High-Order Flux Correction for Viscous Flows on Arbitrary Unstructured Grids

A novel high-order method, termed flux correction, previously formulated for inviscid flows, is extended to viscous flows on arbitrary triangular grids. The correction method involves the addition of truncation error-canceling terms to the second-order linear Galerkin (node-centered finite volume) scheme to produce a third-order inviscid and fourth-order viscous scheme. The correction requires minimal modification of the underlying second-order scheme. As such, the method retains many of the advantages of traditional finite volume schemes, including robust shock capturing, low algorithmic complexity, and solver efficiency. In addition, we extend the scheme to unsteady flows. Verification and validation studies in two dimensions are presented. Significant improvement in accuracy is observed in all cases, with between 30–70 % increase in computational cost over a second-order finite volume method.     

Algebraic and Parametric Solvers for the Power Flow Problem: Towards Real-Time and Accuracy-Guaranteed Simulation of Electric Systems

The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers for the power flow equations, in both algebraic and parametric version. The application of the Alternating Search Direction method to the power flow problem is also detailed. This results in a family of iterative solvers that combined with Proper Generalized Decomposition technique allows to solve the parametric version of the equations. Once the solution is computed using this strategy, analyzing the network state or solving optimization problems, with inclusion of generation in real-time, becomes a straightforward procedure since the parametric solution is available. Complementing this approach, an error strategy is implemented at each step of the iterative solver. Thus, error indicators are used as an stopping criteria controlling the accuracy of the approximation during the construction process. The application of these methods to the model IEEE 57-bus network is taken as a numerical illustration.     

基于压力的高阶精度隐式格式的研究及应用

１．引 言 在研究自然界和工程应用领域的物理流动问题时，一般根据流体流动的马赫数范围，采用不同的数学模型．相应地，在计算流体力学（ＣＦＤ：Ｃｏｍｐｕｔａｔｉｏｎａｌ Ｆｌｕｉｄ Ｄｙｎａｍｉｃｓ）中，针对不同的数学模型，建立和发展了差分格式和求解算法．其中时间推进法与其它方法相比，在数学的严密性、物理特性和所能采用的差分格式等方面具有许多优点，是亚音速、跨音速和超音速等可压缩流动问题中应用非常广泛的算法．尤其自七十年代以来，随着高分辨率高精度差分格式和高计算效率时间推进格式的建立和发展，时间推进法在…     

Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations

We propose a new third-order multidimensional upwind algorithm for the solution of the flow equations on tetrahedral cells unstructured grids. This algorithm has a compact stencil (cell-based computations) and uses a finite element idea when computing the residual over the cell to achieve its third-order (spatial) accuracy. The construction of the new scheme is presented. The asymptotic accuracy for steady or unsteady, inviscid or viscous flow situations is proved using numerical experiments. The new high-order discretization proves to have excellent parallel scalability. Our studies show the advantages of the new compact third-order scheme when compared with the classical second-order multidimensional upwind schemes.     

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.     

双曲型守恒律的一种三阶松弛格式

对一维双曲型守恒律,给出了一种形式更简单、计算量更小的三阶松弛格式.该格式以三阶WENO重构和三阶显隐式Runge-Kutta方法为基础.由于不用求解Riemann问题和计算非线性通量函数的雅可比矩阵,所以本文格式保持了松弛格式简单的优点.数值试验表明:该方法具有较高的分辨率.     

Adjoint-based error estimation and grid adaptation for functional outputs: Application to two-dimensional, inviscid, incompressible flows

Adjoint-based error estimation and grid adaptive procedures are investigated for their robustness and effectiveness in improving the accuracy of functional outputs such as lift and drag. The adjoint error estimates relate the global error in the output function to the local residual errors in the flow solution via adjoint variables as weight functions. These error estimates are used as a correction to produce improved functional estimates. Based on this error correction procedure, two output-based grid adaptive approaches are implemented and compared. While both approaches strive to improve the accuracy of the computed output, the means by which the adaptation parameters are formed differ. The first approach strives to improve the computable error estimates by forming adaptation parameters based on the level of error in the computable error estimates. The second approach uses the computable error estimates as adaptation parameters. Grid adaptation is performed with h-refinement and results are presented for two-dimensional, inviscid, incompressible flows.     

A Not-a-Knot meshless method using radial basis functions and predictor–corrector scheme to the numerical solution of improved Boussinesq equation

A numerical simulation of the improved Boussinesq (IBq) equation is obtained using collocation and approximating the solution by radial basis functions (RBFs) based on the third-order time discretization. To avoid solving the nonlinear system, a predictor–corrector scheme is proposed and the Not-a-Knot method is used to improve the accuracy in the boundary. The method is tested on two problems taken from the literature: propagation of a solitary wave and interaction of two solitary waves. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the new scheme presented in this paper.     

