Fast high-accuracy compact conservative difference schemes for solving the nonlinear Schrödinger equation

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

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.

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

Higher-order compact finite difference scheme with multigrid algorithm is applied in this paper for solving one-dimensional and two-dimensional inhomogeneous Helmholtz equations. In two-dimensional case, the suggested scheme has the stencil of twenty one points. An efficient solver multigrid method yields eighth-order accurate approximation on both fine and coarse grids. For the Neumann boundary condition, an eighth-order accurate representation is also developed. The accuracy and efficiency of eighth-order compact difference scheme are exhibited through graphical illustrations and computed results are drafted in tabular form to validate the numerical experiments.

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

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.

A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The $$L1$$ discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is $$\mathcal{O }(\tau ^{3-\alpha }+h^4)$$ in the maximum norm, where $$\tau $$ is the temporal grid size and $$h$$ is the spatial grid size, respectively. In addition, a Crank---Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of $$\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)$$ is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank---Nicolson scheme are presented to show the effectiveness of the compact scheme.

A combined compact finite difference scheme for predicting the evolution of a mean curvature driven interface

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

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

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear Schrodinger equation. The proposed scheme not only conserves the total mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h 4 +� 2 ) in the discrete L 1 -norm with time stepand mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme. Mathematics subject classification: 65M06, 65M12.