Second-order Accuracy Research Articles

Second-Order Accuracy in Time of Finite Difference Methods for Computational Aeroacoustics

The recently developed second-order accuracy in time finite difference method suitable for computational aeroacoustics (CAA) is introduced. Although, it is straight forward to compute the coefficients for finite-difference method of any order of accuracy using the Taylor series and to then further optimize them to enhance their wavenumber preserving properties, there are difficult questions concerning their numerical stability The goal of this work is to develop an effective numerical technique that includes both linear and nonlinear wave propagation in order to solve acoustics problems in time and space. It also aims to evaluate the accuracy, effectiveness, and stability of the new technique. In 1-D linear and nonlinear computational aeroacoustics, the novel techniques were used. The findings of the conventional methods (square wave (FTCS) technique and step wave lax approach) are presented in this paper, and it is shown that the FTCS method is typically unstable for hyperbolic situations and cannot be employed. Unfortunately, the FTCS equation has very little practical application. It is an unstable method, which can be used only (if at all) to study waves for a short fraction of one oscillation period. Nonlinear instability and shock formation are thus somewhat controlled by numerical viscosity such as that discussed in connection with Lax method equation. The second-order accuracy in time finite difference method is more efficient than the (square wave (FTCS), step wave lax) methods and is faster than the step wave lax method.

Conservative higher-order finite difference scheme for the coupled nonlinear Schrödinger equations

This paper introduces a conservative higher-order finite difference scheme for solving the coupled nonlinear Schrödinger equations. The Crank–Nicolson method is employed to discretize time derivatives and the sixth-order difference operator is used to discretize space derivatives, correspondingly, the resulting difference scheme has second-order accuracy in time and sixth-order accuracy in space. By utilizing the discrete energy method, the conservation of discrete mass and energy, the boundedness, existence and uniqueness of solution, unconditional stability and the convergence of the new scheme are proved. Then making use of Richardson extrapolation, the time accuracy is increased to the fourth order. Finally, the numerical experiments are conducted to validate the theoretical results presented in the paper.

The subdivision-based IGA-EIEQ numerical scheme for the Cahn–Hilliard–Darcy system of two-phase Hele-Shaw flow on complex curved surfaces

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces.

Разностная схема повышенного порядка аппроксимации для уравнения Аллера с переменными коэффициентами

The initial boundary value problem for the one-dimensional Aller’s equation with variable coefficients and boundary conditions of the first kind is studied. The problem under study describes the processes of heat transfer in a heterogeneous environment, moisture transfer in soils, and fluid filtration in fractured porous media. To numerically solve the problem posed, a difference scheme of high order of accuracy was constructed - fourth order of accuracy in h and second order of accuracy in τ. Using the method of energy inequalities, an a priori estimate of the solution in a difference treatment is obtained. From this estimate it follows that the solution is unique and stable with respect to the right-hand side and initial data. Under the assumption of the existence of an exact solution to the original differential problem in the class of sufficiently smooth functions, and also due to the linearity of the problem under consideration, the obtained a priori estimate implies that the solution of the constructed difference problem converges to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme. The goal and scientific novelty of the work is to obtain a new numerical scheme of a higher order of approximation when solving the Dirichlet problem for the Aller’s equation with variable coefficients.

Stress-strain state of a porous flexible rectangular FGM size-dependent plate subjected to different types of transverse loading: Analysis and numerical solution using several alternative methods

In this study, a mathematical model of flexible porous functionally graded (PFG) Kirchhoff size-dependent plates on a rectangular plane is constructed. Hamilton's principle provides new size-dependent governing equations, taking into account geometrical nonlinearity according to the model of Theodor von Kármán, as well as boundary conditions and initial conditions for the displacement of the plates. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. Numerical investigations analysing the stress-strain state of flexible Kirchhoff functionally graded size-dependent porous plates on a rectangular plane are based on the application of the Variational Iteration Method (VIM) or the Extended Kantorovich Method (EKM). The efficiency and high accuracy of the VIM method are demonstrated. The validity and reliability of the solutions obtained by VIM are discussed and compared with solutions obtained by other methods, such as the BGM in higher approximations, FDM of second-order accuracy and the FEM. Additionally, the solutions are compared with results obtained by other authors in the study of porous functionally graded macro, micro, and nanoplates. The solutions obtained are considered accurate. The results include an analysis of the influence of size-dependent parameters, type of material porosity, porosity index, functionally graded index, and different types of boundary conditions on the stress-strain state of plates under the action of different types of transverse loading.

