Compact Difference Research Articles

A fourth-order block-centered compact difference scheme for nonlinear contaminant transport equations with adsorption

Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth order compact difference scheme of solution and its flux is derived. We prove the mass conservation of the proposed scheme and its unconditional stability. We analyze the convergence and obtain the fourth-order error estimate under the smooth regularity of exact solution. Numerical experiments are presented to show the performance of the schemes.

Fourth-order compact difference schemes for the two-dimensional nonlinear fractional mobile/immobile transport models

In this article, we develop fourth-order compact difference schemes based on the linearized generalized BDF2-θ to solve the two-dimensional nonlinear fractional mobile/immobile (M/I) transport equations. We derive theoretical results, including unconditional stability and error estimates associated with solution regularity. Finally, we provide extensive numerical examples with smooth solutions to demonstrate the effectiveness and accuracy of the schemes and develop the corrected compact difference scheme to recovery convergence results with nonsmooth solutions.

Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Compact difference of composition operators with smooth symbols on the unit ball

A long-standing problem raised by Shapiro and Sundberg in 1990, the characterization for compact differences of composition operators acting on the Hardy space over the unit disk, has been recently obtained in terms of certain Carleson measures ([4]). The function theoretic characterization for compact differences still remains open in the Hardy space case even on the unit disc and the situation is much more complicated for the several variables case. In this article, we investigate a function theoretic characterization of the compact difference on the Hardy or the Bergman spaces over the unit ball. The condition ρ(φ(z),ψ(z))→0 as max⁡{|φ(z)|,|ψ(z)|}→1 is shown to be sufficient for the difference, Cφ−Cψ, of two composition operators to be compact on the Hardy or the Bergman spaces over the unit ball when each single composition operator is bounded. We show that this condition is also a necessary condition if the symbols are of class Lip1(B).

Stability analysis of inverse Lax–Wendroff boundary treatment of high order compact difference schemes for parabolic equations

In this paper, we study the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The compact finite difference schemes could achieve very high order accuracy with relatively small stencils. To match the convergence order of the compact schemes in the interior domain, we take the simplified inverse Lax–Wendroff (SILW) procedure (Tan et al., 2012; Li et al., 2017) as our numerical boundary treatment. The third order total variation diminishing (TVD) Runge–Kutta method (Shu and Osher, 1988) is taken as our time-stepping method in the fully-discrete case. Two analysis techniques are adopted to check the algorithm’s stability, one is based on the Godunov–Ryabenkii theory, and the other is the eigenvalue spectrum visualization method (Vilar and Shu, 2015). Both the semi-discrete and fully-discrete cases are investigated, and these two different analysis techniques yield consistent results. Several numerical experimental results are shown to validate the theoretical results.

Local states of chromatin compaction at transcription start sites control transcription levels.

The ‘open’ and ‘compact’ regions of chromatin are considered to be regions of active and silent transcription, respectively. However, individual genes produce transcripts at different levels, suggesting that transcription output does not depend on the simple open-compact conversion of chromatin, but on structural variations in chromatin itself, which so far have remained elusive. In this study, weakly crosslinked chromatin was subjected to sedimentation velocity centrifugation, which fractionated the chromatin according to its degree of compaction. Open chromatin remained in upper fractions, while compact chromatin sedimented to lower fractions depending on the level of nucleosome assembly. Although nucleosomes were evenly detected in all fractions, histone H1 was more highly enriched in the lower fractions. H1 was found to self-associate and crosslinked to histone H3, suggesting that H1 bound to H3 interacts with another H1 in an adjacent nucleosome to form compact chromatin. Genome-wide analyses revealed that nearly the entire genome consists of compact chromatin without differences in compaction between repeat and non-repeat sequences; however, active transcription start sites (TSSs) were rarely found in compact chromatin. Considering the inverse correlation between chromatin compaction and RNA polymerase binding at TSSs, it appears that local states of chromatin compaction determine transcription levels.

Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Bounded and compact differences of two composition operators acting from the weighted Bergman space A^p_omega to the Lebesgue space L^q_nu , where 0<q<p<infty and omega belongs to the class of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for A^p_omega , with p>q and , involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space A^p_alpha with -1<alpha <infty to the setting of doubling weights. The case is also briefly discussed and an open problem concerning this case is posed.

