A low numerical dissipation immersed interface method for the compressible Navier–Stokes equations

A numerical method to solve the compressible Navier–Stokes equations around objects of arbitrary shape using Cartesian grids is described. The approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite-difference discretization. The method is suitable for compressible flows without shocks and can be classified as an immersed interface method. The objects are sharply captured by the Cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes, located around the boundary. In contrast with available methods, no jump conditions are used or explicitly derived from the boundary conditions, although a number of elements are adopted from previous immersed interface approaches. A new element in the present approach is the use of the summation-by-parts formalism to develop stable non-stiff first-order derivative approximations at the irregular grid points. Second-order derivative approximations, as those appearing in the transport terms, can be stiff when irregular grid points are located too close to the boundary. This is addressed using a semi-implicit time integration method. Moreover, it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties. Convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different Mach and Reynolds numbers.     

Curvilinear finite-volume schemes using high-order compact interpolation

During the last years, the need of high fidelity simulations on complex geometries for aeroacoustics predictions has grown. Most of high fidelity numerical schemes, in terms of low dissipative and low dispersive effects, lie on finite-difference (FD) approach. But for industrial applications, FD schemes are less robust compared to finite-volume (FV) ones. Thus the present study focuses on the development of a new compact FV scheme for two- and three-dimensional applications.The proposed schemes are formulated in the physical space and not in the computational space as it is the case in most of the known works. Therefore, they are more appropriate for general grids. They are based on compact interpolation to approximate interface-averaged field values using known cell-averaged values. For each interface, the interpolation coefficients are determined by matching Taylor series expansions around the interface center. Two types of schemes can be distinguished. The first one uses only the curvilinear abscissa along a mesh line to derive a sixth-order compact interpolation formulae while the second, more general, uses coordinates in a spatial three-dimensional frame well chosen. This latter is formally sixth-order accurate in a preferred direction almost orthogonal to the interface and at most fourth-order accurate in transversal directions.For non-linear problems, different approaches can be used to keep the high-order scheme. However, in the present paper, a MUSCL-like formulation was sufficient to address the presented test cases.All schemes have been modified to treat multiblock and periodic interfaces in such a way that high-order accuracy, stability, good spectral resolution, conservativeness and low computational costs are guaranteed. This is a first step to insure good scalability of the schemes although parallel performances issues are not addressed. As high frequency waves, badly resolved, could be amplified and then destabilize the scheme, compact filtering operators have been used.Numerous test cases as the linear convection of a Gaussian wave, the convection of a Lamb–Oseen vortex and the diffraction of an acoustic wave on a plane have been realized to validate the schemes. The most efficient schemes are shown to be at least fifth-order accurate on linear and non-linear convection problems. They are also less dissipative and less dispersive on non-uniform curvilinear grids than schemes using implicit interpolation with constant coefficients of the same order on uniform cartesian grids.     

Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities

This paper reports the three-dimensional (3D) generalization of our previous 2D higher-order matched interface and boundary (MIB) method for solving elliptic equations with discontinuous coefficients and non-smooth interfaces. New MIB algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier MIB scheme for treating interfaces with sharp geometric singularities, such as sharp edges, sharp wedges and sharp tips. The resulting 3D MIB schemes are of second-order accuracy for arbitrarily complex interfaces with sharp geometric singularities, of fourth-order accuracy for complex interfaces with moderate geometric singularities, and of sixth-order accuracy for curved smooth interfaces. A systematical procedure is introduced to make the MIB matrix optimally symmetric and banded by appropriately choosing auxiliary grid points. Consequently, the new MIB linear algebraic equations can be solved with fewer number of iterations. The proposed MIB method makes use of Cartesian grids, standard finite difference schemes, lowest order interface jump conditions and fictitious values. The interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation. While a pair of fictitious values are determined along a mesh at a time, an iterative procedure is proposed to determine all the required fictitious values for higher-order schemes by repeatedly using the lowest order jump conditions. A variety of MIB techniques are developed to overcome geometric constraints. The essential strategy of the MIB method is to locally reduce a 2D or a 3D interface problem into 1D-like ones. The proposed MIB method is extensively validated in terms of the order of accuracy, the speed of convergence, the number of iterations and CPU time. Numerical experiments are carried out to complex interfaces, including the molecular surfaces of a protein, a missile interface, and van der Waals surfaces of intersecting spheres.     

