High-order Accurate Finite Difference Research Articles

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.

Local-basis difference potentials method for elliptic PDEs in complex geometry

We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the difference potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allows unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard, centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically.

Numerical study on the scattering of acoustic waves by a compact vortex

A new family of compact vortex models is developed and taken as base vortical flows to numerically study the acoustic scattering by solving the two-dimensional Euler equations in the time domain with high-order accurate finite-difference methods and nonreflecting boundary conditions. The computations of scattered fields with very small amplitude are found to be in excellent agreement with a benchmark provided by previous studies. Simulations for the scattering from a Taylor vortex reveal that the amplitude of the scattered fields is strongly influenced by two dimensionless quantities, the vortex strength Mv based on the maximal velocity of the vortex, and the acoustic length-scale ratio λ/R defined as the acoustic wavelength relative to the vortex core size. To have a deep understanding of the roles played by these two quantities, another significant quantity used for describing quantitatively the total amount of scattering, namely, scattered sound power, is introduced. Thereupon, on the basis of a global analysis of scale effects of these two dimensionless quantities on the scattered sound power, the scattering defined in a physical coordinate system with Mv and λ/R is divided into three domains, long-wave domain, resonance domain, and geometrical-acoustics domain. For each domain, we examine the influence of Mv and λ/R in detail and derive the explicit scaling laws involved in the strength of the scattered field and these two dimensionless quantities separately. Furthermore, the computations for the scattering from a high-order compact vortex are conducted at a wide range of Mv and λ/R and compared with the results from the Taylor vortex in each domain to gain some insights into the acoustic scattering by a compact vortex.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

Modal analysis of planar waveguides by the immersed interface method

Modal analysis of optical waveguides is a basic task in the design of advanced waveguide devices and optical circuits. How to deal with the problem of electromagnetic heterogeneous interface and absorption boundary condition are two major difficulties in implementing efficient numerical analysis of optical waveguides. Existing high-order accurate finite-difference modal analysis methods do not take into consideration the absorption boundary problem, which, thus, makes it difficult to accurately simulate leakage and radiation modes. Based on the immersed interface method and perfectly matched layer absorption boundary condition, a finite-difference method with the second- and fourth-order accuracy is proposed in this work. By using this method, the single-interface plasmonic waveguide mode, planar symmetric waveguide mode, and one-dimensional photonic crystal waveguide mode are analyzed. Numerico-experimental results show that the convergence rate of the second- and fourth-order algorithm are consistent with the anticipated order of the guided mode, leakage mode and radiation mode. The second-order algorithm provides an ultimate accuracy of about <inline-formula><tex-math id="M5">\begin{document}$10^{-9}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M5.png"/></alternatives></inline-formula> for the relative error of effective refractive index, when the normalized step size is <inline-formula><tex-math id="M6">\begin{document}$10^{-4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M6.png"/></alternatives></inline-formula>. The fourth-order algorithm provides an ultimate accuracy of about <inline-formula><tex-math id="M7">\begin{document}$10^{-10}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M7.png"/></alternatives></inline-formula> for the relative error of effective refractive index, when the normalized step size is <inline-formula><tex-math id="M8">\begin{document}$10^{-3}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230595_M8.png"/></alternatives></inline-formula>. Through the study of field distribution of guided mode and cladding mode in a one-dimensional photonic crystal waveguide, we show that the continuity of the field of transverse electric mode and its first derivative across the interface, and the continuity of the field of transverse magnetic mode and the discontinuity of its first derivative across interface, can be analyzed accurately. The method proposed in this paper can be used to calculate any mode for any refractive index profile, only by using the value of refractive index, independent of the specific functional representation of modal fields. The method provides a simple and efficient tool for implementing the modal analysis of step-index planar waveguides.

Effect of anisotropic resistance characteristic on boundary-layer transitional flow

It is observed that the feather surface exhibits anisotropic resistances for the streamwise and spanwise flows. To obtain a qualitative understanding about the effect of this anisotropic resistance feature of surface on the boundary-layer transitional flow over a flat plate, a simple phenomenological model for the anisotropic resistance is established in this paper. By means of the large eddy simulation (LES) with high-order accurate finite difference method, the numerical investigations are conducted. The numerical results show that with the spanwise resistance hindering the formation of vortexes, the transition from laminar flow to turbulent flow can be delayed, and turbulence is weakened when the flow becomes fully turbulent, which leads to significant drag reduction for the plate. On the contrary, the streamwise resistance renders the flow less stable, which leads to the earlier transition and enhances turbulence in the turbulent region, causing a drag increase for the plate. Thus, it is indicated that a surface with large resistance for spanwise flow and small resistance for streamwise flow can achieve significant drag reduction. The present results highlight the anisotropic resistance characteristic near the feather surface for drag reduction, and shed a light on the study of bird’s efficient flight.

Local Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs

We describe a new approach to derive numerical approximations of boundary conditions for high-order accurate finite-difference approximations. The approach, called the Local Compatibility Boundary Condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified for a wide range of sample problems, both time-dependent and time-independent, in two space dimensions and for schemes up to sixth-order accuracy.