Impact of a discretization scheme on an autoignition time in LES of a reacting droplet-laden mixing layer

We analyse an autoignition process in a two-phase flow in a temporally evolving mixing layer formed between streams of a cold liquid fuel (heptane at 300 K) and a hot oxidizer (air at 1000 K) flowing in opposite directions. We focus on the influence of a discretization method on the prediction of the autoignition time and evolution of the flame in its early development phase. We use a high-order code based on the 6th order compact difference method for the Navier–Stokes and continuity equations combined with the 2nd order Total Variation Diminishing (TVD) and 5th order Weighted Essentially Non-Oscillatory (WENO) schemes applied for the discretization of the advection terms in the scalar transport equations. The obtained results show that the autoignition time is more dependent on the discretization method than on the flow initial conditions, i.e., the Reynolds number and the initial turbulence intensity. In terms of mean values, the autoignition occurs approximately 15% earlier when the TVD scheme is used. In this case, the ignition phase characterizes a sharp peak in the temporal evolution of the maximum temperature. The observed differences are attributed to a more dissipative character of the TVD scheme. Its usage leads to a higher mean level of the fuel in the gaseous form and a smoother distribution of species resulting in a lower level of the scalar dissipation rate, which facilitates the autoignition process.

Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

We consider compact difference schemes of approximation order 4+2 on a three-point spatial stencil for theKlein–Gordon equations with constant and variable coefficients. New compact schemes areproposed for one type of second-order quasilinear hyperbolic equations. In the case of constantcoefficients, we prove the strong stability of the difference solution under small perturbations ofthe initial conditions, the right-hand side, and the coefficients of the equation. A priori estimatesare obtained for the stability and convergence of the difference solution in strong mesh norms.

A fourth-order compact difference algorithm for numerical solution of natural convection in an inclined square enclosure

In this work, a fourth-order compact finite difference (FD) algorithm on the nine-point 2-D stencil is proposed to simulate numerically natural convection in an inclined square enclosure using the vorticity-stream function form of the Naiver-Stokes equations. The key feature of the proposed algorithm is that for all Rayleigh numbers Ra of physical interest, the point-successive overrelaxation or point-successive underrelaxation (SOR) iteration can be used. The numerical capability of the presented algorithm is demonstrated by the application to natural convection in an inclined square enclosure. The angle of inclination of the cavity is varied from (heated from above) to (heated from below) in steps of Computations are performed for Rayleigh numbers equal to 103, 104, 105, 106 and 107 while the Prandtl number is kept constant (Pr = 0.71). Test results, which are presented in terms of streamlines, isotherms, isovorticities and local and average Nusselt number, indicate that the present algorithm could predict the benchmark results for temperature and flow fields on relatively coarser grids.

A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection–Diffusion Problems with Boundary/Interior Layers

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection–diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does/do not contain boundary or interior layer(s).

Scalar mixing in a Kelvin-Helmholtz shear layer and implications for Reynolds-averaged Navier-Stokes modeling of mixing layers.

Large-eddy simulation of a temporally evolving Kelvin-Helmholtz (KH) mixing layer is performed with the tenth-order compact difference code miranda to examine the steady-state behavior of a passive scalar in a shear-driven mixing layer. It is shown that the integral behavior of scalar variance in a KH mixing layer behaves similarly to the integral behavior of scalar variance in a Rayleigh-Taylor (RT) mixing layer, and mixedness of the simulated KH shear layer tends towards a value of about 0.8. It is further shown that if the k-L-a-V Reynolds-averaged Navier-Stokes (RANS) model [B. E. Morgan et al., Phys. Rev. E 98, 033111 (2018)2470-004510.1103/PhysRevE.98.033111], calibrated to reproduce steady-state mixing in an RT layer, is applied to simulate a KH mixing layer, the RANS model will significantly overpredict the magnitude of scalar variance in the KH layer. A straightforward addition to the k-L-a-V model is then suggested, and self-similarity analysis is applied to determine constraints on model coefficients. It is shown that with the addition of a buoyancy production term in the model equation for scalar variance, it becomes possible to eliminate the model deficiency and match steady-state mixedness in simulations of both RT and KH mixing layers with a single model calibration.

Numerical investigation of double-diffusive convection in rectangular cavities with different aspect ratio I: High-accuracy numerical method

In this paper, being based on the idea of dispersion-relation-preserving optimization, a class of high-order high-resolution upwind compact schemes with a free parameter and their consistent boundary scheme are proposed for the solution of the governing equations of the 2D double-diffusive convection. The first derivative and the second derivative, in the vorticity equation, the temperature equation and the concentration equation, are discretized by using the optimal upwind compact scheme on a uniform mesh and the fourth-order symmetrical Padé compact scheme, respectively. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point two dimensional stencil. The fourth-order Runge-Kutta method is utilized for the temporal discretization. To assess numerical capability of the proposed algorithm, particularly its spatial behavior, the problems about the convection diffusion equation, the nonlinear Burgers equation and the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are computed by using the newly proposed algorithm. The numerical results are in excellent agreement with the benchmark solutions and some of the accurate results available in the literature, and show that effectiveness, accuracy and the advantage of better resolution of the present method. After that, steady and unsteady solutions for the double-diffusive convection in a rectangular cavity are also used to assess the efficiency of the present algorithm. The period of oscillation and flow field profiles are in great agreement with the data in the literature for buoyancy ratio, λ=1. The typical separation and secondary vortices at the bottom corner of the cavity as well as the top corner can be captured well for various buoyancy ratio. These indicate that the present method is suitable for simulating effectively the unsteady double-diffusive convection.

