Analysis of numerical dispersion properties of SFDTD and MRTD schemes

In this paper, the Maxwell's equations are written as Hamilton canonical equations by using Hamilton functional variation method. Maxwell's equations can be discretized with symplectic propagation technique combined with high-order difference schemes approximations to construct symplectic finite difference time domain (SFDTD) method. The high-order dispersion equations of the scheme for space is deduced. The numerical dispersion analysis is included, and it is compared with the multiresolution time-domain (MRTD) method based on the Daubechies scaling functions. Numerical results show high efficiency and accuracy of the SFDTD method.     

A central conservative scheme for general rectangular grids

We present an extension of the genuinely multi-dimensional semi-discrete central scheme developed in [A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740.] to arbitrary orthogonal grids. The presented algorithm is constructed to yield the geometric scaling factors in a self-consistent way.Additionally, the order of the scheme is not fixed during the derivation of the basic algorithm. Based on the resulting general scheme it is possible to construct methods of any desired order, just by considering the corresponding reconstruction polynomial. We demonstrate how a second order scheme in plane polar coordinates and cylindrical coordinates can be derived from our general formulation. Finally, we demonstrate the correctness of this second order scheme through application to several numerical experiments.     

Explicit and implicit finite difference schemes for fractional Cattaneo equation

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor–corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor–corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.     

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.     

MIB method for elliptic equations with multi-material interfaces

Elliptic partial differential equations (PDEs) are widely used to model real-world problems. Due to the heterogeneous characteristics of many naturally occurring materials and man-made structures, devices, and equipments, one frequently needs to solve elliptic PDEs with discontinuous coefficients and singular sources. The development of high-order elliptic interface schemes has been an active research field for decades. However, challenges remain in the construction of high-order schemes and particularly, for nonsmooth interfaces, i.e., interfaces with geometric singularities. The challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other. High-order methods for elliptic equations with multi-material interfaces have not been reported in the literature to our knowledge. The present work develops matched interface and boundary (MIB) method based schemes for solving two-dimensional (2D) elliptic PDEs with geometric singularities of multi-material interfaces. A number of new MIB schemes are constructed to account for all possible topological variations due to two-material interfaces. The geometric singularities of three-material interfaces are significantly more difficult to handle. Three new MIB schemes are designed to handle a variety of geometric situations and topological variations, although not all of them. The performance of the proposed new MIB schemes is validated by numerical experiments with a wide range of coefficient contrasts, geometric singularities, and solution types. Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges.     

Hybrid Flux-Splitting Schemes for a Two-Phase Flow Model

In this paper we deal with the construction of hybrid flux-vector-splitting (FVS) schemes and flux-difference-splitting (FDS) schemes for a two-phase model for one-dimensional flow. The model consists of two mass conservation equations (one for each phase) and a common momentum equation. The complexity of this model, as far as numerical computation is concerned, is related to the fact that the flux cannot be expressed in terms of its conservative variables. This is the motivation for studying numerical schemes which are not based on (approximate) Riemann solvers and/or calculations of Jacobian matrix. This work concerns the extension of an FVS type scheme, a Van Leer type scheme, and an advection upstream splitting method (AUSM) type scheme to the current two-phase model. Our schemes are obtained through natural extensions of corresponding schemes studied by Y. Wada and M.-S. Liou (1997, SIAM J. Sci. Comput.18, 633–657) for Euler equations. We explore the various schemes for flow cases which involve both fast and slow transients. In particular, we demonstrate that the FVS scheme is able to capture fast-propagating acoustic waves in a monotone way, while it introduces an excessive numerical dissipation at volume fraction contact (steady and moving) discontinuities. On the other hand, the AUSM scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves. This motivates us to propose other hybrid FVS/FDS schemes obtained by removing numerical dissipation at contact discontinuities in the FVS and Van Leer schemes.     

