A multi-dimensional solver for the steady Euler equations

In this paper a numerical algorithm for the solution of the multi-dimensional steady Euler equations in conservative and non-conservative form is presented. Most existing standard and multi-dimensional schemes use flux balances with assumed constant distribution of variables along each cell edge, which interfaces two grid cells. This assumption is believed to be one of the main reasons for the limited advantage gained from multi-dimensional high order discretisations compared to standard one-dimensional ones. The present algorithm is based on the optimisation of polynomials describing the distribution of flow variables in grid cells, where only polynomials that satisfy the Euler equations in the entire grid cell can be selected. The global solution is achieved if all polynomials and by that the flow variables are continuous along edges interfacing neighbouring grid cells. A discrete approximation of a given spatial order is converged if the deviation between polynomial distributions of adjacent grid cells along the interfacing edge of the cells is minimal. Results from the present scheme between first and fifth order spatial accuracy are compared to standard first and second order Roe computations for simple test cases demonstrating the gain in accuracy for a number of sub- and supersonic flow problems.     

Some observations on multigrid convergence for convection–diffusion equations

This paper is concerned with the convergence behaviour of multigrid methods for two- dimensional discrete convection-diffusion equations. In Elman and Ramage (BIT 46:283–299, 2006), we showed that for constant coefficient problems with grid-aligned flow and semiperiodic boundary conditions, the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4 × 4 blocks, enabling a trivial computation of the norm of the iteration matrix. Here we use a similar Fourier analysis technique to investigate the individual contributions from the smoothing and approximation property matrices which form the basis of many standard multigrid analyses. As well as the theoretical results in the semiperiodic case, we present numerical results for a corresponding Dirichlet problem and examine the correlation between the two cases.     

A modified full multigrid algorithm for the Navier-Stokes equations

A modified full multigrid (FMG) method for the solution of the Navier-Stokes equations is presented. The method proposed is based on a V-cycle omitting the restriction procedure for dependent variables but retaining it for the residuals. This modification avoids possible mismatches between the mass fluxes and the restricted velocities as well as the turbulent viscosity and the turbulence quantities on the coarse grid. In addition, the pressure on the coarse grid can be constructed in the same way as the velocities. These features simplify the multigrid strategy and corresponding programming efforts. This algorithm is applied to accelerate the convergence of the solution of the Navier-Stokes equations for both laminar and high-Reynolds number turbulent flows. Numerical simulations of academic and practical engineering problems show that the modified algorithm is much more efficient than the FMG-FAS (Full Approximation Storage) method.     

An optimal one-way multigrid algorithm for discrete-time stochasticcontrol

The numerical solution of discrete-time stationary infinite-horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately, within a desired accuracy. After a discussion of problem discretization, the authors introduce a multigrid version of the successive approximation algorithm that proceeds `one way' from coarse to fine grids, and analyze its computational requirements as a function of the desired accuracy and of the discount factor. They also study the effects of a certain mixing (ergodicity) condition on the algorithm's performance. It is shown that the one-way multigrid algorithm improves upon the complexity of its single-grid variant and is, in a certain sense, optimal     

A note on convergence of semi-implicit Euler methods for stochastic pantograph equations

In the literature [1] [Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equation, J. Math. Anal. Appl. 325 (2007) 1142–1159], Fan and Liu investigated the existence and uniqueness of the solution for stochastic pantograph equation and proved the convergence of the semi-implicit Euler methods under the Lipschitz condition and the linear growth condition. Unfortunately, the main result of convergence derived by the conditions is somewhat restrictive for the purpose of practical application, because there are many stochastic pantograph equations that only satisfy the local Lipschitz condition. In this note we improve the corresponding results in the above-mentioned reference.     

Accelerating the convergence of steady adjoint equations by dynamic mode decomposition

Structural and Multidisciplinary Optimization - To improve the efficiency of adjoint-based optimization algorithms, dynamic mode decomposition (DMD) technology is used to accelerate the convergence...     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

三维Euler方程的隐式间断有限元算法

为了求解三维欧拉方程,对隐式时间离散格式间断有限元方法进行了研究。根据间断Galerkin有限元方法思想,构造内迭代SOR-LU-SGS隐式时间离散格式,结合当地时间步长技术、多重网格方法,实现了三维流场的计算。数值计算了ONERAM6机翼、大攻角尖前缘三角翼以及DLR-F4翼身组合体的亚声速绕流问题。结果表明,加入SOR内迭代步的LU-SGS隐式算法具有较大的优势,相较于GMRES算法所占用的内存少且收敛速度相当,是LU-SGS算法的三倍以上。针对三维算例,具有较好的稳定性和较高的收敛速度,能够给出准确的流场信息。与原方法相比,SOR-LU-SGS方法无论是在迭代步数上还是在CPU时间上,效率均有明显提高,适合于三维复杂流场计算。     

On the convergence of the Bl-LSQR algorithm for solving matrix equations

Karimi and Toutounian [The block least squares method for solving nonsymmetric linear system with multiple right-hand sides, Appl. Math. Comput. 177 (2006), pp. 852–862], proposed a block version of the LSQR algorithm, say Bl-LSQR, for solving linear system of equations with multiple right-hand sides. In this paper, the convergence of the Bl-LSQR algorithm is studied. We deal with some computational aspects of the Bl-LSQR algorithm for solving matrix equations. Some numerical experiments on test matrices are presented.     