A decoupled, linearly implicit and high-order structure-preserving scheme for Euler–Poincaré equations

It is challenging to develop high-order structure-preserving finite difference schemes for the modified two-component Euler–Poincaré equations due to the nonlinear terms and high-order derivative terms. To overcome the difficulties, we introduce a bi-variate function and carefully choose the intermediate average variable in the temporal discretization. Then, we obtain a decoupled and linearly implicit scheme. It is shown that the fully-discrete scheme can keep both the discrete mass and energy conserved. And the fully-discrete scheme has fourth-order accuracy in the spatial direction and second-order accuracy in the temporal direction. Several numerical examples are given to confirm the theoretical results.

Alternating direction implicit approach for the two-dimensional time fractional nonlinear Klein–Gordon and Sine–Gordon problems

The aim of this article is to establish a numerical scheme to achieve the theoretical accuracy near the weak singularity at t=0 in solving the two-dimensional time-fractional nonlinear mixed diffusion-wave equation (TFNMDWE). The governing problem involves diffusion term with time-fractional Caputo derivative (TFCD) of order ν1(0<ν1<1), and wave term with TFCD of order ν2(1<ν2<2). To handle the singularity at t=0, we use linearized L1 method to discretize both the TFCDs on nonuniform time meshes. By using the nonuniform L1 method to approximate TFCD and central difference operator for the space derivative approximation, the considered problem is converted to an equivalent system of equations. Then, we use the Alternating Direction Implicit (ADI) approach to develop a numerical scheme for solving the resulting system of coupled equations. Further, we prove stability analysis of the scheme. Numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFNMDWEs with smooth and non-smooth exact solutions to describe the accuracy of numerical scheme. The illustrated examples confirm that the scheme has second-order accuracy in space, and order of convergence (OC) in time direction is min(2−ν1,3−ν2,γν1,γ(ν2−1)), where γ is the mesh grading parameter used in construction of the nonuniform meshes. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Derivation of a 2D PCCU-AENO method for nonconservative problems

In this paper, we introduce a methodology to design genuinely two-dimensional (2D) second-order well-balanced path-conservative central-upwind (PCCU) schemes. The scheme studies dam-break with high sediment concentration over abrupt moving topography quickly spatially variable even in the presence of the resonance. This study is possible via a 2D hyperbolic sediment transport model (including arbitrarily sloping sediment beds and associated energy and entropy) in new generalized Shallow Water equations derived with associated energy and entropy in this work. We expose some properties of the model and we establish an existence result of global weak solutions. In addition, we show the convergence of a sequence of solutions of the proposed new model. The second-order accuracy of the PCCU scheme is achieved using a new extension AENO (Averaging Essentially Non-Oscillatory) reconstruction developed in the 2D version of this work. We prove that the derived 2D scheme is well-balanced, positivity-preserving and steady states capturing. Several dam break tests are made to show the ability and the superb performances of the proposed numerical modeling. The proposed method can help to address many engineering problems.

A Linearized Implicit Scheme for Solving Logarithmic Nonlinear Schrödinger’s Equation

In this research, we will develop a numerical scheme for solving the nonlinear logarithmic Schrödinger’s equation using the Linearized implicit scheme, that has second-order accuracy in space and time, with significant savings in computational time. We then compare the results to those obtained previously using the Crank-Nicolson scheme of the finite difference method. The stability and precision of this method will be evaluated before it is implemented. Precise solution and conserved quantities will be used to prove the efficiency as well as reliability of the method that has been proposed. Additionally, tests are conducted to explore the interactions that take place between two and three solitons. The numerical findings of our investigation demonstrated that the behavior of interactions is elastic.

Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

An efficient Hermite-Galerkin spectral scheme for three-dimensional incompressible Hall-magnetohydrodynamic system on infinite domain

The Hall-magnetohydrodynamic (Hall-MHD) system plays a significant role in the fluid description of an astrophysical plasma that features fast magnetic reconnection. Compared to the classical MHD system, the appearance of Hall term brings much more challenges in the field of numerical treatment. The aim of this paper is to construct an efficient spectral scheme for incompressible Hall-MHD system on three-dimensional infinite domain R3. Combining with a novel (HN+2,HN)-Hermite-Galerkin spectral scheme for spatial approximation, a new non-zero constant function approach for the nonlinear terms, a second-order projection method for the momentum equations, and BDF2 scheme for the decoupled system, here we establish, for the first time, a fully decoupled, completely linearized, and unconditionally energy stable scheme with directly spatial approximation in R3 and second-order temporal accuracy for incompressible Hall-MHD system. Numerical results are presented to illustrate the features of the proposed scheme, including convergence/stability tests and simulations of O- and X-points appearing in fast magnetic reconnection within the Hall-MHD regime.

