Second-order Accuracy In Time Research Articles

Discrete Maximum Principle and Energy Stability of the Compact Difference Scheme for Two-Dimensional Allen-Cahn Equation

The Allen-Cahn model is discussed mainly in the phase field simulation. The compact difference method will be used to numerically approximate the two-dimensional nonlinear Allen-Cahn equation with initial and boundary value conditions, and then, a fully discrete compact difference scheme with second-order accuracy in time and fourth-order in space is established. And its numerical solution satisfies the discrete maximum principle under the constraints of reasonable space and time steps. On this basis, the energy stability of the scheme is investigated. Finally, numerical examples are given to illustrate the theoretical results.

A second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-Nematic model for two-phase complex fluids confined in the Hele-Shaw cell

We consider the numerical approximation of the binary immiscible mixture of nematic liquid crystals and viscous Newtonian fluids confined in a Hele-Shaw cell, where the free interface motion is simulated by using the phase-field approach via the energy variational method. The governing system is highly complicated nonlinear and coupled, consisting of the Darcy equations for the flow field, the Cahn-Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. The numerical scheme developed in this paper is the first “ideal” scheme, namely, it not only has the following characteristics: linearity, second-order time accuracy, unconditional energy stability, and decoupling structure, but also at each time step, only needs to solve a few elliptic equations with constant coefficients. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical experiments are further carried out to prove the effectiveness of the scheme, in which the influence of the initial orientations and anchoring elastic energy of the liquid crystal on the Saffman-Taylor fingering instability are studied.

Time–space domain scalar wave modeling by a novel hybrid staggered-grid finite-difference method with high temporal and spatial accuracies

Staggered-grid finite-difference (SFD) scheme is favored in wave equation simulation due to its superior accuracy and stability to center-grid finite-difference (CFD) scheme. However, for the scalar wave equation (SWE) modeling, the conventional SFD (CSFD) scheme only reaches second-order accuracy in space and time for the discrete SWE; the recently developed modified SFD (MSFD) scheme improves the accuracy to (2N)th-order (N<4) but is costly because the MSFD scheme is coined by adding many extra grid points to the CSFD scheme. To tackle these issues, we develop a cost-effective hybrid SFD (HSFD) scheme, which combines the features of the CSFD and MSFD schemes; we prove that the new HSFD scheme can simultaneously reach (2N)th-order accuracy in space and time for the discrete wave equation. In addition, to deal with the optimization difficulties due to the nonlinear dispersion relation of the SFD schemes, we propose a two-step linear optimization method to improve the accuracy of the new HSFD scheme. The analyses on dispersion, stability properties and numerical simulation examples demonstrate that the new HSFD scheme owns better accuracy and stability than the CSFD scheme. Computational cost analysis shows that the HSFD scheme can be more efficient than the MSFD scheme because it only requires half the additional grid points.

A massively parallel time-domain coupled electrodynamics–micromagnetics solver

We present a high-performance coupled electrodynamics–micromagnetics solver for full physical modeling of signals in microelectronic circuitry. The overall strategy couples a finite-difference time-domain approach for Maxwell’s equations to a magnetization model described by the Landau–Lifshitz–Gilbert equation. The algorithm is implemented in the Exascale Computing Project software framework, AMReX, which provides effective scalability on manycore and GPU-based supercomputing architectures. Furthermore, the code leverages ongoing developments of the Exascale Application Code, WarpX, which is primarily being developed for plasma wakefield accelerator modeling. Our temporal coupling scheme provides second-order accuracy in space and time by combining the integration steps for the magnetic field and magnetization into an iterative sub-step that includes a trapezoidal temporal discretization for the magnetization. The performance of the algorithm is demonstrated by the excellent scaling results on NERSC multicore and GPU systems, with a significant (59×) speedup on the GPU using a node-by-node comparison. We demonstrate the utility of our code by performing simulations of an electromagnetic waveguide and a magnetically tunable filter.

Defect-deferred correction method for the non-stationary coupled Stokes/Darcy model

This paper develops the defect-deferred correction method to solve the non-stationary coupled Stokes/Darcy model. This method is a combination of defect correction method and deferred correction method. And it can not only achieve the second order accuracy in time, but also be applied to the problem with small viscosity and hydraulic conductivity coefficients. The theoretical proof of the stability and the second order accuracy in time are shown. Some numerical experiments are given to verify the error convergence order in time. In addition, compared with the standard Galerkin finite element method, the advantages of the presented method in calculating small viscosity and hydraulic conductivity coefficients will also be reflected in the numerical experiments.

