler-Lagrange equations of spinning compact binaries.

A high-order fully discrete scheme for the Korteweg–de Vries equation with a time-stepping procedure of Runge–Kutta-composition type

Abstract We consider the periodic initial-value problem for the Korteweg–de Vries equation that we discretize in space by a spectral Fourier–Galerkin method and in time by an implicit, high-order, Runge–Kutta scheme of composition type based on the implicit midpoint rule. We prove $L^{2}$ error estimates for the resulting semidiscrete and the fully discrete approximations. Some numerical experiments illustrate the results.

Symplectic discrete-time energy-based control for nonlinear mechanical systems

We present a novel approach for discrete-time state feedback control implementation which reduces the deteriorating effects of sampling on stability and performance in digitally controlled nonlinear mechanical systems. We translate the argument of energy shaping to discrete time by using the symplectic implicit midpoint rule. The method is motivated by recent results for linear systems, where feedback imposes closed-loop behavior that exactly represents the symplectic discretization of a desired target system. For the nonlinear case, the sampled system and the target dynamics are approximated with second order accuracy using the implicit midpoint rule. The implicit nature of the resulting state feedback requires the numerical solution of an in general nonlinear system of algebraic equations in every sampling interval. For an implementation with pure position feedback, the velocities/momenta have to be approximated in the sampling instants, which gives a clear interpretation of our approach in terms of the Störmer–Verlet integration scheme on a staggered grid. Both the Hamiltonian and the Lagrangian perspective are adopted. We present discrete-time versions of impedance or energy shaping plus damping injection control as well as computed torque tracking control in the simulation examples to illustrate the performance and stability gain compared to the quasi-continuous implementation. We discuss computational aspects and show the structural advantages of the implicit midpoint rule compared to other integration schemes in the appendix.

Comment on “A magnetic velocity Verlet method” [Am. J. Phys. 88, 1075 (2020)

The implicit midpoint method described by Chambliss and Franklin [Am. J. Phys. 88, 1075 (2020)] is known not to exactly conserve energy for a spatially varying magnetic field, nor does it conserve angular momentum in a constant magnetic field. It is, therefore, not competitive with the widely used Boris solver in plasma physics, which conserves both.

Majorant series for the N-body problem

As a follow-up of a previous work of the authors, this work considers uniform global time-renormalization functions for the gravitational N-body problem. It improves on the estimates of the radii of convergence obtained therein by using a completely different technique, both for the solution to the original equations and for the solution of the renormalized ones. The aforementioned technique which the new estimates are built upon is known as majorants and allows for an easy application of simple operations on power series. The new radii of convergence so-obtained are approximately doubled with respect to our previous estimates. In addition, we show that majorants may also be constructed to estimate the local error of the implicit midpoint rule (and similarly for Runge-Kutta methods) when applied to the time-renormalized N-body equations and illustrate the interest of our results for numerical simulations of the Solar System.

Two linearly implicit energy preserving exponential scalar auxiliary variable approaches for multi-dimensional fractional nonlinear Schrödinger equations

In this paper, we propose two linearly implicit energy preserving schemes with constant coefficient matrix for multi-dimensional fractional nonlinear Schrödinger equations. By introducing an exponential auxiliary variable for the nonlinear energy, we first reformulate the original system into an equivalent system, which admits mass and energy conservation laws. Further, by virtue of the Lawson transformation, we rewrite the modified system into another equivalent system, which possesses both the mass and energy conservation laws. As for the two kinds of modified systems, we construct two classes of linearly implicit energy preserving schemes by combining the implicit midpoint method and the extrapolation strategy. Moreover, the proposed schemes enjoy the second-order accuracy in time and spectral accuracy in space, respectively. Numerical results are reported to demonstrate that the proposed numerical schemes have high efficiency for energy preservation in long-time computations.

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.