Second-order Numerical Scheme Research Articles

Constraint-preserved numerical schemes with decoupling structure for the Ericksen–Leslie model with variable density

In this paper, first- and second-order numerical schemes are developed for the Ericksen–Leslie model with variable density. The influence of variable density leads to some conflicts between decoupling and unconditional energy stability. We overcome it by special discretization and extra-fractional-step method. The fully decoupled structures are obtained by using the scalar auxiliary variable(SAV) method with only one SAV. This allows only one additional ODE to be solved so that the schemes are computationally cheaper. The non-convex constraint |b|=1 is preserved by rewriting the orientation field with polar coordinates. It also improves computational efficiency because the vector function of the orientation field is replaced by scalar function. We first reformulate the Ericksen–Leslie model as an equivalent new form by introducing a SAV and polar coordinate form of the orientation field. Secondly, two linear schemes and the corresponding unconditional energy stabilities are established. Then, we show the detailed implementations for proving that the schemes are fully decoupled and uniquely solvable. Finally, the convergence rates and energy dissipations are tested by performing some numerical experiments. The evolutionary simulations are also given.

Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O(τ2) in H1- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

High-order energy stable variable-step schemes for the time-fractional Cahn–Hilliard model

Based on the Alikhanov formula of the Caputo derivative and the exponential scalar auxiliary variable approach, two different second-order stable numerical schemes are constructed for the time-fractional Cahn–Hilliard model, including the nonlinear L2-1σ scheme and the linear L2-1σ-ESAV scheme. Applying the discrete gradient structure, we construct the asymptotically compatible energies and the associated energy dissipation laws of these two schemes for the time-fractional Cahn–Hilliard model. Several experiments are carried out to demonstrate the accuracy of our schemes and compare their computational efficiency, as well as to investigate the coarsening dynamics of the time-fractional Cahn–Hilliard model with different adaptive time-stepping strategies. In the long-time simulations, the linear L2-1σ-ESAV scheme may be more costly than the nonlinear L2-1σ scheme although the linear scheme may take less time at each time level. It is observed that the energy of the L2-1σ-ESAV scheme prones to produce small fluctuations when large time steps are used.

Midpoint geometric integrators for inertial magnetization dynamics

We consider the numerical solution of the inertial version of Landau-Lifshitz-Gilbert equation (iLLG), which describes high-frequency nutation on top of magnetization precession due to angular momentum relaxation. The iLLG equation defines a higher-order nonlinear dynamical system with very different nature compared to the classical LLG equation, requiring twice as many degrees of freedom for space-time discretization. It exhibits essential conservation properties, namely magnetization amplitude preservation, magnetization projection conservation, and a balance equation for generalized free energy, leading to a Lyapunov structure (i.e. the free energy is a decreasing function of time) when the external magnetic field is constant in time. We propose two second-order numerical schemes for integrating the iLLG dynamics over time, both based on implicit midpoint rule. The first scheme unconditionally preserves all the conservation properties, making it the preferred choice for simulating inertial magnetization dynamics. However, it implies doubling the number of unknowns, necessitating significant changes in numerical micromagnetic codes and increasing computational costs especially for spatially inhomogeneous dynamics simulations. To address this issue, we present a second time-stepping method that retains the same computational cost as the implicit midpoint rule for classical LLG dynamics while unconditionally preserving magnetization amplitude and projection. Special quasi-Newton techniques are developed for solving the nonlinear system of equations required at each time step due to the implicit nature of both time-steppings. The numerical schemes are validated on analytical solution for macrospin terahertz frequency response and the effectiveness of the second scheme is demonstrated with full micromagnetic simulation of inertial spin waves propagation in a magnetic thin-film.

电荷输运相场模型二阶、 线性、 解捐、 能量稳 定的数值格式

电荷输运相场模型二阶、 线性、 解捐、 能量稳 定的数值格式

Convergence of a second-order scheme for non-local conservation laws

In this article, we present the convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws. We combine a MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme. The resulting scheme is shown to converge to a weak solution by establishing the maximum principle, bounded variation estimates and L1Lipschitz continuity in time. Further, using a space-step dependent slope limiter, we prove its convergence to the entropy solution. We also propose a MUSCL-Hancock type second-order scheme which requires only one intermediate stage unlike the Runge-Kutta schemes and is easier to implement. The performance of the proposed second-order schemes in comparison to a first-order scheme is demonstrated through several numerical experiments.

A Second-Order Scheme for the Generalized Time-Fractional Burgers' Equation

Abstract A second-order numerical scheme is proposed to solve the generalized time-fractional Burgers' equation. The time-fractional derivative is considered in the Caputo sense. First, the quasi-linearization process is used to linearize the time-fractional Burgers' equation, which gives a sequence of linear partial differential equations (PDEs). The Crank–Nicolson scheme is used to discretize the sequence of PDEs in the temporal direction, followed by the central difference formulae for both the first and second-order spatial derivatives. The established error bounds (in the L2− norm) obtained through the meticulous theoretical analysis show that the method is second-order convergent in space and time. The technique is also shown to be conditionally stable. Some numerical experiments are presented to confirm the theoretical results.

Error analysis of second-order IEQ numerical schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation

In this article, we derive error estimates for two second-order numerical schemes for solving the viscous Cahn-Hilliard equation with hyperbolic relaxation, one based on the second-order Crank-Nicolson time marching method and the other on the backward differentiation formula. In both schemes, the nonlinear potential is discretized by the Invariant Energy Quadratization (IEQ) method, which employs an auxiliary variable to produce a linear and unconditional energy-stable structure. After assuming some reasonable boundedness and continuity conditions for the nonlinear function, the optimal error estimate is rigorously derived using mathematical induction. Finally, several numerical experiments are carried out to verify the theoretical predictions of the convergence rate and energy stability of the algorithms.