High-order accurate finite difference discretisations on fully unstructured dual quadrilateral meshes

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives of the original function. The method generalises traditional finite difference methods to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy. We demonstrate the accuracy of the numerical scheme using dual quadrilateral meshes and a refinement method based on subdivision surfaces. The scheme is applied to the solution of a range of partial differential equations, including both linear and nonlinear, second and fourth order equations, and a time-dependent first order equation.

A novel higher order compact-immersed interface approach for elliptic problems

We present a new higher-order accurate finite difference explicit jump Immersed Interface Method for solving two-dimensional elliptic problems with singular source and discontinuous coefficients in the irregular region on a compact Cartesian mesh. We propose a new strategy for discretizing the solution at irregular points on a nine-point compact stencil such that the higher-order compactness is maintained throughout the whole computational domain. The scheme is employed to solve four problems embedded with circular- and star-shaped interfaces in a rectangular region having analytical solutions and varied discontinuities across the interface in source and the coefficient terms. We also simulate a plethora of fluid flow problems past bluff bodies in complex flow situations which are governed by the Navier–Stokes equations; they include problems involving multiple bodies immersed in the flow as well. In the process, we show the superiority of the proposed strategy over the explicit jump finite difference immersed interface method and other existing immersed interface methods by establishing the rate of convergence and grid independence of the computed solutions. In all the cases, our computed results are extremely close to the available numerical and experimental results.

A High Order Accurate Finite Difference Method for the Drinfel\u2019d\u2013Sokolov\u2013Wilson Equation

We analyse numerically the periodic problem and the initial boundary value problem of the Korteweg-de Vries equation and the Drindfeld–Sokolov–Wilson equation using the summation-by-parts simultaneous-approximation-term method. Two sets of boundary conditions are derived for each equation of which stability is shown using the energy method. Numerical analysis is done when the solution interacts with the boundaries. Results show the benefit of higher order SBP operators.

Effects of Variable Thermal Conductivity and Grashof Number on Non-Darcian Natural Convection Flow of Viscoelastic Fluids with Non Linear Radiation and Dissipations

Effects of variable thermal conductivity, porosity and Grashof number on natural convection of viscoelastic fluid flow and heat transfer are studied and discussed. The nonlinear radiation and dissipations are taken into consideration. The fluid flows through non-Darcy porous medium which lies between two heated vertical plates that are kept at constant, but different, temperatures. High order accurate finite difference schemes are introduced to solve the governing equations. The coupled nonlinear differential equations are linearized and iterations are used to approximate that linearized terms. The finite difference method transforms the coupled linearized differential (momentum and energy) equations to a linear system of algebraic equations. An error analysis is introduced by refinement of mesh size and comparisons with available previous results. Effects of fluid and heat parameters on the velocity field, temperature, skin friction factor and Nusselt number are tabulated and plotted. The present results and comparisons show that the numerical solution is of excellent agreement with previous analytical and numerical solution.

A high-order accurate finite difference scheme for the KdV equation with time-periodic boundary forcing

In this article, we present an accurate finite difference scheme for solving the KdV equation with time-periodic forcing at boundary. We first employ a variable transformation to change the time-periodic boundary to a homogeneous boundary. We then develop a three-level fourth-order accurate finite difference scheme for solving the transformed KdV problem. The stability, existence and uniqueness of the numerical solution are analyzed. Numerical examples are provided to confirm the accuracy of the scheme, and the effectiveness for time-periodic simulation.

A high-order finite-difference scheme to model the fluid-structure interaction in pneumatic seismic sources

A high-order accurate finite-difference scheme modeling the fluid-structure interaction inside a pneumatic seismic source is presented. The model consists of two deforming fluid compartments separated by a moving shuttle. The fluid is governed by the 1D Euler equations. Well-posedness of the continuous problem is analyzed and proven in the frozen coefficient case. A stable discretization is derived using summation-by-parts operators with the boundary conditions imposed weakly using the simultaneous-approximation-term method. The theoretical convergence rate of the numerical scheme is verified by numerical experiments. Simulation results are compared to pressure measurements from inside a pneumatic seismic source and capture many of the features observed in the data.

Construction of invariant compact finite-difference schemes.

In this paper we propose a method, which is based on equivariant moving frames, for development of high-order accurate invariant compact finite-difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of partial differential equations (PDEs) are used to transform independent and dependent variables and derivative terms of compact finite-difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames. In some cases, owing to selection of convenient moving frames, numerical representation of invariant (or symmetry-preserving) compact numerical schemes is found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite-difference algorithms. Application of the proposed method is demonstrated through construction of invariant compact finite-difference schemes for some common linear and nonlinear PDEs (including the linear advection-diffusion equation in one or two dimensions, the inviscid Burgers' equation in one dimension, viscous Burgers' equation in one or two dimensions, spherical Burgers' equation in one dimension, and shallow water equations in two dimensions). Results obtained from our numerical simulations indicate that invariant compact finite-difference schemes not only inherit selected symmetry properties of underlying PDEs, but are also comparably more accurate than the standard, noninvariant base numerical schemes considered here.

