High-order Finite Difference Method Research Articles

A sixth-order finite difference method for the two-dimensional nonlinear advection diffusion reaction equation

A novel higher-order finite difference method is developed to solve the two-dimensional nonlinear advection diffusion reaction equation. Firstly, a new five-point sixth-order finite difference scheme is proposed to discretize spatial derivatives and the sixth-order backward difference method is applied to teat temporal derivative. As a result, a new fully implicit scheme with sixth-order accuracy in both time and space is proposed. Subsequently, for the calculation of the starting step of the sixth-order backward difference method, the Crank-Nicolson method combined with the Richardson extrapolation technique is employed to obtain the equivalent temporal accuracy as the main scheme. Finally, the accuracy and reliability of the presented method are validated through various numerical examples.

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art eN prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

A space marching method for sonic boom near field predictions

The conventional sonic boom propagation prediction method widely adopted in supersonic aircraft design involves a two-step procedure. In the first step, a compressible viscous or inviscid Computational Fluid Dynamics analysis is applied to the aircraft geometry at flight conditions to produce a flow field solution near the aircraft. Then in the second step, a one dimensional nonlinear Burgers’ equation model is used to propagate sonic boom signature traces from the near field solution at flight altitude to the ground along a ray path. For an accurate ground signature prediction the near field signature must be accurately modeled at a sufficient distance from the aircraft flight path in order to minimize errors in the one dimensional propagation model. This is a very challenging task for general purpose CFD tools in a design environment because the cost of maintaining highly accurate off-body solutions increases dramatically as the radial distance is enlarged in the computational domain. It is also particularly difficult to apply these tools for wave propagation because the algorithms are normally lower order and numerically dissipative and dispersive. In this work a space marching procedure based on an optimized higher order finite difference method is developed and applied in conjunction with a CFD solution concentrated in the close vicinity of the aircraft. This new approach is much more efficient, compared to previous methods, in providing highly accurate near field signatures for full carpet ground predictions. Results indicate that near field signatures retain more waveform shape information farther from the aircraft geometry while reducing the CFD cost significantly. The predicted ground signature is also shown to converge in shape as the radial distance of the near field signature grows, which is indicative of a more ideal initial condition being supplied to the one dimensional wave propagation to ground. This feature is very difficult to replicate in a tractable manner with common CFD approaches used in design.

A Time-Domain Wavenumber Integration Model for Underwater Acoustics Based on the High-Order Finite Difference Method

Simulating the acoustic field excited by pulse sound sources holds significant practical value in computational ocean acoustics. Two primary methods exist for modeling underwater acoustic propagation in the time domain: the Fourier synthesis technique based on frequency decomposition and the time-domain underwater acoustic propagation model (TD-UAPM). TD-UAPMs solve the wave equation in the time domain without requiring frequency decomposition, providing a more intuitive explanation of the physical process of sound energy propagation over time. However, time-stepping numerical methods can accumulate numerical errors, making it crucial to improve the algorithm’s accuracy for TD-UAPMs. Herein, the time-domain wavenumber integration model SPARC was improved by replacing the second-order finite element method (FEM) with the high-order accuracy finite difference method (FDM). Furthermore, the matched interface and boundary (MIB) method was used to process the seabed more accurately. The improved model was validated using three classic underwater acoustic benchmarks. By comparing the acoustic solutions obtained using the FDM and the FEM, it is evident that the improved model requires fewer grid points while maintaining the same level of accuracy, leading to lower computational costs and faster processing compared to the original model.

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

A high-order finite difference method for moving immersed domain boundaries and material interfaces

We present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

Higher-order compact finite difference method for a free boundary problem of ductal carcinoma in situ

A higher-order compact finite difference scheme is established to investigate the model of ductal carcinoma in situ, a free boundary value problem. For the first time, to determine the growth of a tumor numerically, a compact finite difference method is applied in the spatial direction, and the Crank-Nicolson scheme is used in the temporal direction. Through stability analysis, it has been demonstrated that the proposed numerical scheme is unconditionally stable. The convergence of the numerical scheme is analyzed theoretically by von Neumann's method. It has been proved that the proposed numerical method is second-order convergent in time and fourth-order convergent in space in the L2-norm. Two numerical examples are provided to verify the scheme's efficiency and validate the theoretical results. The tabular and graphical representations confirm the high accuracy and adaptability of the scheme.

High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

An immersed boundary method for the thermo–fluid–structure interaction in rarefied gas flows

An immersed boundary method for the thermo–fluid–structure interaction in rarefied gas flows is presented. In this method, the slip model is incorporated with the penalty feedback immersed boundary method to address the velocity and temperature jump conditions at the fluid–structure interface in rarefied gas flows within the slip-flow regime. In addition, the compressible flows governed by the Navier–Stokes equations are solved by using a high-order finite difference method; the elastic solid is solved by using the finite element method; the fluid and solid dynamics are solved independently, and the thermo–fluid–structure interaction is achieved by using a penalty feedback method in a partitioned way. To model the local rarefaction in the supersonic flow, an artificial viscosity is proposed by introducing the local Knudsen number to diffuse the sharp transition at the shock wave front. Several validations are conducted: the Poiseuille flow in a channel, the flow around a two-dimensional airfoil, a moving square cylinder in a channel, the flow around a sphere, and a moving sphere in quiescent flow. The numerical results from the present method show very good agreements with the previous published data obtained by other methods, confirming the good ability of the proposed method in handling the thermo–fluid–structure interaction in both weakly and highly compressible rarefied gas flows. Inspired by the micro/unmanned aerial vehicles in Martian exploration, the proposed method is applied to the aerodynamics of a flapping wing in rarefied gas flows in both two-dimensional and three-dimensional spaces to demonstrate the versatility of the proposed method for modeling flows involving large deformation and fluid–structure interaction.

