Convergence acceleration for the steady-state Euler equations

We consider the iterative solution of systems of equations arising from discretizations of the non-linear Euler equations governing compressible flow. The differential equation is discretized on a structured grid, and the steady-state solution is computed by a time-marching method.A convergence acceleration technique based on semicirculant approximations of the difference operator or the Jacobian is used. Implementation issues and variants of the scheme allowing for a reduction of the arithmetic complexity and memory requirement are discussed. The technique can be combined with a variety of iterative solvers, but we focus on non-linear explicit Runge-Kutta time-integration schemes. The results show that the single-stage forward Euler method can be used, and that the time step is not limited by a CFL-criterion. This results in that the arithmetic work required for computing the solution is equivalent to the work required for a fixed number of residual evaluations.     

Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography

Many natural terrains have complicated surface topography. The simulation of steep-fronted flows that occur after heavy rainfall flash floods or as inundation from dyke breaches is usually based on the non-linear shallow water equations in hyperbolic conservation form. Particular challenges to numerical modellers are posed by the need to balance correctly the flux gradient and source terms in Godunov-type finite volume shock-capturing schemes and by the moving wet-dry boundary as the flood rises or falls. This paper presents a Godunov-type shallow flow solver on adaptive quadtree grids aimed at simulating flood flows as they travel over natural terrain. By choosing the stage and discharge as dependent variables in the hyperbolic non-linear shallow water equations, a new deviatoric formulation is derived that mathematically balances the flux gradient and source terms in cases where there are wet-dry fronts. The new formulation is more general in application than previous a priori approaches. Three benchmark tests are used to validate the solver, and include steady flow over a submerged hump, flow disturbances propagating over an elliptical-shaped hump, and free surface sloshing motions in a vessel with a parabolic bed. The model is also used to simulate the propagation of a flood due to a dam break over an initially dry floodplain containing three humps.     

A finite element first-order equation formulation for the small-disturbance transonic flow problem

The nonlinear, mixed elliptic hyperbolic equation describing a steady transonic flow is considered. The original equation is replaced by a system of first-order equations that are hyperbolic in time and defined in terms of velocity components. Parabolic regularization terms are added to capture shock wave solutions and to damp iterative solution algorithms. A finite element Galerkin method in space and a Crank-Nicolson finite difference method in iterative time are used to reduce the problem to the solution of a system of algebraic equations. Stability and convergence characteristics of the iterative method are discussed. The numerical implementation of the method is explained, and numerical results are presented.     

Least-Squares-Lösungen und Dämpfung bei unterbestimmten nichtlinearen Gleichungssystemen

Least-squares-solutions are defined and appropriate iterative methods are derived for the numerical solution of under-determined systems of non-linear equations. Especially an heuristic derivation of a damped Newton-like method is given in which the multiplicative damping factors are considered as additional unknowns to be computed during the iterations in the least-squares sense. A semilocal convergency theorem is proved for a related modification.     

Technically oriented algorithms for unsteady pipe flow

The two best-known integration methods for nonlinear hyperbolic systems of partial differential equations, i.e. the method of characteristics and the explicit difference method, are examined with respect to systematic calculation of complicated technical problems of instationary compressible pipe flow, especially with regard to their adaptability to computer solution. Although difference methods are much easier to program than characteristic methods, they generally tend to produce unpredictable errors, at least in their current form. Therefore, it is the aim of this paper to develop and test difference algorithms for some of the most usual boundary conditions of technical pipe flow, including such essentially two-dimensional configurations as the flow through bends and branches.     

Discrete Fundamental Solution Preconditioning for Hyperbolic Systems of PDE

We present a new preconditioner for the iterative solution of systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electromagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple one-stage iterative method. As an example of a more involved problem, we consider the steady state solution of the non-linear Euler equations in a two-dimensional, non-axisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.     

Multi-dimensional semi-Lagrangian scheme that guarantees exact conservation

A new numerical method that guarantees exact mass conservation is proposed to solve multi-dimensional hyperbolic equations in semi-Lagrangian form without directional splitting. The method is based on a concept of CIP scheme and keep the many good characteristics of the original CIP scheme. The CIP strategy is applied to the integral form of variable. Although the advection and non-advection terms are separately treated, the mass conservation is kept in a form of spatial profile inside a grid cell. Therefore, it retains various advantages of the semi-Lagrangian schemes with exact conservation that has been beyond the capability of conventional semi-Lagrangian schemes.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

