An improved CE/SE scheme and its application to dilute gas–particle flows

An improved space–time Conservation Element and Solution Element (CE/SE) scheme is constructed by proposing a new structure of solution elements and conservation elements based on the rectangular mesh. Furthermore, the improved CE/SE scheme was applied to dilute gas–particle two-phase flows. A two-fluid model and two corresponding chemical reaction models, i.e., two-step reaction model and detailed chemical reaction model, were used to describe the physical and chemical characteristics in the two-phase flows. Shock wave reflection in gas, shock wave diffraction in air–sand mixture, explosive synthesis of TiO₂ nanoparticle and air–fuel two-phase detonation were simulated by the improved CE/SE scheme and appropriate physical and chemical models. All the numerical results were compared and discussed carefully. The results show that the improved CE/SE scheme is clear in physical concept, easy to be implemented and high accurate for the above-mentioned problems. Thus, the improved CE/SE scheme can be applied to gas–particle flows widely.     

Extension of CE/SE method to 2D viscous flows

The space–time conservation element–solution element (CE/SE) method is extended to two-dimensional viscous flow problems. The formulation is presented and discussed. The extended CE/SE method is applied to several 2D viscous flow problems, such as boundary layer flow, entrance of channel flow, backward facing step flow and cavity flow. The numerical results and their comparison with analytical result and experimental data are presented. The capability of this method for predicting viscous flow feature and heat transfer is demonstrated. Also the capability of the method to solve both high-speed flows and low-Mach number flows is demonstrated.     

An improved CE/SE scheme for numerical simulation of gaseous and two-phase detonations

A new structure of solution elements and conservation elements based on rectangular mesh was proposed and an improved space-time conservation element and solution element (CE/SE) scheme with second-order accuracy was constructed. Furthermore, the application of improved CE/SE scheme was extended to detonation simulation. Three models were used for chemical reaction in gaseous detonation. And a two-fluid model was used for two-phase (gas-droplet) detonation. Shock reflections were simulated by the improved CE/SE scheme and the numerical results were compared with those obtained by other different numerical schemes. Gaseous and gas-droplet planar detonations were simulated and the numerical results were carefully compared with the experimental data and theoretical results based on C-J theory. Mach reflection of a cellular detonation was also simulated, and the numerical cellular patterns were compared with experimental ones. Comparisons show that the improved CE/SE scheme is clear in physical concept, easy to be implemented and high accurate for above-mentioned problems.     

Linearized Conservative Finite Element Methods for the Nernst–Planck–Poisson Equations

The aim of this paper is to present and study new linearized conservative schemes with finite element approximations for the Nernst–Planck–Poisson equations. For the linearized backward Euler FEM, an optimal \(L^2\) error estimate is provided almost unconditionally (i.e., when the mesh size h and time step \(\tau \) are less than a small constant). Global mass conservation and electric energy decay of the schemes are also proved. Extension to second-order time discretizations is given. Numerical results in both two- and three-dimensional spaces are provided to confirm our theoretical analysis and show the optimal convergence, unconditional stability, global mass conservation and electric energy decay properties of the proposed schemes.     

Numerical Methods for Euler Equations with Self-similar and Quasi Self-similar Solutions

Some problems of Euler equations have self-similar solutions which can be solved by more accurate method. The current paper proposes two new numerical methods for Euler equations with self-similar and quasi self-similar solutions respectively, which can use existing difference schemes for conservation laws and do not need to redesign specified schemes. Numerical experiments are implemented on one dimensional shock tube problems, two dimensional Riemann problems, shock reflection from a solid wedge, and shock refraction at a gaseous interface. For self-similar equations, one-dimensional results are almost equal to the exact solutions, and two-dimensional results also exhibit considerable high resolution. For quasi self-similar equations, the method can solve solutions that are not but close to self-similar, i.e. quasi self-similar, and this method can also achieve very high resolution when computing time is long enough. Numerical simulations to self-similar and quasi self-similar Euler equations have important implications on the study of self-similar problems, development of high resolution schemes, even the research for exact solutions of Euler equations.     

SMB色谱模型数值计算的时空守恒元解元方法

为了探寻出一种求解SMB色谱模型的快速数值求解方法,并试图通过比较得出时空守恒元/解元(CE/SE)方法确实是快速数值求解方法,因而采用该方法对SMB色谱模型进行数值求解,并在数值方法的计算效率和精确度两个方面与有限差分法和正交配置有限元法进行了比较,最终得出了CE/SE方法是具有高计算效率和高精确度特性的快速数值求解方法.通过两个实例的模拟仿真,结果表明了该方法在高计算效率和高精确度方面的优越性.     