一种模拟倾斜折射率界面光波导的新方法

针对传统离散格式下，束传播方法(BPM)模拟倾斜折射率界面出现的问题，提出了一种简明且易于编程实现的改进方案. 在横向上，通过坐标系变换和插值处理，以新颖的7点差分格式代替传统的5点差分方式；在纵向上，以四阶显式Runge-Kutta方法(RKBPM)代替二阶Crank-Nicholson算法(CNBPM)，避免了求解不规则矩阵方程，从而使计算效率显著提高. 关键词： 束传播方法 Runge-Kutta方法 光波导 脊形波导     

Optimized prefactored compact schemes

The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and dissipation errors. In this paper we describe a strategy for developing high-order accurate prefactored compact schemes, requiring very small stencil support. These schemes require fewer boundary stencils and offer simpler boundary condition implementation than existing compact schemes. The prefactorization strategy splits the central implicit schemes into forward and backward biased operators. Using Fourier analysis, we show it is possible to select the coefficients of the biased operators such that their dispersion characteristics match those of the original central compact scheme and their numerical wavenumbers have equal and opposite imaginary components. This ensures that when the forward and backward stencils are added, the original central compact scheme is recovered. To extend the resolution characteristic of the schemes, an optimization strategy is employed in which formal order of accuracy is sacrificed in preference to enhanced resolution characteristics across the range of wavenumbers realizable on a given mesh. The resulting optimized schemes yield improved dispersion characteristics compared to the standard sixth- and eighth-order compact schemes making them more suitable for high-resolution numerical simulations in gas dynamics and computational aeroacoustics. The efficiency, accuracy and convergence characteristics of the new optimized prefactored compact schemes are demonstrated by their application to several test problems.     

Full-vectorial analysis of optical waveguides by the finite difference method based on polynomial interpolation

Based on the polynomial interpolation, a new finite difference (FD) method in solving the full-vectorial guided-modes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present FD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.     

Numerical methods for solid mechanics on overlapping grids: Linear elasticity

This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids.     

解三维扩散方程的积分内插法格式

研究三维扩散方程的数值模拟.在非正规六面体网格上,使用积分内插法建立扩散方程差分格式,涉及到27个相邻网格,适用于大变形网格上带间断系数的拟线性扩散方程的计算.叙述差分格式的建立,推导通量流和网格顶点温度的计算公式,给出了数值试验结果.     

A High-Order Discontinuous Galerkin Method for the Two-Dimensional Time-Domain Maxwell's Equations on Curved Mesh

In this paper, a DG (Discontinuous Galerkin) method which has been widely employed in CFD (Computational Fluid Dynamics) is used to solve the two-dimensional time-domain Maxwell's equations for complex geometries on unstructured mesh. The element interfaces on solid boundary are treated in both curved way and straight way. Numerical tests are performed for both benchmark problems and complex cases with varying orders on a series of grids, where the high-order convergence in accuracy can be observed. Both the curved and the straight solid boundary implementation can give accurate RCS (Radar Cross-Section) results with sufficiently small mesh size, but the curved solid boundary implementation can significantly improve the accuracy when using relatively large mesh size. More importantly, this CFD-based high-order DG method for the Maxwell's equations is very suitable for complex geometries.     

Rib waveguide analysis by theimaginary-distance vectorialbeam-propagation method based on Yee's mesh

The effective index and the mode profile of rib waveguides are calculated using the vectorial beam-propagation method using the imaginary-distance procedure. Yee's mesh originally developed in the finite-difference time-domain method is employed to discretize a waveguide geometry. The refractive index at the interface between two different media is determined by Ampere's circuital law. The field discontinuities at the interface and singularities at the corners of the waveguide are clearly displayed.     

Boundary closures for fourth-order energy stable weighted essentially non-oscillatory finite-difference schemes

A general strategy was presented in 2009 by Yamaleev and Carpenter for constructing energy stable weighted essentially non-oscillatory (ESWENO) finite-difference schemes on periodic domains. ESWENO schemes up to eighth order were developed that are stable in the energy norm for systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain, wherever possible, the WENO stencil biasing properties and satisfy the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L₂ norm. Second-order and third-order boundary closures are developed that are stable in diagonal and block norms, respectively, and achieve third- and fourth-order global accuracy for hyperbolic systems. A novel set of nonuniform flux interpolation points is necessary near the boundaries to simultaneously achieve (1) accuracy, (2) the SBP convention, and (3) WENO stencil biasing mechanics.     

Output error estimation for summation-by-parts finite-difference schemes

The paper develops a posteriori error estimates of integral output functionals for summation-by-parts finite-difference methods. The error estimates are based on the adjoint-weighted residual method and take advantage of a variational interpretation of summation-by-parts discretizations. The estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators. For smooth boundary-value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined. The theory is verified using linear boundary-value problems and the Euler equations.     

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain(LOD-CFDTD) method is presented for solving two-dimensional(2D) electromagnetic(EM) scattering problems. The formulation for the 2D transverse-electric(TE) case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit(ADI) CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field(TF/SF) boundary and the perfectly matched layer(PML), the radar cross section(RCS) of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.     