High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres

Wave propagation in idealized stellar atmospheres is modeled by the equations of ideal MHD, together with the gravity source term. The waves are modeled as small perturbations of isothermal steady states of the system. We consider a formulation of ideal MHD based on the Godunov–Powell form, with an embedded potential magnetic field appearing as a parameter. The equations are discretized by finite volume schemes based on approximate Riemann solvers of the HLL type and upwind discretizations of the Godunov–Powell source terms. Local hydrostatic reconstructions and suitable discretization of the gravity source term lead to a well-balanced scheme, i.e., a scheme which exactly preserves a discrete version of the relevant steady states. Higher order of accuracy is obtained by employing suitable minmod, ENO and WENO reconstructions, based on the equilibrium variables, to construct a well-balanced scheme. The resulting high order well-balanced schemes are validated on a suite of numerical experiments involving complex magnetic fields. The schemes are observed to be robust and resolve the complex physics well.     

二维浅水波方程的数值激波不稳定性

研究二维浅水波方程的数值激波不稳定性问题.线性稳定性分析和数值实验表明,格式的临界稳定性与数值激波的不稳定现象有重要的联系.基于扰动量的增长矩阵分析,本文将高分辨率的数值格式和HLL格式进行特定的加权,设计一类新的混合型数值格式.其中可以调节非线性波速的HLLC与HLL的混合格式,数值试验展示了消除浅水波方程激波不稳定现象的有效性和鲁棒性.     

A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

Moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data. In particular, they are often employed when solving partial differential equations on unstructured meshes, which are typically needed when the geometry defining the domain is complex. It is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements, however little research has specifically addressed this issue. In this paper, a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions, and the precise conditions under which singularities occur are identified. The theory is used to develop a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner, while attempting to maintain stencil symmetry as closely as possible. Finally, the scheme is applied in a convection–diffusion equation solver and an incompressible Navier–Stokes solver, and the results are shown to compare favourably with known analytical solutions and previously published results.     

气相爆轰高阶中心差分-WENO组合格式自适应网格方法

研究一种高阶中心差分-WENO组合格式,并采用自适应网格方法进行二维和三维气相爆轰波的数值模拟.采用ZND爆轰模型的控制方程为包含化学反应源项的Euler方程组.组合格式在大梯度区采用WENO格式捕捉间断,在光滑区采用高阶中心差分格式提高计算效率.采用一种基于流场结构特征的自适应网格.计算结果,表明这种方法同时具有高精度、高分辨率和高效率的特点.     

A geometrically-conservative,synchronized, flux-corrected remap for arbitrary Lagrangian–Eulerian computations with nodal finite elements

This article describes a conservative synchronized remap algorithm applicable to arbitrary Lagrangian–Eulerian computations with nodal finite elements. In the proposed approach, ideas derived from flux-corrected transport (FCT) methods are extended to conservative remap. Unique to the proposed method is the direct incorporation of the geometric conservation law (GCL) in the resulting numerical scheme. It is shown here that the geometric conservation law allows the method to inherit the positivity preserving and local extrema diminishing (LED) properties typical of FCT schemes. The proposed framework is extended to the systems of equations that typically arise in meteorological and compressible flow computations. The proposed algorithm remaps the vector fields associated with these problems by means of a synchronized strategy. The present paper also complements and extends the work of the second author on nodal-based methods for shock hydrodynamics, delivering a fully integrated suite of Lagrangian/remap algorithms for computations of compressible materials under extreme load conditions. Extensive testing in one, two, and three dimensions shows that the method is robust and accurate under typical computational scenarios.     

一种基于特征理论模拟凝聚炸药爆轰的单元中心型Lagrange方法

提出一种数值模拟凝聚炸药爆轰问题的单元中心型Lagrange方法.利用有限体积离散爆轰反应流动方程组,基于双曲型偏微分方程组的特征理论获得离散网格节点的速度与压力,获得的网格节点速度与压力用于更新网格节点位置以及计算网格单元边的数值通量.以这种方式获得的网格节点解是一种"真正多维"的理论解,是一维Godunov格式在二维Riemann问题的推广.有限体积离散得到的爆轰反应流动的半离散系统使用一种显-隐Runge-Kutta格式来离散求解：显式格式处理对流项,隐式格式处理化学反应刚性源项.算例表明,提出的单元中心型Lagrange方法能够较好地模拟凝聚炸药的爆轰反应流动.     

L-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant–Friedrichs–Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.     