High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations

&lt;abstract&gt;&lt;p&gt;In this paper, a high order compact finite difference is established for the time multi-term fractional sub-diffusion equation. The derived numerical differential formula can achieve second order accuracy in time and four order accuracy in space. A unconditionally stable and convergent difference scheme is presented, and a rigorous proof for the stability and convergence is given. Numerical results demonstrate the efficiency of the proposed difference schemes.&lt;/p&gt;&lt;/abstract&gt;

Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations

&lt;abstract&gt;&lt;p&gt;This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.&lt;/p&gt;&lt;/abstract&gt;

Numerical scheme for solving the nonuniformly forced cubic and quintic Swift–Hohenberg equations strictly respecting the Lyapunov functional

Computational modeling of pattern formation in nonequilibrium systems is a fundamental tool for studying complex phenomena in biology, chemistry, materials and engineering sciences. The pursuit for theoretical descriptions of some among those physical problems led to the Swift–Hohenberg equation (SH3) which describes pattern selection in the vicinity of instabilities. A finite differences scheme, known as Stabilizing Correction (Christov and Pontes, 2002), developed to integrate the cubic Swift–Hohenberg equation in two dimensions, is reviewed and extended in the present paper. The original scheme features Generalized Dirichlet boundary conditions (GDBC), forcings with a spatial ramp of the control parameter, strict implementation of the associated Lyapunov functional, and second-order representation of all derivatives. We now extend these results by including periodic boundary conditions (PBC), forcings with Gaussian distributions of the control parameter and the quintic Swift–Hohenberg (SH35) model. The present scheme also features a strict implementation of the functional for all test cases. A code verification was accomplished, showing unconditional stability, along with second-order accuracy in both time and space. Test cases confirmed the monotonic decay of the Lyapunov functional and all numerical experiments exhibit the main physical features: highly nonlinear behavior, wavelength filter and competition between bulk and boundary effects.

Fully-discrete Spectral-Galerkin scheme with second-order time-accuracy and unconditionally energy stability for the volume-conserved phase-field lipid vesicle model

In this work, for the phase-field model of lipid vesicles with the property of accurate volume conservation, we construct an effective fully-discrete numerical scheme, in which, the time marching method is based on a novel splitting type technique, and space is discretized by using the Spectral-Galerkin method. The advantage of this scheme is its high efficiency and ease of implementation. Specifically, although the model is highly nonlinear, just by solving two independent linear biharmonic equations with constant coefficients at each time step, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. The essence of the scheme is to introduce several additional auxiliary variables and use the specially designed ordinary differential equations to reformulate the system. In this way, energy stability can be obtained unconditionally, while avoiding the calculation of variable-coefficient systems. We strictly prove that the energy stability in the fully-discrete form that the scheme holds and give a detailed implementation process. Numerical experiments in 2D and 3D are further carried out to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

A second-order numerical method for Landau-Lifshitz-Gilbert equation with large damping parameters

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with coefficient matrices independent of the magnetization, and fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.

A finite point method for the fractional cable equation using meshless smoothed gradients

This paper presents a meshless finite point method (FPM) for the numerical analysis of the fractional cable equation. A second-order time discrete scheme is proposed to approximate both integer-order and fractional-order time derivatives. Then, based on the stabilized moving least squares approximation and the meshless smoothed gradients, a new implementation of the FPM is provided to enhance the accuracy and convergence rate in space. Theoretical error of the FPM is analyzed. Numerical results verify the efficiency of the method and show that the method can gain second-order accuracy in time and fourth-order accuracy in space.

A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations

For highly coupled nonlinear incompressible magnetohydrodynamic (MHD) system, a well-known numerical challenge is how to establish an unconditionally energy stable linearized numerical scheme which also has a fully decoupled structure and second-order time accuracy. This paper simultaneously reaches all of these requirements for the first time by developing an effective numerical scheme, which combines a novel decoupling technique based on the “zero-energy-contribution” feature satisfied by the coupled nonlinear terms, the second-order projection method for dealing with the fluid momentum equations, and a finite element method for spatial discretization. The implementation of the scheme is very efficient, because only a few independent linear elliptic equations with constant coefficients need to be solved by the finite element method at each time step. The unconditional energy stability and well-posedness of the scheme are proved. Various 2D and 3D numerical simulations are carried out to illustrate the developed scheme, including convergence/stability tests and some benchmark MHD problems, such as the hydromagnetic Kelvin-Helmholtz instability, and driven cavity problems.

Uniform error bounds of exponential wave integrator methods for the long-time dynamics of the Dirac equation with small potentials

Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods are presented and analyzed for the long-time dynamics of the Dirac equation with small potentials characterized by ε∈(0,1] a dimensionless parameter. Based on the (symmetric) exponential wave integrator for temporal derivatives in phase space followed by applying the Fourier pseudospectral discretization for spatial derivatives, the EWI-FP methods are explicit and of spectral accuracy in space and second-order accuracy in time for any fixed ε=ε0. Uniform error bounds are rigorously carried out at O(hm0+τ2) up to the time at O(1/ε) with the mesh size h, time step τ and m0 an integer depending on the regularity of the solution. Extensive numerical results are reported to confirm our error bounds and comparisons of two methods are shown. Finally, dynamics of the Dirac equation in 2D are presented to validate the numerical schemes.