Analysis of a second-order numerical scheme for time-fractional partial integro-differential equations with a weakly singular kernel

In this paper, a second-order numerical scheme is developed and analyzed to approximate the solution of a class of time-fractional Volterra integro-differential equations with a weakly singular kernel on a rectangular domain. In the present scheme, we first discretize the domain on a uniform grid. The L1-2 technique is used to discretize the Caputo derivative of order α with α∈(0,1) in the temporal direction. A weakly singular kernel makes it more challenging to approximate the integral and reduces the accuracy of the classical numerical integration method. In order to improve the convergence rate, the composite product trapezoidal formula is employed to approximate the fractional integral. Secondly, a classical central difference technique is used for spatial discretization. Using a maximum norm, the stability and convergence analysis is carried out under suitable regularity assumptions. It is proved that the scheme achieves second-order accuracy. In addition, we extend the proposed difference scheme to solve the corresponding semilinear problem. The well-known Newton’s linearization technique is used to deal with semilinearity. Numerical examples demonstrate the efficiency and applicability of the proposed scheme.

A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.

An Unconditionally Stable Difference Scheme for the Two-Dimensional Modified Fisher–Kolmogorov–Petrovsky–Piscounov Equation

In this article, we develop an unconditionally stable numerical scheme for the modified Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation modeling population dynamics in two-dimensional space. The Fisher–KPP equation models the process of interaction between reaction and diffusion. The new solution algorithm is based on an alternating direction implicit (ADI) method and an interpolation method so that it is unconditionally stable. The proposed finite difference method is second-order accurate in time and space variables. Therefore, the main purpose of this study is to propose the novel Fisher–KPP equation with a nonlinear growth term and develop an unconditionally stable second-order numerical scheme. The novelty of our method is that it is a numerical method with second-order accuracy using interpolation and ADI methods in two dimensions. We demonstrate the performance of the proposed scheme through computational tests such as convergence and stability tests and the effects of model parameters and initial conditions.

A highly efficient variant of scalar auxiliary variable (SAV) approach for the phase-field fluid-surfactant model

In this work, we study a phase-field fluid-surfactant system by applying two conservative Allen-Cahn equations and Navier-Stokes equations. A highly efficient variant of scalar auxiliary variable approach is applied in designing a second-order time-accurate numerical scheme. By introducing several scalar auxiliary variables, all variables are decoupled with each other and they can be updated in a step-by-step manner. The proposed method satisfies the mass conservation law, unique solvability and modified energy dissipation property. Several tests are performed to investigate the surfactant-laden phase-separation and the effect of surfactant in two dimensional and three dimensional space.

Convergence analysis of a temporally second-order accurate finite element scheme for the Cahn–Hilliard-Magnetohydrodynamics system of equations

In this paper we propose and analyze a temporally second-order accurate numerical scheme for the Cahn–Hilliard-Magnetohydrodynamics system of equations. The scheme is based on a modified Crank–Nicolson-type approximation for the time discretization and a mixed finite element method for the spatial discretization. The modified Crank–Nicolson approximation enables us to carry out the mass conservation and the energy stability analysis. Error estimates are derived for the phase field in the Lτ∞(0,T;H1) norm, and for the velocity and the magnetic fields in the Lτ∞(0,T;L2) norm, respectively. Numerical examples are presented to validate the theoretical results of the proposed scheme.

Fully decoupled linear BDF2 scheme for the penalty incompressible Ericksen–Leslie equations

This work focuses on the numerical approximation of the penalized Ericksen–Leslie equations. A second-order numerical scheme which has the advantages of full decoupling, linearization and unconditional stability in energy is constructed. Firstly, an equivalent new system is established by introducing two scalar auxiliary variables. One is added to the convective term in the Navier–Stokes equation and the other is added to the rest nonlinear terms. Secondly, a second-order BDF(backward differentiation formula) scheme for the new system is constructed while the pressure-correction method is used and the unconditional stability in energy is subsequently proved. Thirdly, the detailed implementation process is given to clarify that the scheme is fully decoupled and linear. Finally, several numerical simulations are shown to illustrate the accuracy and effectiveness of the scheme. Moreover, the annihilation phenomena of singularities are also performed.

Second order stabilized semi-implicit scheme for the Cahn–Hilliard model with dynamic boundary conditions

We study the numerical algorithm and error analysis for the Cahn–Hilliard equation with dynamic boundary conditions. A second-order in time, linear and energy stable scheme is proposed, which is an extension of the first-order stabilized approach. The corresponding energy stability and convergence analysis of the scheme are derived theoretically. Some numerical experiments are performed to verify the effectiveness and accuracy of the second-order numerical scheme, including numerical simulations under various initial conditions and energy potential functions, and comparisons with the literature works.

Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

A second-order BDF convex splitting numerical scheme for the Ericksen-Leslie equation

A second-order BDF convex splitting numerical scheme for the Ericksen-Leslie equation

A second-order ADI method for pricing options under fractional regime-switching models

&lt;abstract&gt;&lt;p&gt;Fractional regime-switching option models have recently attracted much attention because they can capture the sudden state movement of the market, and deal with the non-stationary behavior. A second-order numerical scheme is proposed to solve the regime-switching option pricing models with fractional derivatives in space. The sufficient conditions of the stability and convergence of the proposed scheme are studied in details. An alternating direction implicit (ADI) method is implemented to accelerate the computation in every time layer. Numerical experiments are presented to verify the convergence and efficiency of the proposed method, compared with classical Krylov subspace solvers.&lt;/p&gt;&lt;/abstract&gt;

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.

Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity

This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank–Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann–Liouville fractional integral term. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at t=0 so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, the stability and convergence are proved by means of the energy argument. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.