Verification of RANS solvers with manufactured solutions

This paper discusses code verification of Reynolds-Averaged Navier Stokes (RANS) solvers with the method of manufactured solutions (MMS). Examples of manufactured solutions (MSs) for a two-dimensional, steady, wall-bounded, incompressible, turbulent flow are presented including the specification of the turbulence quantities incorporated in several popular eddy-viscosity turbulence models. A wall-function approach for the MMS is also described. The flexiblity and usefulness of the MS is illustrated with calculations performed in three different exercises: the calculation of the flow field using the manufactured eddy-viscosity; the calculation of the eddy-viscosity using the manufactured velocity field; the calculation of the complete flow field coupling flow and turbulence variables. The results show that the numerical performance of the flow solvers is model dependent and that the solution of the complete problem may exhibit different orders of accuracy than in the exercises with no coupling between the flow and turbulence variables.     

Adaptive multigrid for finite element computations in plasticity

The solution of the system of equilibrium equations is the most time-consuming part in large-scale finite element computations of plasticity problems. The development of efficient solution methods are therefore of utmost importance to the field of computational plasticity. Traditionally, direct solvers have most frequently been used. However, recent developments of iterative solvers and preconditioners may impose a change. In particular, preconditioning by the multigrid technique is especially favorable in FE applications.The multigrid preconditioner uses a number of nested grid levels to improve the convergence of the iterative solver. Prolongation of fine-grid residual forces is done to coarser grids and computed corrections are interpolated to the fine grid such that the fine-grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time at low memory requirements. By means of a posteriori error estimates the computational grid could successively be refined (adapted) until the solution fulfils a predefined accuracy level. In contrast to procedures where the preceding grids are erased, the previously generated grids are used in the multigrid algorithm to speed up the solution process.The paper presents results using the adaptive multigrid procedure to plasticity problems. In particular, different error indicators are tested.     

A two-level computational graph method for the adjoint of a finite volume based compressible unsteady flow solver

The adjoint method is a useful tool for finding gradients of design objectives with respect to system parameters for fluid dynamics simulations. But the utility of this method is hampered by the difficulty in writing an efficient implementation for the adjoint flow solver, especially one that scales to thousands of cores. This paper demonstrates a Python library, called adFVM, that can be used to construct an explicit unsteady flow solver and derive the corresponding discrete adjoint flow solver using automatic differentiation (AD). The library uses a two-level computational graph method for representing the structure of both solvers. The library translates this structure into a sequence of optimized kernels, significantly reducing its execution time and memory footprint. Kernels can be generated for heterogeneous architectures including distributed memory, shared memory and accelerator based systems. The library is used to write a finite volume based compressible flow solver. A wall clock time comparison between different flow solvers and adjoint flow solvers built using this library and state of the art graph based AD libraries is presented on a turbomachinery flow problem. Performance analysis of the flow solvers is carried out for CPUs and GPUs. Results of strong and weak scaling of the flow solver and its adjoint are demonstrated on subsonic flow in a periodic box.     

Uniformly Effective Numerical Methods for Hyperbolic Systems

A unified method to compute compressible and incompressible flows is presented. Accuracy and efficiency do not degrade as the Mach number tends to zero. A staggered scheme solved with a pressure correction method is used. The equation of state is arbitrary. A Riemann problem for the barotropic Euler equations with nonconvex equation of state is solved exactly and numericaly. A hydrodynamic flow with cavitation in which the Mach number varies between 10⁻³ and 20 is computed. Unified methods for compressible and incompressible flows are further discussed for the flow of a perfect gas. The staggered scheme with pressure correction is found to have Mach-uniform accuracy and efficiency, and for the fully compressible case the accuracy is comparable with that of established schemes for compressible flows. Received October 20, 1999; revised May 26, 2000     

A Fourth Order Scheme for Incompressible Boussinesq Equations

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.     

An adaptive method of lines solution of the Korteweg-de Vries equation

Following a method of lines formulation, the Korteweg-de Vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle, which allows the number of nodes to be significantly reduced as compared to a fixed-grid solution. Several finite difference schemes, including direct and stagewise procedures, are compared and the results of a large number of computational experiments are presented, which demonstrate that the selection of a spatial approximation scheme for the third-order derivative term is the primary determinant of solution accuracy.     