Spatial-temporal high-order rotated-staggered-grid finite-difference scheme of elastic wave equations for TTI medium

In the finite-difference (FD) solution of the elastic wave equation for TTI medium, the traditional FD scheme can improve the computational accuracy of the spatial derivatives. However, the difference discretization on the temporal partial derivatives still uses the FD scheme of second-order accuracy, which cannot guarantee the computational accuracy of the temporal partial derivatives and often causes the temporal dispersion problem. The temporal high-order FD scheme can improve the computational accuracy of the temporal partial derivatives. Nevertheless, due to the complexity of the elastic wave equation in TTI medium, using the conventional staggered-grid technique to solve the elastic wave equation, some spatial derivatives need to be approximated by spatial interpolation, which will become new error and reduce the accuracy of the solution of the equations. Therefore, the currently commonly used temporal high-order FD method for the elastic wave equation in isotropic medium cannot be directly applied to the simulation of anisotropic medium. To solve the above problems, we propose a new temporal high-order rotated-staggered-grid FD stencil and derive a new temporal high-order FD scheme under this new stencil for the TTI medium elastic wave equation. Then we estimated the FD coefficients of the new stencil using Taylor series expansions (TE) and optimized the FD coefficients based on least squares (LS). Numerical experiments of the homogeneous TTI medium model and the modified BP model demonstrate that the TE + LS-based temporal high-order rotated-staggered-grid FD scheme can well solve the temporal dispersion in the numerical simulation of TTI medium.

Inverse scattering transform algorithm for the Manakov system

A numerical algorithm is described for solving the inverse spectral scattering problem associated with the Manakov model of the vector nonlinear Schrödinger equation. This model of wave processes simultaneously considers dispersion, nonlinearity and polarization effects. It is in demand in nonlinear physical optics and is especially perspective for describing optical radiation propagation through the fiber communication lines. In the presented algorithm, the solution to the inverse scattering problem based on the inversion of a set of nested matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations, using a block version of the Levinson-type Toeplitz bordering algorithm. Numerical tests carried out by comparing calculations with known exact analytical solutions confirm the stability and second order of accuracy of the proposed algorithm. We also give an example of the algorithm application to simulate the collision of a differently polarized pair of Manakov optical vector solitons.

An efficient matrix factorization within the projection framework for ameliorating the surface tension time step constraint in interfacial flows

An efficient matrix factorization approach is proposed for simulation of interfacial flows involving surface tension forces within the projection framework. The level set method is used to capture the interface of different phases and a projection based fluid solver is used for solving the variable density problem on a staggered grid. The governing equations for the two-phase flow, together with the convection equation for the interface, are organized into a matrix representation. Instead of explicitly partitioning the system or adopting non-linear iterative strategy, the system is factorized into separate equations using nested approximate LU decomposition with resort to the projection approach. Second-order temporal accuracy and implicit property are maintained throughout the decomposition. Numerical tests showed that the current scheme is second-order accurate in time, and can maintain numerical stability even when the time step is 80 times of the capillary time step constraint.

Angular Momentum Conserving Limiters for the Euler Equations

Limiters that are used to stabilise the discontinuous Galerkin method fail to preserve its angular momentum conservation property. We propose a simple technique that restores the conservation of angular momentum while preserving the second-order accuracy of the scheme when paired with an explicit time integrator. Numerical experiments confirm the conservation property of the technique paired with two limiters [Giuliani and Krivodonova. 2018. “Analysis of Slope Limiters on Unstructured Triangular Meshes.” Journal of Computational Physics 374: 1–26; Giuliani and Krivodonova. 2019. “A Moment Limiter for the Discontinuous Galerkin Method on Unstructured Triangular Meshes.” SIAM Journal on Scientific Computing 41 (1): A508–A537]. We show that the technique eliminates numerical artifacts that arise from limiting solutions to the Euler equations.

A class of unconditionally energy stable relaxation schemes for gradient flows

In this paper, a novel class of unconditionally energy stable schemes are constructed for solving gradient flow models by combining the relaxed scalar auxiliary variable (SAV) approach with the linear multistep technique. The main idea is first to rewrite the original model as an equivalent system, and then the linear multistep method in time, together with the relaxation strategy is employed to compute the solution of the reformulated system. The main benefit of the proposed schemes is that it can achieve second-order temporal accuracy and strictly unconditional energy stability, as well as a quite close to, and in many cases equal to, the original energy than the baseline SAV approaches. Finally, ample numerical experiments for the Allen–Cahn equation and Cahn–Hilliard equation are presented to confirm the efficiency, accuracy and strictly unconditional energy stability of the proposed schemes.