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.

High-order entropy stable schemes based on a local adaptive WENO viscosity for degenerate convection-diffusion equations

In this work, high-order entropy stable finite difference methods are constructed to solve systems of degenerate convection-diffusion partial differential equations. The approximation of the convective term is based on high-order entropy stable fluxes under a TeCNO scheme formulation and for the diffusive term high-order entropy conservative fluxes are computed by considering linear combinations of second-order approximations. Numerical simulations show that the proposed high-order entropy conservative schemes provide a convergence rate better than conventional WENO methods. To obtain entropy stability, a local adaptive artificial viscosity is included using weighted essentially nonoscillatory reconstructions satisfying a local sign property at discontinuities. Different WENO interpolations for the local artificial viscosity are numerically analyzed being the fast and optimal WENO interpolation (FOWENO), the one that introduces less spurious oscillations with a lower computational cost.

Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

Structural Behavior of Water Flow Glazing: Stress and Elastic Deformation Considering Hydrostatic Pressure

The fluid inside a Water Flow Glazing (WFG) panel creates a linear pressure distribution along the vertical dimension. Tension stress can cause problems with the sealant; compression stress can cause deflections in the glass panel. Increasing the glass thickness until the deflection is below the required limit implies more weight and cost of the glazing. Another solution is to limit glass deflection by inserting pillars or stripes into the cavity between two glass panes. The novelty of this article was to test a High-Order Finite Difference Method for linear and non-linear models to evaluate the effect of hydrostatic pressure produced by the fluid chamber on WFG panels. The methodology was tested on two case studies to assess the tension and deflection of the glass panes to guarantee the structural stability of WFG. The main conclusion drawn was that a linear plate model was sufficient to dimension the width, length, and thickness of the WFG panel. Furthermore, the mathematical model provided criteria to keep the glass tension below 45 MPa for tempered glass and the maximum deflection as the minimum between 1‰ of the glass height and 10% of the water chamber thickness. Introducing pillars or stripes solved hydrostatic pressure problems when the panel’s height was above 1.5 m.

A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier–Stokes equations.

Thermogravitational Convection in a Multiple Baffled Enclosure Filled with Magneto-Hybrid Nanofluid Subjected to Magnetic Field Dependent Viscosity

This study explores the significant impacts of thin baffles and magnetic field dependent viscosity on magnetohydrodynamic (MHD) thermogravitational convection of Cu-Al2O3 (50%–50%) water hybrid nanoliquid in a cavity. Considering different arrangements of baffle sticks on both the vertical walls, four geometrical configurations (Case-I, Case-II, Case-III and Case-IV) have been analyzed. Numerical simulation has been performed for the governing Navier-Stokes (N-S) equations in streamfunction - vorticity form having energy equation. These coupled equations are solved by proposing a higher-order compact finite difference method. The combination of five important aspects (hybrid nanofluid, multiple baffles, magnetic field dependent viscosity (MFDV), magnetic field and compact computation) signifies the novelty of this work. Fluid flow and transportation of thermal energy within the stipulated domain are presented for various flow pertinent parameters. The outcomes show that the increase in number of baffles diminishes the average Nusselt number values. It is concluded here that an increase in Hartmann number from 0 to 90 leads to a decrease in average Nusselt number up to 23.7% for Case-I, 23.8% for Case-II, 21.2% for Case-III and 28% for Case-IV in presence of MFDV effects.

A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

A conservative splitting high-order finite difference method for coupled Gross–Pitaevskii equations in 2D

A conservative splitting high-order finite difference method for coupled Gross–Pitaevskii equations in 2D

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

A uniformly convergent numerical scheme is proposed to solve a singularly perturbed convection-diffusion problem with a large time delay. The diffusion term of the problem is multiplied by a perturbation parameter, ε. For a small ε, the problem exhibits a boundary layer, which makes it challenging to solve it analytically or using standard numerical methods. As a result, the backward Euler scheme is applied in the temporal direction. Non-symmetric finite difference schemes are applied for approximating the first-order derivative terms, and a higher-order finite difference method is applied for approximating the second-order derivative term. Furthermore, an exponential fitting factor is computed and induced in the difference scheme to handle the effect of the small parameter. Using the discrete maximum principle, the stability of the scheme is examined and analyzed. The developed scheme is parameter-uniform with a linear order of convergence in both space and time. To examine the accuracy of the method, two model examples are considered. Further, the boundary layer behavior of the solutions is given graphically.

A robust higher-order finite difference technique for a time-fractional singularly perturbed problem

A higher-order finite difference method is developed to solve the variable coefficients convection–diffusion singularly perturbed problems (SPPs) involving fractional-order time derivative with the order α∈(0,1). The solution to this problem class possesses a typical weak singularity at the initial time t=0 and an exponential boundary layer at the right lateral surface as the perturbation parameter ɛ→0. Alikhanov’s L2−1σ approximation is applied in the temporal direction on a suitable graded mesh, and the spatial variable is discretized on a piecewise uniform Shishkin mesh using a combination of midpoint upwind and central finite difference operators. Stability estimates and the convergence analysis of the fully discrete scheme are provided. It is shown that the fully discrete scheme is uniformly convergent with a rate of O(M−p+N−2(logN)2), where p=min{2,rα}, r is the graded mesh parameter and M,N are the number of mesh points in the time and space direction, respectively. Two numerical examples are taken in counter to confirm the sharpness of the theoretical estimates.