Design of maximum thrust plug nozzles with variable inlet geometry

An optimization analysis is presented for axisymmetric plug nozzles with varible inlet geometry. The analysis is based on the governing gas dynamic relations for a rotational flow of a frozen or equilibrium gas mixture. The problem is formulated to maximize the axial thrust produced by the plug nozzle for a general isoperimetric constraint, such as constant nozzle length or constant nozzle surface area. The effects of base pressure and ambient pressure are included in the thrust expression to be maximized. The governing gas dynamic equations and the differential and integral constraints that the solution must satisfy are incorporated into the formulation by means of Lagrange multiples. The formalism of the calculus of variations is applied to the resulting functional to be maximized. The results of the optimization analysis are a set of partial differential equations for determining the Lagrange multipliers in the region of interest and a set of equations for determining the necessary boundary conditions for the solution. The complete set of equations for the gas dynamic properties and the Lagrange multipliers are system of first order, quasi-linear, non-homogeneous partial differential equations of the hyperbolic type, which can be treated by the method of charac- teristics. The characteristic and compatibility equations for the system are presented. A numerical solution procedure is presented to determine wether or not a given plug nozzle geometry is an optimal solution. An iteration technique is developed which systematically adjusts the plug nozzle geometry until the optimal solution is obtained. Selected parametric studies are presented. These studies illustrate the effect of the specific heat ratio, the design pressure ratio and the base pressure model on the thrust peformance and nozzle geometry of optimal, fixed length, plug nozzles.     

液体通过滑阀的阶跃响应的仿真研究

精确地了解滑阀的动态特性对液压控制系统的合理设计是很重要的。可是，滑阀的瞬变流体结构由于其非稳定性的复杂所以有关的信息较少。该文对流体通过滑阀的阶跃响应进行了数值解析计算。该研究所用的数值解析方法是先用等间隔的交错矩形网格。基于有限体积法将基础方程式离散化，然后用类似于SIMPLER法的迭代算法解离散化所得的差分方程组。压力差阶跃变化的情况下，做出了流动场随时间变化的流线图和流量及再附着点距离的时间变化图。数值计算结果说明了流体通过滑阀的阶跃响应可用具有不同的时间常数的指数函数来模拟。该文证实了理论模型和数值计算结果一致。     

A Modified Double Gyre with Ground Truth Hyperbolic Trajectories for Flow Visualization

The model of a Double Gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the Double Gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the Double Gyre , which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.     

Convergence Acceleration for Hyperbolic Systems Using Semicirculant Approximations

The iterative solution of systems of equations arising from systems of hyperbolic, time-independent partial differential equations (PDEs) is studied. The PDEs are discretized using a finite volume or finite difference approximation on a structured grid. A convergence acceleration technique where a semicirculant approximation of the spatial difference operator is employed as preconditioner is considered. The spectrum of the preconditioned coefficient matrix is analyzed for a model problem. It is shown that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of grid points and the artificial viscosity parameter. By linearizing the Euler equations around an approximate solution, a system of linear PDEs with variable coefficients is formed. When utilizing the semicirculant (SC) preconditioner for this problem, which has properties very similar to the full nonlinear equations, numerical experiments show that the favorable convergence properties hold also here. We compare the results for the SC method to those of a multigrid (MG) scheme. The number of iterations and the arithmetic complexities are considered, and it is clear that the SC method is more efficient for the problems studied. Also, the MG scheme is sensitive to the amount of artificial dissipation added, while the SC method is not.     

Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method

The polynomial based differential quadrature and the Fourier expansion based differential quadrature method are applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of a transverse external oblique magnetic field. Numerical solution for velocity and induced magnetic field is obtained for the steady-state, fully developed, incompressible flow of a conducting fluid inside of the duct. Equal and unequal grid point discretizations are both used in the domain and it is found that the polynomial based differential quadrature method with a reasonable number of unequally spaced grid points gives accurate numerical solution of the MHD flow problem. Some graphs are presented showing the behaviours of the velocity and the induced magnetic field for several values of Hartmann number, number of grid points and the direction of the applied magnetic field.     

Closed-form solution of an idealized,optimal, highly eccentric hyperbolic rendezvous problem

Closed-form solutions of optimal impulsive rendezvous problems are difficult to obtain, even using linearized equations. An idealized model of a highly eccentric hyperbolic impulsive rendezvous problem is constructed by considering the limit as the eccentricity tends to infinity. In this manner, a closed- form solution of an idealized hyperbolic rendezvous problem can be obtained. This idealized solution establishes an arbitrarily good approximation to a solution of the problem of optimal impulsive rendezvous of a spacecraft near a satellite in hyperbolic orbit of sufficiently high eccentricity.     

Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.     

Exponential difference schemes with double integral transformation for solving convection-diffusion equations

Convection-diffusion equations are studied. These equations are used for describing many nonlinear processes in solids, liquids, and gases. Although many works deal with solving them, they are still challenging in terms of theoretical and numerical analysis. In this work, the grid approach based on the method of finite differences for solving equations of this kind is considered. In order to make it easier, the one-dimensional version of such an equation was chosen. However, the equation preserves its principal properties; i.e., it is non-monotonic and non-linear. To solve boundary-value problems for such equations, a special variant of the non-monotonic sweep procedure is proposed.     

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

A conservative TVD scheme is adopted to solve the equations governing the three-dimensional flow of a nonideal compressible conducting fluid in a magnetic field.The eight-wave equations for magnetohydrodynamics(MHD) are proved to be a non-strict hyperbolic system,therefore it is difficult to develop its eigenstructure.Powell developed a new set of equations which cannot be numerically simulated by conservative TVD scheme directly due to its non-conservative form.A conservative TVD scheme augmented with a ne...     

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

In numerous industrial CFD applications, it is usual to use two (or more) different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman–Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.