Optical frequency comb Fourier transform spectroscopy of 14N216O at 7.8 µm

We use a Fourier transform spectrometer based on a compact mid-infrared difference frequency generation comb source to perform broadband high-resolution measurements of nitrous oxide, 14N216O, and retrieve line center frequencies of the ν1 fundamental band and the ν1 + ν2 – ν2 hot band. The spectrum spans 90 cm−1 around 1285 cm−1 with a sample point spacing of 3 × 10−4 cm−1 (9 MHz). We report line positions of 72 lines in the ν1 fundamental band between P(37) and R(38), and 112 lines in the ν1 + ν2 – ν2 hot band (split into two components with e/f rotationless parity) between P(34) and R(33), with uncertainties in the range of 90-600 kHz. We derive upper state constants of both bands from a fit of the effective ro-vibrational Hamiltonian to the line center positions. For the fundamental band, we observe excellent agreement in the retrieved line positions and upper state constants with those reported in a recent study by AlSaif et al. using a comb-referenced quantum cascade laser [J Quant Spectrosc Radiat Transf, 2018;211:172-178]. We determine the origin of the hot band with precision one order of magnitude better than previous work based on FTIR measurements by Toth [http://mark4sun.jpl.nasa.gov/n2o.html], which is the source of the HITRAN2016 data for these bands.

High Order Compact Block-Centered Finite Difference Schemes for Elliptic and Parabolic Problems

Based on the combination of block-centered and compact difference methods, fourth order compact block-centered finite difference schemes for the numerical solutions of one-dimensional and two-dimensional elliptic and parabolic problems with variable coefficients are derived and analyzed. Stability and optimal fourth-order error estimates are proved for both the solution and flux. Numerical experiments for model problems are presented to confirm the theoretical results and superior performance of the proposed schemes.

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ and h are, respectively, the temporal and spatial stepsizes. We also consider the method of adding correction terms to efficiently deal with the nonsmooth problems. Numerical examples are provided to verify the effectiveness of the proposed scheme.

Difference of Composition Operators on Weighted Bergman Spaces with Doubling Weights

In this paper, some characterizations for the compact difference of composition operators on weighted Bergman spaces $$A^p_\omega $$ with doubling weights are given, which extend Moorhouse’s characterization for the difference of composition operators on the weighted Bergman space $$A^2_\alpha $$ .

Numerical solution of nonlinear advection diffusion reaction equation using high-order compact difference method

In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order Padé formula of the first-order derivative, and the time derivative term is discretized by using the fourth-order backward differencing formula. An unconditionally stable five-step fourth-order fully implicit compact difference scheme is developed. This scheme has fourth-order accuracy in both time and space. Secondly, for the calculations of the start-up time steps, the time derivative term is discretized by the Crank-Nicolson method, and Richardson extrapolation formula is used to improve the accuracy in time direction from the second-order to the fourth-order. Thirdly, convergence and stability of the difference scheme in H1 seminorm, L∞ and L2 norms, existence and uniqueness of the numerical solutions are proved, respectively. Fourthly, the Thomas algorithm is used to solve the nonlinear algebraic equations at each time step, and a time advancement algorithm with linearized iteration strategy is established. Finally, the accuracy, stability and efficiency of the present approach are verified by some numerical experiments.

Matrix-free TriGlobal adjoint stability analysis of compressible Navier–Stokes equations

A numerical method for TriGlobal (i.e., fully three-dimensional) adjoint stability analysis for compressible flows was developed and is presented in this paper. The developed method solves the adjoint stability problem using a matrix-free method based on Krylov-Schur method and a time-stepping approach. Because of the low memory (RAM) requirement of the matrix-free approach, the developed method can analyze fully three-dimensional flows that are difficult to analyze with conventional matrix-based methods. To perform time-stepping on the adjoint variables, the adjoint equations, including appropriate boundary conditions for the compressible Navier–Stokes equations, were derived. The equations were discretized using a finite compact difference method, and time integration was conducted using the three-step third-order Runge–Kutta method. A flow field around a two-dimensional square cylinder and a cubic cavity flow were analyzed using the developed method, and it was confirmed that the method reproduces the dominant instabilities reported in the literature. In addition, in the square cylinder flow analysis, the receptivity and sensitivity regions of the secondary wake mode, which corresponds to far wake instability, were clarified. Finally, the TriGlobal direct and adjoint stabilities of compressible flows over a finite width cavity were analyzed for the first time, and it was proven that large-scale adjoint stability analysis can be performed with the developed method. The results also show that instability phenomena similar to those obtained with BiGlobal stability analysis appear, but sidewall effects exist.