A New High Order Accurate, Finite Difference Method on Quasi-variable Meshes for the Numerical Solution of Three Dimensional Poisson Equation

In this paper, a compact high order accurate finite difference scheme on non-uniform meshes network to solve the 3D Poisson equation is presented. We considered here the development of classical Numerov’s method in combination with nonuniform quasi-variable meshes network. The theoretical validation of the present scheme is considered by means of irreducibility and monotonicity criteria of the coefficient matrix obtained from difference equations. Numerical tests are presented to validate the theoretical prediction of the proposed scheme.

Accurate gradient preserved method for solving heat conduction equations in double layers

Analyzing heat transfer in layered structures is important for the design and operation of devices and the optimization of thermal processing of materials. Existing numerical methods dealing with layered structures if using only three grid points across the interface usually provide only a second-order truncation error. Obtaining a numerical scheme using three grid points across the interface so that the overall numerical scheme is unconditionally stable and convergent with higher-order accuracy is mathematically challenging. In this study, we develop a higher-order accurate finite difference method using three grid points across the interface by preserving the first-order derivative, ux, in the interfacial condition and/or the boundary condition. As such, when the compact fourth-order accurate Padé scheme is used at the interior points, the overall scheme maintains to be higher-order accurate. Stability and convergence of the scheme are analyzed. Finally, four examples are given to test the obtained numerical method. Results show that the convergence order in space is close to 4.0, which coincides with the theoretical analysis.

An improved projection method

Strictly stable high-order accurate finite difference approximations are derived, for linear initial boundary value problems. The boundary closures are based on the diagonal-norm summation-by-parts framework and the boundary conditions are imposed using an improved projection method. The improved projection method removes some of the numerical issues with the previously derived (here referred to as traditional) projection method. The finite difference approximations lead to fully explicit ODE systems. The accuracy and stability properties are demonstrated for linear hyperbolic and parabolic problems in 1D and 2D. In particular we study the problem where initial and boundary data are inconsistent, that causes the traditional projection method to fail. Comparisons are made also with weak (penalty) enforcement of boundary conditions.

Towards simulating natural transition in hypersonic boundary layers via random inflow disturbances

A random forcing approach was implemented into a high-order accurate finite-difference code in order to investigate ‘natural’ laminar–turbulent transition in hypersonic boundary layers. In hypersonic transition wind-tunnel experiments, transition is caused ‘naturally’, by free-stream disturbances even when so-called quiet tunnels are employed such as the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University. The nature and composition of the free-stream disturbance environment in high-speed transition experiments is difficult to assess and therefore largely unknown. Consequently, in the direct numerical simulations (DNS) presented here, the free-stream disturbance environment is simply modelled by random pressure (acoustic) disturbances with a broad spectrum of frequencies and a wide range of azimuthal wavenumbers. Results of a high-resolution DNS for a flared cone at Mach 6, using the random forcing approach, are presented and compared to a fundamental breakdown simulation using a ‘controlled’ disturbance input (with a specified frequency and azimuthal wavenumber). The DNS results with random forcing clearly exhibit the ‘primary’ and ‘secondary’ streak pattern, which has previously been observed in our ‘controlled’ breakdown simulations and the experiments in the BAM6QT. In particular, the spanwise spacing of the ‘primary’ streaks for the random forcing case is identical to the spacing obtained from the ‘controlled’ fundamental breakdown simulation. A comparison of the wall pressure disturbance signals between the random forcing DNS and experimental data shows remarkable agreement. The random forcing approach seems to be a promising strategy to investigate nonlinear breakdown in hypersonic boundary layers without introducing any bias towards a distinct nonlinear breakdown mechanism and/or the selection of specific frequencies or wavenumbers that is required in the ‘controlled’ breakdown simulations.

Higher-order accurate two-step finite difference schemes for the many-dimensional wave equation

The accurate simulation of wave propagation is a problem of longstanding interest. In this article, the focus is on higher-order accurate finite difference schemes for the wave equation in any number of spatial dimensions. In particular, two-step schemes (which operate over three time levels) are studied. A novel approach to the construction of schemes exhibiting both isotropy and accuracy is presented using modified equation techniques, and allowing for the specification of precise stencils of operation for the scheme, and thus direct control over stencil size and thus operation counts per time step. Both implicit and explicit schemes are presented, as well as parameterised families of such schemes under conditions specifying the order of isotropy and accuracy. Such conditions are framed in terms of a set of coupled constraints which are nonlinear in general, but linear for a fixed Courant number. Depending on the particular choice of stencils, it is often possible to develop schemes for which the traditional Courant–Friedrichs–Lewy condition is exceeded. A wide variety of families of such schemes is presented in one, two and three spatial dimensions, and accompanied by illustrations of numerical dispersion as well as convergence results confirming higher-order accuracy up to eighth order.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Scalar Convection Diffusion Equations

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak...