An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

On Semi-implicit Splitting Schemes for the Beltrami Color Image Filtering

The Beltrami flow is an efficient nonlinear filter, that was shown to be effective for color image processing. The corresponding anisotropic diffusion operator strongly couples the spectral components. Usually, this flow is implemented by explicit schemes, that are stable only for very small time steps and therefore require many iterations. In this paper we introduce a semi-implicit Crank-Nicolson scheme based on locally one-dimensional (LOD)/additive operator splitting (AOS) for implementing the anisotropic Beltrami operator. The mixed spatial derivatives are treated explicitly, while the non-mixed derivatives are approximated in an implicit manner. In case of constant coefficients, the LOD splitting scheme is proven to be unconditionally stable. Numerical experiments indicate that the proposed scheme is also stable in more general settings. Stability, accuracy, and efficiency of the splitting schemes are tested in applications such as the Beltrami-based scale-space, Beltrami denoising and Beltrami deblurring. In order to further accelerate the convergence of the numerical scheme, the reduced rank extrapolation (RRE) vector extrapolation technique is employed.     

Sensors’ optimal dimensionality compression matrix in estimation fusion

When there exists the limitation of communication bandwidth between sensors and a fusion center, one needs to optimally pre-compress sensor outputs–sensor observations or estimates before sensors’ transmission to obtain a constrained optimal estimation at the fusion center in terms of the linear minimum error variance criterion. This paper will give an analytic solution of the optimal linear dimensionality compression matrix for the single sensor case and analyze the existence of the optimal linear dimensionality compression matrix for the multisensor case, as well as how to implement a Gauss–Seidel algorithm to search for an optimal solution to linear dimensionality compression matrix.     

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

In this paper we deal with iterative numerical methods to solve linear systems arising in continuous-time Markov chain (CTMC) models. We develop an algorithm to dynamically tune the relaxation parameter of the successive over-relaxation method. We give a sufficient condition for the Gauss–Seidel method to converge when computing the steady-state probability vector of a finite irreducible CTMC, and a sufficient condition for the generalized minimal residual projection method not to converge to the trivial solution 0 when computing that vector. Finally, we compare several splitting-based iterative methods and a variant of the generalized minimal residual projection method.     

Integrating synchronization with priority into a Kronecker representation

The compositional representation of a Markov chain using Kronecker algebra, according to a compositional model representation as a superposed generalized stochastic Petri net or a stochastic automata network, has been studied for a while. In this paper we describe a Kronecker expression and associated data structures, that allows to handle nets with synchronization over activities of different levels of priority. New algorithms for these structures are provided to perform an iterative solution method of Jacobi or Gauss–Seidel type. These algorithms are implemented in the APNN Toolbox. We use this implementation in combination with GreatSPN and exercise an example that illustrates characteristics of the presented algorithms.     

Implicit multigrid schemes for challenging aerodynamic simulations on block-structured grids

In this paper, the use of implicit multigrid smoothers for challenging aerodynamic simulations is explored. The block lower–upper symmetric Gauss–Seidel (LU-SGS) and hybrid Runge–Kutta/LU-SGS schemes are implemented in Bombardier’s multiblock Navier–Stokes solver, FANSC. The schemes are compared to the existing Runge–Kutta and point-Jacobi preconditioned explicit multistage smoothers. Through tests ranging from 2D airfoils to 3D wing-body-engine cases, the computational speed-up and robustness of the implicit schemes are evaluated. It is shown that the implicit smoothers present a computational speed-up of at least two, and are significantly more robust, especially for flow problems involving the “power-on” engine boundary condition.     

High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.     

An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation

An alternating direction implicit (ADI) fractional trapezoidal rule (FTR) type difference scheme is formulated and analysed for a two-dimensional fractional evolution equation. In this method, standard central difference approximation used for the spatial discretization and the time stepping – an ADI scheme based on FTR, combined with chosen second-order fractional quadrature rule suggested by Lubich, are considered. The L², H¹-stability and convergence are derived. Numerical experiments in total agreement with our analysis are reported.     

A rational high-order compact ADI method for unsteady convection–diffusion equations

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.     

Approximate factorization for a viscous wave equation

A general procedure to construct ADI methods for multidimensional problems was originated by Beam and Warming using the method of approximate factorization. In this paper, we extend the method of approximate factorization to solve a viscous wave equation. The method can be combined with any implicit linear multistep method for the time integration of the wave equation. The stability of the factored schemes and their underlying schemes is analyzed based on a discrete Fourier analysis and the energy method. Convergence proofs are presented and numerical results supporting our analysis are provided.     

Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation,stability, and error analysis

Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat). The approximation at the grid points near the fine and coarse grid interface is based on the approach proposed by the authors in a previous paper for selfadjoint elliptic equations. The proposed schemes are implicit of backward Euler type and are shown to be unconditionally stable. Error analysis is also presented.     

A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.     

Improved accuracy for the approximate factorization of parabolic equations

A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.     

Tailored Finite Point Method for First Order Wave Equation

Following the idea of the tailored finite point method proposed in Han and Huang (J. Comput. Math. 26:728–739, 2008) and Huang (Netw. Heterog. Media 4:91–106, 2009), a series of efficient numerical schemes are developed for the one dimensional scalar wave equation within various types of media. Stability and accuracy are analyzed and numerically verified. In particular we can obtain unconditionally stable implicit schemes that can be solved explicitly for boundary value problems. We can also deal with the propagation of discontinuity and highly oscillatory waves efficiently. The generalization to higher order schemes is straightforward.     

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

In this paper, a combined compact finite difference method (CCD) together with alternating direction implicit (ADI) scheme is developed for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. The proposed CCD-ADI method is second-order accurate in time variable and sixth-order accurate in space variable. For the linear hyperbolic equation, the CCD-ADI method is shown to be unconditionally stable by using the Von Neumann stability analysis. Numerical results for both linear and nonlinear hyperbolic equations are presented to illustrate the high accuracy of the proposed method.     

Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation

In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann–Oden and for the symmetric DG method, we give a detailed analysis of the convergence for cell- and point-wise block-relaxation strategies.We show that, for a suitably constructed two-dimensional polynomial basis, point-wise block partitioning gives much better results than the classical cell-wise partitioning. Independent of the mesh size, for Poissons equation, simple MG cycles with block-Gauss–Seidel or symmetric block-Gauss–Seidel smoothing, yield a convergence rate of 0.4–0.6 per iteration sweep for both DG-methods studied.     

An implicit gas-kinetic scheme for turbulent flow on unstructured hybrid mesh

In this study, an implicit scheme for the gas-kinetic scheme (GKS) on the unstructured hybrid mesh is proposed. The Spalart–Allmaras (SA) one equation turbulence model is incorporated into the implicit gas-kinetic scheme (IGKS) to predict the effects of turbulence. The implicit macroscopic governing equations are constructed and solved by the matrix-free lower-upper symmetric-Gauss–Seidel (LU-SGS) method. To reduce the number of cells and computational cost, the hybrid mesh is applied. A modified non-manifold hybrid mesh data(NHMD) is used for both unstructured hybrid mesh and uniform grid. Numerical investigations are performed on different 2D laminar and turbulent flows. The convergence property and the computational efficiency of the present IGKS method are investigated. Much better performance is obtained compared with the standard explicit gas-kinetic scheme. Also, our numerical results are found to be in good agreement with experiment data and other numerical solutions, demonstrating the good applicability and high efficiency of the present IGKS for the simulations of laminar and turbulent flows.