An efficient algorithm for finding ideal schedules

We study the problem of scheduling unit execution time jobs with release dates and precedence constraints on two identical processors. We say that a schedule is ideal if it minimizes both maximum and total completion time simultaneously. We give an instance of the problem where the min-max completion time is exceeded in every preemptive schedule that minimizes total completion time for that instance, even if the precedence constraints form an intree. This proves that ideal schedules do not exist in general when preemptions are allowed. On the other hand, we prove that, when preemptions are not allowed, then ideal schedules do exist for general precedence constraints, and we provide an algorithm for finding ideal schedules in O(n ³) time, where n is the number of jobs. In finding such ideal schedules we resolve a conjecture of Baptiste and Timkovsky (Math. Methods Oper. Res. 60(1):145–153, 2004) Further, our algorithm for finding min-max completion-time schedules requires only O(n ³) time, while the most efficient solution to date has required O(n ⁹) time.     

An efficient multigrid preconditioner for Maxwell’s equations in micromagnetism

We consider a system of Maxwell’s and Landau-Lifshitz-Gilbert equations describing magnetization dynamics in micromagnetism. The problem is discretized by a convergent, unconditionally stable finite element method. A multigrid preconditioned Uzawa type method for the solution of the algebraic system resulting from the discretized Maxwell’s equations is constructed. The efficiency of the method is demonstrated on numerical experiments and the results are compared to those obtained by simplified models.     

An unfactored implicit scheme with multigrid acceleration for the solution of the Navier-Stokes equations

An unfactored implicit difference scheme for the steady state solution of the multidimensional Navier-Stokes equations of a compressible fluid is presented. The hyperbolic part is approximated by a high resolution scheme based on flux-vector splitting and upwind-biased differences to avoid the necessity of artificial dissipation terms and to construct a diagonal dominant solution matrix. Consequently, an iterative inversion of the solution matrix can be performed without any time step restriction. The rate of convergence is improved by using the indirect multigrid concept in form of the FAS scheme. The method is formulated for a body-fitted, curvilinear coordinate system. The computational results for laminar subsonic, transonic and supersonic steady-state flows which are compared with analytical and other numerical results as well as with experimental data illustrate the efficiency and the accuracy of the algorithm.     

Global convergence of a filter-trust-region algorithm for solving nonsmooth equations

In this paper, we present a new algorithm for solving nonsmooth equations, where the function is locally Lipschitzian. The algorithm attempts to combine the efficiency of filter techniques and the robustness of trust-region method. Global convergence for this algorithm is established under reasonable assumptions.     

Parallel multigrid method for solving elliptic equations

The proposed algorithm represents an efficient parallel implementation of the Fedorenko multigrid method and is intended for solving three-dimensional elliptic equations. Scalability is provided by the use of the Chebyshev iterations for solution of the coarsest grid equations and for construction of the smoothing procedures. The calculation results are given: they confirm the efficiency of the algorithm and scalability of the parallel code.     

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.     

Compact high-order schemes for the Euler equations

An implicit approximate factorization (AF) algorithm is constructed, which has the following characteristics.

On the convergence of spectral multigrid methods for solving periodic problems

The spectral multigrid method combines the efficiencies of the spectral method and the multigrid method. In this paper, we show that various spectral multigrid methods have constant convergence rates (independent of the number of unknowns in the linear system, to be solved) in their multilevel iterations for solving periodic problems.     

Fast Euler solver for steady,one-dimensional flows

A numerical technique to solve the Euler equations for steady, one-dimensional flows is presented. The technique is essentially implicit, but is structured as a sequence of explicit solutions for each Riemann variable separately. Each solution is obtained by integrating in the direction prescribed by the propagation of the Riemann variables. The technique is second-order accurate. It requires very few steps for convergence, and each step requires a minimal number of operations. Therefore, it is three orders of magnitude more efficient than a standard time-dependent technique. The technique works well for transonic flows and provides shock fitting with errors as small as 0.001. Results are presented for subsonic and transonic problems. Errors are evaluated by comparison with exact solutions.     

A multigrid solver for two-dimensional stochastic diffusion equations

Steady and unsteady diffusion equations, with stochastic diffusivity coefficient and forcing term, are modeled in two dimensions by means of stochastic spectral representations. Problem data and solution variables are expanded using the Polynomial Chaos system. The approach leads to a set of coupled problems for the stochastic modes. Spatial finite-difference discretization of these coupled problems results in a large system of equations, whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time. To accelerate the convergence of the iterative technique, a multigrid method, based on spatial coarsening, is implemented. Numerical experiments show good scaling properties of the method, both with respect to the number of spatial grid points and the stochastic resolution level.     

Moment preserving schemes for Euler equations

A high order accurate finite difference scheme is proposed for one-dimensional Euler equations. In the scheme a set of first three moments of each signal are preserved during the updating. The scheme is one of 5th order in space and 4th order in time. This feature is different from that in typical existing methods in which the use of the first three polynomials results in only 3rd order accuracy in space. The scheme has different features from the existing high order schemes, and the most noticeable are the simultaneous discretization both in space and time, and the use of moments of Riemann invariants instead of primitive physical variables. Numerical examples are given to show the accuracy of the scheme and its robustness for the flows involving shocks.