Lattice Boltzmann simulations of homogeneous shear turbulence laden with finite-size particles

As a powerful tool to investigate turbulent flows laden with finite-size particles, the lattice Boltzmann method (LBM)'s capability to contribute to a particle-resolved simulation of particle-laden homogeneous shear turbulence (HST) has yet to be realized. To capture the turbulence-particle interactions more accurately, the interpolated-bounce-back schemes combined with the momentum-exchange method that possesses second-order accuracy are adopted to treat the no-slip condition on particle surfaces and compute the hydrodynamic force/torque. However, this choice brings extra difficulties in enforcing the shear periodic boundaries due to the absence of fluid information in solid regions. The present study resolves these difficulties by designing a scheme to construct the distribution functions in the particle gaps based on the lubrication theory and an algorithm for the “refilling” process near the shear periodic boundaries. With these developments, a direct numerical simulation of particle-laden HST is achieved with LBM. The turbulence modulation effects induced by particles are briefly discussed.

A pressure Poisson equation-based second-order method for solving two-dimensional moving contact line problems with topological changes

We develop a second-order Cartesian grid based numerical method to solve two-dimensional moving contact line problems, which are modeled by the incompressible Navier–Stokes equations with the Navier-slip condition and the contact angle condition (CAC). The solutions of the flow field and the interface motion are decoupled in an alternating way. For a given interface, the velocity field is solved via a pressure Poisson equation formulation of the incompressible Navier–Stokes equations, which is numerically discretized by the immersed interface method. Once the velocity field is obtained, the interfacial kinematics together with the CAC is reformulated into a variational system, which is solved by the parametric finite element method (FEM). With this hybrid method, we detect topological changes in the interface by the inconsistency of neighboring normal vectors, which are directly computed through the parametric FEM. Second-order accuracy of the proposed method in both the interface and the contact line positions before and after topological changes has been numerically validated. Moreover, with the help of the numerical method, the merging and collision dynamics of droplets on the substrates are easily investigated.

Vertex-Centered Mixed Finite Element–Finite Volume scheme for 2D anisotropic hybrid mesh adaptation

The scope of this paper is to propose an appropriate numerical strategy to extend the Mixed Finite Element–Finite Volume (MEV) scheme to adapted meshes composed of both triangular and quadrangular elements, called hybrid meshes (Kallinderis and Kavouklis, 2005; Ito et al., 2013) (also named mixed-element meshes (Marcum and Gaither, 1999; Mavriplis, 2000)). Convective, diffusive and source terms require specific gradient discretization to maintain a global second-order accuracy. The major contributions of this paper concern the specification of these gradients, the convective fluxes’ discretization and an innovative framework for anisotropic metric-based adaptation on hybrid meshes. Particularly, the V4 scheme, traditionally developed for triangular elements, is extended to quadrilateral elements. This method maintains the same order of accuracy as the baseline version when upwind/downwind quadrilateral elements are aligned with the attached edge. Viscous fluxes are discretized using the APproximated Finite Element (APFE) method (Puigt et al., 2010), which is second-order accurate on regular quadrilaterals. Nodal gradients, which are considered at boundaries and for source terms, are discretized with a generalization of Clement’s L2-operator on hybrid meshes. The mesh adaptation relies on a specific metric gradation process to favor the clustering of structured elements at wall boundaries. The proposed scheme is validated on a set of CFD simulations.

A three-dimensional Petrov-Galerkin finite element interface method for solving inhomogeneous anisotropic Maxwell's equations in irregular regions

In this paper, a three-dimensional Petrov-Galerkin finite element interface (3D PGFEI) method with non-body-fitted grids is proposed for solving the inhomogeneous anisotropic time-harmonic Maxwell's equations in irregular regions. We derive the weak variational form of the anisotropic Maxwell interface problem. The irregular region is discretized by uniform tetrahedral grids while the media interface is not aligned with the grids. The level-set functions are applied to capture the discontinuity on the anisotropic media interface. Meanwhile, we construct the special basis functions according to the jump conditions across the inhomogeneous interface, where the permittivity and permeability are discontinuous three-dimensional tensors. The resulting linear system is a banded sparse matrix and demands fewer memory resources than dense matrix. Numerical experiments illustrate the efficiency and accuracy of the proposed 3D PGFEI method and the solutions can achieve second-order accuracy in terms of the relative L∞(Ω) and relative L2(Ω) error norms.