Two level algorithms for partitioned fluid-structure interaction computations

In this paper we use the multigrid algorithm - commonly used to improve the efficiency of the flow solver - to improve the efficiency of partitioned fluid-structure interaction iterations. Coupling not only the structure with the fine flow mesh, but also with the coarse flow mesh (often present due to the multigrid scheme) leads to a significant efficiency improvement. As solution of the flow equations typically takes much longer than the structure solve, and as multigrid is not standard in structure solvers, we do not coarsen the structure or the interface. As a result, the two level method can be easily implemented into existing solvers.Two types of two level algorithms were implemented: (1) coarse grid correction of the partitioning error and (2) coarse grid prediction or full multigrid to generate a better initial guess. The resulting schemes are combined with a fourth-order Runge-Kutta implicit time integration scheme. For the linear, one-dimensional piston problem with compressible flow the superior stability, accuracy and efficiency of the two level algorithms is shown. The parameters of the piston problem were chosen such that both a weak and a strong interaction case were obtained.Even the strong interaction case, with a flexible structure, could be solved with our new two level partitioned scheme with just one iteration on the fine grid. This is a major accomplishment as most weakly coupled methods fail in this case. Of the two algorithms the coarse grid prediction or full multigrid method was found to perform best. The resulting efficiency gain for our one-dimensional problem is around a factor of ten for the coarse to intermediate time steps at which the high-order time integration methods should be run. For two- and three-dimensional problems the efficiency gain is expected to be even larger.     

A streamlined approximation and duality scheme for structural optimization

An efficient and commonly used approach to structural optimization is to solve a sequence of approximate design problems that are constructed iteratively. As is well-known, a major part of the computational burden of this scheme lies in the sensitivity analysis needed to state the approximate problems. We propose a possibility for reducing this burden by streamlining the calculations in a combined approximation and duality scheme for structural optimization. The difference between this scheme and the traditional one is that, instead of calculating all the constraint gradients to state an approximate design problem explicitly, linear combinations of these gradients are generated as they are needed during the solution of the approximate problem by the dual method. We show, by analysing some typical scenarios of problem characteristics, that this rearrangement of the calculations may be a computationally viable alternative to the traditional scheme. An advantage of streamlining the calculations is that there is no need to incorporate an active set strategy in the scheme, as is usually done, since all the design constraints may be taken into consideration without any loss of computational efficiency. This may, clearly, enhance the practical rate of convergence of the overall approximation scheme. Moreover, the proposed rearrangement of the calculations may make it computationally viable to apply iterative equation solvers to the structural analysis problem. Numerical results with direct as well as iterative equation solvers show that the streamlined scheme is a feasible and promising approach to structural optimization.     

A new L-stable Simpson-type rule for the diffusion equation

The well-known Simpson rule is an optimal two-step fourth-order method which is unconditionally unstable. In the present paper we describe a new L-stable version of the method. A suitable combination of the arithmetic average approximation with the explicit backward Euler formula provides a third-order approximation at the midpoint which, when plugged into the Simpson rule, gives a third-order L-stable scheme. The L-stable Simpson-type rule (LSIMP3) obtained is then employed to derive a third-order time integration scheme for the diffusion equation. Numerical illustrations are provided to compare the performance of the new LSIMP3 scheme with the Crank–Nicolson scheme. While the Crank–Nicolson scheme can produce unacceptable oscillations in the computed solution, the present LSIMP3 scheme can provide both stable and accurate approximations.     

Implementation of density-based solver for all speeds in the framework of OpenFOAM

In the framework of open source CFD code OpenFOAM, a density-based solver for all speeds flow field is developed. In this solver the preconditioned all speeds AUSM+(P) scheme is adopted and the dual time scheme is implemented to complete the unsteady process. Parallel computation could be implemented to accelerate the solving process. Different interface reconstruction algorithms are implemented, and their accuracy with respect to convection is compared. Three benchmark tests of lid-driven cavity flow, flow crossing over a bump, and flow over a forward-facing step are presented to show the accuracy of the AUSM+(P) solver for low-speed incompressible flow, transonic flow, and supersonic/hypersonic flow. Firstly, for the lid driven cavity flow, the computational results obtained by different interface reconstruction algorithms are compared. It is indicated that the one dimensional reconstruction scheme adopted in this solver possesses high accuracy and the solver developed in this paper can effectively catch the features of low incompressible flow. Then via the test cases regarding the flow crossing over bump and over forward step, the ability to capture characteristics of the transonic and supersonic/hypersonic flows are confirmed. The forward-facing step proves to be the most challenging for the preconditioned solvers with and without the dual time scheme. Nonetheless, the solvers described in this paper reproduce the main features of this flow, including the evolution of the initial transient.     

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier-Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.     

Wave Equation Numerical Resolution: A Comprehensive Mechanized Proof of a C Program

We formally prove correct a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. Such an implementation introduces errors at several levels: the numerical scheme introduces method errors, and floating-point computations lead to round-off errors. We annotate this C program to specify both method error and round-off error. We use Frama-C to generate theorems that guarantee the soundness of the code. We discharge these theorems using SMT solvers, Gappa, and Coq. This involves a large Coq development to prove the adequacy of the C program to the numerical scheme and to bound errors. To our knowledge, this is the first time such a numerical analysis program is fully machine-checked.