Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.

High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications

To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS),optimized flux difference schemes are proposed.The disadvantages in previous optimization routines,i.e.,reducing formal orders,or extending stencil widths,are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes.Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation(ADR)for nonlinear schemes.Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators.The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes,in terms of accuracy and resolution for smooth wave and vortex,especially for long-time simulations.Using optimized schemes increases the total CPU time by less than 4%.

High-order targeted essentially non-oscillatory scheme for two-fluid plasma model

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

High-order schemes for predicting computational aeroacoustic propagation with adaptive mesh refinement

High-order schemes based on block-structured adaptive mesh refinement method are prepared to solve computational aeroacoustic （CAA） problems with an aim at improving computational efficiency. A number of numerical issues associated with high-order schemes on an adaptively refined mesh, such as stability and accuracy are addressed. Several CAA benchmark problems are used to demonstrate the feasibility and efficiency of the approach.

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A new type of high-order multi-resolution weighted essentially non-oscillatory(WENO)schemes(Zhu and Shu in J Comput Phys,375:659-683,2018)is applied to solve for steady-state problems on structured meshes.Since the classical WENO schemes(Jiang and Shu in J Comput Phys,126:202-228,1996)might suffer from slight post-shock oscillations(which are responsible for the residue to hang at a truncation error level),this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations.This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes,could obtain fifth-order,seventh-order,and ninth-order in smooth regions,and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities.The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one.This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state problems.In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes,the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.

A High-Order Compact Scheme with Square-Conservativity

In order to improve the accuracy of forecasts of atmospheric and oceanic phenomena which possess a wide range of space and time scales, it is crucial to design the high-order and stable schemes. On the basis of the explicit square-conservative scheme, a high-order compact explicit square-conservative scheme is proposed in this paper. This scheme not only keeps the square-conservative characteristics, but also is of high accuracy. The numerical example shows that this scheme has less computing errors and better computational stability, and it could be considered to be tested and used in many atmospheric and oceanic problems.

Properties of High-Order Finite Difference Schemes and Idealized Numerical Testing

Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.

A FAMILY OF HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEMES WITH BRANCHING STABILITY FOR SOLVING 3-D PARABOLIC PARTIAL DIFFERENTIAL EQUATION

A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).

Stability of High-Order Staggered-Grid Schemes for 3D Elastic Wave Equation in Heterogeneous Media

In this paper, we firstly derive the stability conditions of high-order staggered-grid schemes for the three-dimensional (3D) elastic wave equation in heterogeneous media based on the energy method. Moreover, the plane wave analysis yields a sufficient and necessary stability condition by the von Neumann criterion in homogeneous case. Numerical computations for 3D wave simulation with point source excitation are given.

A high-order splitting scheme for the advection diffusion equation

A high-order splitting scheme for the advection-diffusion equation of pollutants is proposed in this paper. The multidimensional advection-diffusion equation is splitted into several one-dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth-order spatial accuracy. Several typically pure advection and advection-diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can he efficiently solved with little programming effort.

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.

A High-order Accuracy Explicit Difference Scheme with Branching Stability for Solving Higher-dimensional Heat-conduction Equation

A high-order accuracy explicit difference scheme for solving 4-dimensional heatconduction equation is constructed. The stability condition is r = △t/△x^2 = △t/△y^2 = △t/△z^2 = △t/△w^2 〈 3/8, and the truncation error is O（△t^2 ＋ △x^4）.

A High-Order Spatiotemporal Precision-Matching Taylor–Li Scheme for Time-Dependent Problems

Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.

A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

Compact finite difference schemes for the backward fractional Feynman–Kac equation with fractional substantial derivative

The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the nonlocal time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

Perfect plane-wave source for a high-order symplectic finite-difference time-domain scheme

The method of splitting a plane-wave finite-difference time-domain （SP-FDTD） algorithm is presented for the initiation of plane-wave source in the total-field / scattered-field （TF/SF） formulation of high-order symplectic finite- difference time-domain （SFDTD） scheme for the first time. By splitting the fields on one-dimensional grid and using the nature of numerical plane-wave in finite-difference time-domain （FDTD）, the identical dispersion relation can be obtained and proved between the one-dimensional and three-dimensional grids. An efficient plane-wave source is simulated on one-dimensional grid and a perfect match can be achieved for a plane-wave propagating at any angle forming an integer grid cell ratio. Numerical simulations show that the method is valid for SFDTD and the residual field in SF region is shrinked down to -300 dB.