A shock-detecting sensor for filtering of high-order compact finite difference schemes

A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is $O(\tau^{2}+h^{4})$ in the $L^{\infty}$ -norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

High-order compact finite difference schemes for solving the regularized long-wave equation

Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations

Fast high-order compact finite difference schemes are investigated for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary conditions. These schemes are convergent of order s in space, where () with second-order temporal accuracy. The discrete conservation laws and convergence of the finite difference schemes are rigorously demonstrated. Thanks to the circulant matrices resulting from spatial discretization, we significantly reduce the computation complexity and storage requirement of the proposed schemes via fast Fourier transform. Numerical examples are presented to show the accuracy and efficiency of these schemes and verify the theoretical analysis. Moreover, we extend the tenth-order scheme to solve the generalized nonlinear Schrödinger equation.

The direct numerical simulation of transitional and turbulent incompressible flows is an area that is increasing with the advance in computational resources. Code parallelization has became a useful tool in these simulations. In the present paper the unsteady two dimensional Navier–Stokes equations are used as physical model. Tollmien–Schlichting waves propagating in a Poiseuille flow is adopted as test case. Three frequencies showing different behavior according to the linear stability theory (LST) are used: unstable, nearly neutral and stable waves. The parallelization is done via domain decomposition in the streamwise direction. The time derivative is integrated by a classical fourth-order Runge–Kutta method. High-order compact finite difference schemes are used for the spatial derivatives discretization. The Poisson equation is solved by multigrid methods. The present paper explores different parallelization techniques for solving the tridiagonal system, and the multigrid method. The results are compared with LST, and also between the different parallel strategies to show the advantages and disadvantages of each one.

This paper presents results of Large Eddy Simulation (LES) of a flow in a channel with an adverse pressure gradient. The applied computational code (SAILOR) is based on a high-order compact finite difference scheme on half-staggered meshes and is combined with an Immersed Boundary (IB) method. The IB method is implemented using the direct forcing approach in a simplified stepwise variant. The results are verified using the experimental data and the general agreement is good. The observed discrepancies are small close to the inlet section and increase along the channel length where strong flow separation occurs.

Efficient modeling of the complex dynamics between flow physics and chemical kinetics during flame propagation is crucial for understanding combustion mechanisms in real-world applications such as propulsion, energy conversion, and emission control. A two-dimensional, high-order, compact finite difference computational fluid dynamics (CFD) code in cylindrical coordinates has been developed to investigate the interaction between flow and chemistry in a laminar non-premixed flame. The numerical framework (implemented in Fortran) employs a fifth-order scheme for convective terms and a fourth-order scheme for viscous terms to enhance the resolution of steep gradients characteristic of reacting flows. Conversely, a fourth-order Runge–Kutta (RK4) time integration scheme improves solution stability and reduces the convergence error by approximately 20% compared to the simpler Euler-forward (EF) method. To accurately capture the intricate coupling between flow and chemistry, the steady flamelet model (SFM) with detailed kinetics is incorporated into the CFD framework, enabling a reduced yet robust representation of chemical kinetics. The model is validated against benchmark experimental and numerical datasets, demonstrating strong agreement and predictive capability, with 60% and 15% lower Root Mean Squared Error (RMSE) for temperature and H2O mole fraction, compared to the species-transport-based numerical results. A systematic parametric study was conducted using a canonical co-flow laminar methane–air diffusion flame configuration. The effects of varying Reynolds number, fuel-to-air velocity ratio, and combustor geometry are analyzed in detail. The study reveals that increasing the combustor length leads to more anchored and spatially confined flame structures. Additionally, the study demonstrates that increasing the inner diameter enhances radial diffusion, resulting in more dispersed flame fronts. These findings underscore the role of combustor geometry in optimizing designs for both efficiency and emissions control. Furthermore, the proposed higher-order scheme advances reactive flow modeling by offering a validated and computationally efficient tool for high-resolution simulations of chemically reacting flows.