On energy conservation for finite element approximation of flow-induced airfoil vibrations

In this paper the numerical approximation of a two-dimensional fluid–structure interaction problem is addressed. The fully coupled formulation of incompressible viscous fluid flow interacting with a flexibly supported airfoil is considered. The flow is described by the incompressible system of Navier–Stokes equations, where large values of the Reynolds number are considered. The Navier–Stokes equations are spatially discretized by the finite element method and stabilized with a modification of the Galerkin Least Squares (GLS) method; cf. [T. Gelhard, G. Lube, M.A. Olshanskii, J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows, Journal of Computational and Applied Mathematics 177 (2005) 243–267]. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian (ALE) method and the stabilizing terms are modified in a consistent way with the ALE formulation.     

Numerical Study of a Modified Time-Stepping θ-Scheme for Incompressible Flow Simulations

In [Turek (1996). Int. J. Numer. Meth. Fluids 22, 987–1011], we had performed numerical comparisons for different time stepping schemes for the incompressible Navier–Stokes equations. In this paper, we present the numerical analysis in the context of the Navier–Stokes equations for a modified time-stepping θ-scheme which has been recently proposed by Glowinski [Glowinski (2003). In: Ciarlet, P. G., and Lions, J. L. (eds.), Handbook of Numerical Analysis, Vol. IX, North-Holland, Amsterdam, pp. 3–1176]. Like the well-known classical Fractional-Step-θ-scheme which had been introduced by Glowinski [Glowinski (1985). In Murman, E. M. and Abarbanel, S. S. (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Birkh?user, Boston MA; Bristeau et al. (1987). Comput. Phys. Rep. 6, 73–187], too, and which is still one of the most popular time stepping schemes, with or without operator splitting techniques, this new scheme consists of 3 substeps with nonequidistant substepping to build one macro time step. However, in contrast to the Fractional-Step-θ-scheme, the second substep can be formulated as an extrapolation step for previously computed data only, and the two remaining substeps look like a Backward Euler step so that no expensive operator evaluations for the right hand side vector with older solutions, as for instance in the Crank–Nicolson scheme, have to be performed. This modified scheme is implicit, strongly A-stable and second order accurate, too, which promises some advantageous behavior, particularly in implicit CFD simulations for the nonstationary Navier–Stokes equations. Representative numerical results, based on the software package FEATFLOW [Turek (2000). FEATFLOW Finite element software for the incompressible Navier–Stokes equations: User Manual, Release 1.2, University of Dortmund] are obtained for typical flow problems with benchmark character which provide a fair rating of the solution schemes, particularly in long time simulations.Dedicated to David Gottlieb on the occasion of his 60th anniversary     

Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.     

Application of a discontinuous Galerkin finite element method to special relativistic hydrodynamic models

A Runge–Kutta discontinuous Galerkin (RKDG) finite element method is proposed for solving the special relativistic hydrodynamic (SRHD) equations and as a limiting case the ultra-relativistic hydrodynamic (URHD) equations. The latter model is obtained by ignoring the rest-mass energy when the internal energy of fluid particles is sufficiently large. Several test problems of SRHD and URHD models are carried out. For validation, the results of DG-method are compared with the staggered central scheme. The numerical results verify the accuracy of the proposed method qualitatively and quantitatively.     

A Discontinuous Galerkin Scheme Based on a Space–Time Expansion. I. Inviscid Compressible Flow in One Space Dimension

In this paper, we propose an explicit discontinuous Galerkin scheme for conservation laws which is of arbitrary order of accuracy in space and time. The basic idea is to use a Taylor expansion in space and time to define a space–time polynomial in each space–time element. The space derivatives are given by the approximate solution at the old time level, the time derivatives and the mixed space–time derivatives are computed from these space derivatives using the so-called Cauchy–Kovalevskaya procedure. The space–time volume integral is approximated by Gauss quadrature with values at the space–time Gaussian points obtained from the Taylor expansion. The flux in the surface integral is approximated by a numerical flux with arguments given by the Taylor expansions from the left and from the right-hand side of the element interface. The locality of the presented method together with the space–time expansion gives the attractive feature that the time steps may be different in each grid cell. Hence, we drop the common global time levels and propose that every grid zone runs with its own time step which is determined by the local stability restriction. In spite of the local time steps the scheme is locally conservative, fully explicit, and arbitrary order accurate in space and time for transient calculations. Numerical results are shown for the one-dimensional Euler equations with orders of accuracy one up to six in space and time.     

Nonlinear Upwind-Biased Free-Stream-Preserving Schemes for Compressible Euler Equations

The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.