Momentum transfer correction for macroscopic-gradient boundary conditions in lattice Boltzmann methods

The boundary conditions used to represent macroscopic-gradient-related effects in arbitrary geometries with the lattice Boltzmann methods need a trade-off between the complexity of the scheme, due to the loss of localness and the difficulties for directly applying link-based approaches, and the accuracy obtained. A generalization of the momentum transfer boundary condition is presented, in which the arbitrary location of the boundary is addressed with link-wise interpolation (used for Dirichlet conditions) and the macroscopic gradient is taken into account with a finite-difference scheme. This leads to a stable approach for arbitrary geometries that can be used to impose Neumann and Robin boundary conditions. The proposal is validated for stress boundary conditions at walls. Two-dimensional steady and unsteady configurations are used as test case: partial-slip flow between two infinite plates and the slip flow past a circular cylinder.     

Electric and magnetic losses modeled by a stable hybrid with explicit–implicit time-stepping for Maxwell’s equations

A stable hybridization of the finite-element method (FEM) and the finite-difference time-domain (FDTD) scheme for Maxwell’s equations with electric and magnetic losses is presented for two-dimensional problems. The hybrid method combines the flexibility of the FEM with the efficiency of the FDTD scheme and it is based directly on Ampère’s and Faraday’s law. The electric and magnetic losses can be treated implicitly by the FEM on an unstructured mesh, which allows for local mesh refinement in order to resolve rapid variations in the material parameters and/or the electromagnetic field. It is also feasible to handle larger homogeneous regions with losses by the explicit FDTD scheme connected to an implicitly time-stepped and lossy FEM region. The hybrid method shows second-order convergence for smooth scatterers. The bistatic radar cross section (RCS) for a circular metal cylinder with a lossy coating converges to the analytical solution and an accuracy of 2% is achieved for about 20 points per wavelength. The monostatic RCS for an airfoil that features sharp corners yields a lower order of convergence and it is found to agree well with what can be expected for singular fields at the sharp corners. A careful convergence study with resolutions from 20 to 140 points per wavelength provides accurate extrapolated results for this non-trivial test case, which makes it possible to use as a reference problem for scattering codes that model both electric and magnetic losses.     

NS方程的各向异性笛卡尔网格法研究

将近年发展起来的用于Euler方程求解的具有局部均匀网格总体非结构特性的笛卡尔网格法推广到NS方程的求解。为了与流场的各向异性相适应、减少网格点数量,提出了一种各向异性网格加密法。另外还研究了分级笛卡尔网格对内点格式稳定性的影响和插值固体边界条件的稳定性。数值结果表明各向异性笛卡尔网格法相对于传统的各向同性网格方法能大量节省网格点数量而且与后者具有同样的精度。     

A massively parallel multi-block hybrid compact–WENO scheme for compressible flows

A new multi-block hybrid compact–WENO finite-difference method for the massively parallel computation of compressible flows is presented. In contrast to earlier methods, our approach breaks the global dependence of compact methods by using explicit finite-difference methods at block interfaces and is fully conservative. The resulting method is fifth- and sixth-order accurate for the convective and diffusive fluxes, respectively. The impact of the explicit interface treatment on the stability and accuracy of the multi-block method is quantified for the advection and diffusion equations. Numerical errors increase slightly as the number of blocks is increased. It is also found that the maximum allowable time steps increase with the number of blocks. The method demonstrates excellent scalability on up to 1264 processors.     

A second order virtual node method for elliptic problems with interfaces and irregular domains

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L^∞.     

Nonuniform time-step Runge–Kutta discontinuous Galerkin method for Computational Aeroacoustics

With many superior features, Runge–Kutta discontinuous Galerkin method (RKDG), which adopts Discontinuous Galerkin method (DG) for space discretization and Runge–Kutta method (RK) for time integration, has been an attractive alternative to the finite difference based high-order Computational Aeroacoustics (CAA) approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain, demanding a high computational cost for obtaining time accurate numerical solutions in CAA. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. In order to ensure correct communication of solutions on the interfaces of grids with different time step sizes, the values at intermediate-stages of the Runge–Kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors. Based upon the general form of an explicit p-stage RK scheme, a linear coupling procedure is proposed, with details on the coefficient matrices and execution steps at common time-levels and intermediate time-levels. Applications of the coupling procedures to Runge–Kutta schemes frequently used in simulation of fluid flow and acoustics are given, including the third-order TVD scheme, and low-storage low dissipation and low dispersion (LDDRK) schemes. In addition, an analysis on the stability of coupling procedures on a nonuniform grid is carried out. For validation, numerical experiments on one-dimensional and two-dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme, as well as the computational benefits it brings. Application to a one-dimensional nonlinear problem is also investigated.     

A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids

A residual-based (RB) scheme relies on the vanishing of residual at the steady-state to design a transient first-order dissipation, which becomes high-order at steady-state. Initially designed within a finite-difference framework for computations of compressible flows on structured grids, the RB schemes displayed good convergence, accuracy and shock-capturing properties which motivated their extension to unstructured grids using a finite volume (FV) method. A second-order formulation of the FV–RB scheme for compressible flows on general unstructured grids was presented in a previous paper. The present paper describes the derivation of a third-order FV–RB scheme and its application to hyperbolic model problems as well as subsonic, transonic and supersonic internal and external inviscid flows.