Most of previous studies focused on the boundary-layer receptivity to the convected disturbances in the free stream interacting with localized wall roughness. Whereas the research on the boundary-layer receptivity induced by localized blowing or localized suction is relatively few. In this paper, we investigate two-dimensional boundary-layer receptivity induced by localized blowing/suction within free-stream turbulence through using direct numerical simulation and fast Fourier transformation. High-order compact finite difference schemes in the y-direction, fast Fourier transformation in the x-direction, and a Runge-Kutta scheme in time domain are used to solve the Navier-Stokes equations. The numerical results show that Tollmien-Schlichting (T-S) wave packets are excited by the free-stream turbulence interacting with localized blowing in the two-dimensional boundary layer, which are superposed by a group of stable, neutral and unstable T-S waves. The dispersion relations, growth rates, amplitude distributions and phase distributions of the excited waves accord well with theoretical solutions of the linear stability theory, thus confirming the existence of the boundary-layer receptivity. And the frequencies of the instability waves are between the upper and lower branches of the neutral stability curves. According to the evolutions of the wave packets, the positions of peaks and valleys are tracked over time to calculate the propagation speed by taking the average. The propagation speeds of the wave packets are approximately one-third of the free-stream velocity, which are in accordance with Dietz's measurements. The propagation speeds of wave packets are also close to the phase speeds of the most unstable waves for the numerical results. The relations of the receptivity response to the forcing amplitude, the blowing intensity, and the blowing width are found to be linear, when the forcing amplitude and the blowing intensity are less than 1% free-steam velocity amplitude and 0.01, respectively. And the maximum amplitudes of the T-S waves can be excited while the blowing length is equal to the resonant wavelength /(TS-FS), where TS is the wave-number of the T-S wave, and FS is the wave-number of the forcing disturbance. These results are similar to those given by Dietz [Dietz A J 1999 J. Fluid Mech. 378 291]. Additionally, T-S waves with the same dispersion relations but opposite phases are generated by localized blowing and localized suction respectively, and the amplitudes of the T-S waves excited by localized blowing are nearly 15% greater than those by localized suction under the same condition. According to this theory, an optimal design of localized suction device is able to enhance or delay the laminar-turbulent transition for turbulent control.

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

Two solution methods for compressible flows and their application in Large-Eddy Simulation (LES) of turbulent flows are discussed. After a summary of the basic equations and subgrid scale models, the applied discretization methods, solution schemes and boundary conditions with their implementation on high-performance computers are presented. The two discretization methods for the advective terms used in the simulations are a second-order mixed central-upwind discretization based on a AUSM formulation and a higher-order compact finite difference scheme. First, results of the second-order scheme are shown for the case of a turbulent channel and pipe flow, which are only weakly influenced by subgrid scale models, but are in good agreement with DNS and experimental data. Higher-order compact schemes are then used for both laminar and turbulent flows and the results are compared to those from simulations with the second-order scheme. It is found that for the laminar flow and the LES of plane turbulent jets the higher-order compact scheme is not of principal advantage. This is primarily related to the sensitivity of the compact scheme to stretched and curvilinear grids, which requires explicit filtering steps to avoid unphysical oscillations in the solution. Further results are then presented, which are obtained with the computationally cheaper second-order scheme. The results include turbulent flows in pipe bends with different Reynolds and Dean numbers. The solutions show swirl switching with a distinct low frequency peak. Finally, turbulent flows in non-trivial geometries are considered. This includes a first step towards the LES of the flow in an internal combustion engine with four valves and a realistic geometry.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

High-order implicit finite difference schemes for the two-dimensional Poisson equation

High order compact finite difference schemes for a system of third order boundary value problem

Introduction A high-order conservative hybrid finite difference scheme was developed for solving compressible turbulent boundary layer DNS problems. The scheme is hybrid in the sense that it contains both high-order central differencing to resolve the large range of length scales that occur in compressible turbulent flow and upwind biased flux vector splitting to provide necessary high wavenumber dissipation. The hybrid scheme, ncfvs (non-compact flux vector splitting), compares favorably with high-order compact schemes in numerical tests but does not contain the costly tridiagonal inversion that is found in compact schemes. When coupled to third-order Runge-Kutta time integration, this hybrid scheme is restricted by stability to a CFL = 0.6. This results in a very small time step due to the fine grid spacing necessary to resolve the length scales occurring in high Reynolds number flows. Although the hybrid scheme is fully parallelizable and optimized to run efficiently on the 512 node partition of the CM-5, a massively parallel supercomputer, solutions may take days due to this small time step restriction. If the time step could be increased by using an implicit method, statistically steady solutions could be found in a much shorter time. Rai, Gatski, and Erlebacher used an approximate factorization method to solve the implicit problem. Although effective, this method is difficult to efficiently implement on a parallel machine because of the complicated communication paths required in the inversion of tridiagonal systems. In this paper, we propose a high-order accurate formulation of the Data Parallel Lower-Upper Relaxation method (DP-LUR). The DP-LUR method has been used to efficiently solve many steady-state problems and is completely parallelizable. We modify this method to include secondorder time accuracy and then we compare the results of a compressible turbulent boundary layer problem to the third-order Runge-Kutta results of the same problem.