     

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space-time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Transport of pollutant in shallow flows: A space–time CE/SE scheme

This paper focuses on implementation of space–time CE/SE scheme for computing the transport of a passive pollutant by a flow. The flow model comprises of the Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. The one-dimensional and two-dimensional flow equations are numerically investigated using the CE/SE scheme. A number of test problems are presented to check the accuracy and efficiency of the proposed scheme. The results of CE/SE scheme are compared with the central scheme. Both the schemes are found to be in close agreement. However, our proposed CE/SE scheme accurately captures shocks and discontinuous profiles.     

Non-Linear Filtering and Limiting in High Order Methods for Ideal and Non-Ideal MHD

The adaptive nonlinear filtering and limiting in spatially high order schemes (Yee et al. J. Comput. Phys. 150, 199–238, (1999), Sjögreen and Yee, J. Scient. Comput. 20, 211–255, (2004)) for the compressible Euler and Navier–Stokes equations have been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations, (Sjögreen and Yee, (2003), Proceedings of the 16th AIAA/CFD conference, June 23–26, Orlando F1; Yee and Sjögreen (2003), Proceedings of the International Conference on High Performance Scientific Computing, March, 10–14, Honai, Vietnam; Yee and Sjögreen (2003), RIACS Technical Report TR03. 10, July, NASA Ames Research Center; Yee and Sjögreen (2004), Proceedings of the ICCF03, July 12–16, Toronto, Canada). The numerical dissipation control in these adaptive filter schemes consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of discontinuity capturing nonlinear numerical dissipation for both the ideal and non-ideal MHD.     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

Local,point-wise rotational transformations of the conservation equations into stream-wise coordinates

In dealing with multidimensional simulations, many authors have shown that a major cause of numerical dispersion errors is due to the flow being skewed to the coordinate axes. Crane and Blunt [1] have shown that the stream-wise transformations can reduce the numerical errors associated with the multidimensional transport equations. However, it has been proven that no transformation can completely diagonalize the multidimension conservation equations.It shall be demonstrated that a subset of the multidimensional Euler equations can be diagonalized, but not the entire set. The formulation of the conservation equations into the local stream-wise coordinate system is extended to the time-dependent, two- and three-dimensional (2D and 3D) conservation equations. At any point in space, there exists a set of local rotations that aligns the fluid velocity vector coincident with the stream-wise coordinate; hence, the fluid velocity components orthogonal to the stream-wise coordinate are identically zero. Such transformations result in a subset of PDEs that are diagonalized, namely, the mass, total energy, and principal momentum density PDEs. However, the orthogonal momentum component conservation PDEs are not diagonalized and are multidimensional; these PDEs are responsible for streamline bending.     

Residual Distribution Schemes on Quadrilateral Meshes

We propose an investigation of the residual distribution schemes for the numerical approximation of two-dimensional hyperbolic systems of conservation laws on general quadrilateral meshes. In comparison to the use of triangular cells, usual basic features are recovered, an extension of the upwinding concept is given, and a Lax–Wendroff type theorem is adapted for consistency. We show how to retrieve many variants of standard first and second-order accurate schemes. They are proven to satisfy this theorem. An important part of this paper is devoted to the validation of these schemes by various numerical tests for scalar equations and the Euler equations system for compressible fluid dynamics on non Cartesian grids. In particular, second-order accuracy is reached by an adaptation of the Linearity preserving property to quadrangle meshes. We discuss several choices as well as the convergence of iterative method to steady state. We also provide examples of schemes that are not constructed from an upwinding principle     

Convection of Concentrated Vortices and Passive Scalars as Solitary Waves

A new version of a computational method, Vorticity Confinement, is described. Vorticity Confinement has been shown to efficiently treat thin features in multi-dimensional incompressible fluid flow, such as vortices and streams of passive scalars, and to convect them over long distances with no spreading due to numerical errors. Outside the features, where the flow is irrotational or the scalar vanishes, the method automatically reduces to conventional discretized finite difference fluid dynamic equations. The features are treated as a type of weak solution and, within the features, a nonlinear difference equation, as opposed to finite difference equation, is solved that does not necessarily represent a Taylor expansion discretization of a simple partial differential equation (PDE). The approach is similar to artificial compression and shock capturing schemes, where conservation laws are satisfied across discontinuities. For the features, the result of this conservation is that integral quantities such as total amplitude and centroid motion are accurately computed. Basically, the features are treated as multi-dimensional nonlinear discrete solitary waves that live on the computational lattice. These obey a confinement relation that is a generalization to multiple dimensions of 1-D discontinuity capturing schemes. A major point is that the method involves a discretization of a rotationally invariant operator, rather than a composition of separate 1-D operators, as in conventional discontinuity capturing schemes. The main objective of this paper is to introduce a new formulation of Vorticity Confinement that, compared to the original formulation, is simpler, allows more detailed analysis, and exactly conserves momentum for vortical flow. First, a short critique of conventional methods for these problems is given. The basic new method is then described. Finally, analysis of the new method and initial results are presented.