Splitting based finite volume schemes for ideal MHD equations

We design finite volume schemes for the equations of ideal magnetohydrodynamics (MHD) and based on splitting these equations into a fluid part and a magnetic induction part. The fluid part leads to an extended Euler system with magnetic forces as source terms. This set of equations are approximated by suitable two- and three-wave HLL solvers. The magnetic part is modeled by the magnetic induction equations which are approximated using stable upwind schemes devised in a recent paper [F. Fuchs, K.H. Karlsen, S. Mishra, N.H. Risebro, Stable upwind schemes for the Magnetic Induction equation. Math. Model. Num. Anal., Available on conservation laws preprint server, submitted for publication, URL: <http://www.math.ntnu.no/conservation/2007/029.html>]. These two sets of schemes can be combined either component by component, or by using an operator splitting procedure to obtain a finite volume scheme for the MHD equations. The resulting schemes are simple to design and implement. These schemes are compared with existing HLL type and Roe type schemes for MHD equations in a series of numerical experiments. These tests reveal that the proposed schemes are robust and have a greater numerical resolution than HLL type solvers, particularly in several space dimensions. In fact, the numerical resolution is comparable to that of the Roe scheme on most test problems with the computational cost being at the level of a HLL type solver. Furthermore, the schemes are remarkably stable even at very fine mesh resolutions and handle the divergence constraint efficiently with low divergence errors.     

Geometric conservation law and applications to high-order finite difference schemes with stationary grids

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators δ₃ have no effects on the SCL, no extra errors can be introduced as δ₃ = δ₂. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator δ₁ which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator δ₁ to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.     

孤立波方程的保结构算法

讨论了孤立波方程的保结构差分算法，以一些经典的孤立波方程为例，如KdV，sine-Gordon，K-P方程，给出了它们的辛和多辛结构，说明辛和多辛算法的可适用性．提出局部守恒算法和广义保结构算法的概念，它们是保结构算法的概念自然推广．还给出一种能系统构造局部守恒格式的复合方法．数值例子说明，保结构数值能很好模拟各种孤立波的演化。     

常微分方程边值问题的高阶三对角OCI差分法

本文给出了二阶线性常微分方程两点边值问题(ODETPBVP)的高阶差分格式构造的基本思想,推导出六阶三对角OCI差分格式,并对端点有奇异性的方程进行了极限值处理,消去了奇异性,对边界层问题采用了非均匀网格上的六阶三对角OCI差分格式。通过大量的数值比较实验表明,这种高阶三对角OCI差分格式能很好地求解奇异性问题,固有不稳定性问题,奇异摄动问题,对生不稳定性问题和振荡性问题。     

DGM-FD: A finite difference scheme based on the discontinuous Galerkin method applied to wave propagation

In this paper we formulate a numerical method that is high order with strong accuracy for numerical wave numbers, and is adaptive to non-uniform grids. Such a method is developed based on the discontinuous Galerkin method (DGM) applied to the hyperbolic equation, resulting in finite difference type schemes applicable to non-uniform grids. The schemes will be referred to as DGM-FD schemes. These schemes inherit naturally some features of the DGM, such as high-order approximations, applicability to non-uniform grids and super-accuracy for wave propagations. Stability of the schemes with boundary closures is investigated and validated. Proposed scheme is demonstrated by numerical examples including the linearized acoustic waves and solutions of non-linear Burger’s equation and the flat-plate boundary layer problem. For non-linear equations, proposed flux finite difference formula requires no explicit upwind and downwind split of the flux. This is in contrast to existing upwind finite difference schemes in the literature.     

守恒律方程组的大时间步长叠波格式

大时间步长叠波格式最初思想为LeVeque提出的大时间步长Godunov格式,通过叠加间断分解发出的强波来构造数值格式.原方法只给出了间断强波的穿越叠加方法,文章对其进行了完善,并推广到多维.针对膨胀波提出了一种网格单元分解法可以自动满足熵条件,避免出现非物理解.给出了格式的具体计算公式,并用单个守恒律方程、一维/多维Euler方程组进行了数值计算.计算结果表明,新格式除了可以采用大时间步长的优点外,在一定范围内随CFL数增加其耗散反而更低,因而对激波接触间断膨胀波的分辨率更高.     

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation,applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.