An unconditionally stable second-order linear scheme for the Cahn-Hilliard-Hele-Shaw system

In this paper, we focus on the numerical approximation of the Cahn-Hilliard-Hele-Shaw system. Firstly, based on the idea of the stabilized method, an unconditionally stable linear scheme with second-order accuracy in time and space is proposed, which is modified from the Crank-Nicolson scheme. Secondly, we derive that the proposed numerical scheme is unconditionally stable, without any restriction for the time step size. After a careful calculation, we get discrete error estimates of the time step size τ and space step size h. Finally, numerical simulations of energy dissipation and spinodal decomposition are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme.

CTCS Schemes for Second Order Wave Equation: Numerical Results and Spectral Analysis

IIn this paper, two finite difference methods are used to solve the one-dimensional second order wave equation with constant coefficients subject to specified initial and boundary conditions. Two numerical experiments are considered. The two methods are Central in Time and Central in Space scheme with second order accuracy in both time and space, abbreviated as CTCS (2,2) and Central in Time and Central in Space scheme with second order accuracy in time and fourth order accuracy in space, abbreviated as CTCS (2,4). Properties such as consistency and stability are studied. We also perform spectral analysis of dispersive and dissipative properties of the two methods. Two numerical experiments are considered, and the numerical results are displayed.

Fully-discrete spectral-Galerkin scheme with decoupled structure and second-order time accuracy for the anisotropic phase-field dendritic crystal growth model

In this work, we consider numerical approximations of the anisotropic phase-field dendritic crystal growth model, which is a highly complex coupled nonlinear system consisting of the anisotropic Allen-Cahn equation and the heat equation. Through the combination of a novel explicit auxiliary variable IEQ approach for temporal discretization and the spectral-Galerkin approach for spatial discretization, we develop the first fully-discrete numerical scheme with linearity, decoupled structure, unconditional energy stability, and second-order time accuracy for the particular phase-field dendritic model. In the process of obtaining a full decoupling structure and maintaining energy stability, the introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role. The designed scheme is highly efficient because only a few elliptic equations with constant coefficients are needed to be solved at each time step. The unconditional energy stability of the scheme has been strictly proved, and the detailed implementation process is given. Through several numerical simulations of 2D and 3D dendritic crystal growth examples, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

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.

A reduced-order finite element method based on POD for the incompressible miscible displacement problem

In this paper, we establish a reduced-order finite element (ROFE) method with very few degrees of freedom for the incompressible miscible displacement problem. Firstly, we construct the finite element (FE) method with second-order accuracy in time, where backward difference formulation is used for the time discretization and classical FE formulation is used for the space discretization. Optimal a priori error estimates for the FE solutions are proved. Secondly, we apply the proper orthogonal decomposition (POD) technique to develop the ROFE method, which can effectively reduce degrees of freedom and CPU time. Optimal a priori error estimates for the ROFE solutions are derived. Besides, we give the algorithm process of the ROFE method. Finally, some numerical examples are presented to verify the behavior of the ROFE method for piecewise linear element. And these examples imply the proposed method is feasible and effective for solving the incompressible miscible displacement problem.

A unified second-order accurate in time MPM formulation for simulating viscoelastic liquids with phase change

We assume that the viscous forces in any liquid are simultaneously local and non-local , and introduce the extended POM-POM model [McLeish and Larson 1998; Oishi et al. 2012; Verbeeten et al. 2001] to computer graphics to design a unified constitutive model for viscosity that generalizes prior models, such as Oldroyd-B, the Upper-convected Maxwell (UCM) model [Sadeghy et al. 2005], and classical Newtonian viscosity under one umbrella, recovering each of them with different parameter values. Implicit discretization of our model via backward Euler recovers the variational Stokes solver of [Larionov et al. 2017] for Newtonian viscosity. For greater accuracy, however, we introduce the second-order accurate Generalized Single Step Single Solve (GS4) scheme [Tamma et al. 2000; Zhou and Tamma 2004] to computer graphics, which recovers all prior second-order accurate time integration schemes to date. Using GS4 and our generalized constitutive model, we present a Material Point Method (MPM) for simulating various viscoelastic liquid behaviors, such as classical liquid rope coiling, buckling, folding, and shear thinning/thickening. In addition, we show how to couple our viscoelastic liquid simulator with the recently introduced non-Fourier heat diffusion solver [Xue et al. 2020] for simulating problems with phase change, such as melting chocolate and digital fabrication with 3D printing. While the discretization of heat diffusion is slightly different within GS4, we show that it can still be efficiently solved using an assembly-free Multigrid-preconditioned Conjugate Gradients solver. We present end-to-end 3D simulations to demonstrate the versatility of our framework.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.