Residual Smoothing Technique Research Articles

Preconditioning methods for compressible flow CFD codes: Revisited

The present article aims to implement and investigate a different preconditioning method based in a three-dimensional in-house compressible CFD code that ensures the robustness and numerical stability to determine the flowfield considering low Mach number flow. The present preconditioning method involve two different methodologies developed by references [8, 32]. The CFD solver was developed to calculate the Euler and Navier-Stokes equations, numerically, for steady-state regime based on the cell-centered finite volume method (FVM) using Reynolds Averaged Navier-Stokes equations (RANS). The centered second-order scheme was used for the discretization of convective terms from momentum equations. The explicit second-order five-step Runge-Kutta scheme was employed for the time-marching procedure, using an implicit residual smoothing technique to enhance the numerical stability. A local preconditioning method was implemented due to its robustness in predicting low Mach number flows in a compressible CFD code environment. However, for low Mach number flows, near stagnation points, numerical perturbations were amplified generating a stiffness in the convergence rate, which provided an inaccurate solution and numerical stability degradation. Aiming to improve the preconditioning robustness, a flux function and a new limiter were applied to operate with the preconditioning technique based on a pressure sensor. Those corrections re-scale the eigenvectors and ensure the locality of the algorithm, which improves the numerical stability and guarantees the convergence for low-speed flows. The inviscid flow over a NACA 0012 airfoil for compressible and incompressible cases shown accurate and robust solutions. For a viscous flow over a flat plate case in the compressible and incompressible cases, the preconditioning technique purposed in this work supplied good numerical solution in agreement with the analytical solution.

A new smoothing approach for accelerating the convergence of power-law preconditioning method in steady and unsteady flows simulation

A new explicit smoothing approach is presented to improve the convergence acceleration of laminar viscous steady-state and unsteady flows simulation over airfoils (with/without considering surface air injection). This approach is developed based on Jameson's residual smoothing technique, and it is called here as a Solution Smoothing Method (SSM). This scheme is accompanied by the progressive locally power-law preconditioning method. The steady/unsteady preconditioned governing equations are solved using Jameson's cell-centered finite volume method in which the stabilization is achieved by means of the second- and fourth-order artificial dissipation terms. An implicit dual-time algorithm is used for simulation of unsteady cases and an explicit four-stage Runge–Kutta time integration with local time step is applied to achieve the steady-state condition in pseudo-time in an inner loop. Stability of SSM is analyzed on a one-dimensional linear model equation, and the effects of different Courant numbers, artificial dissipation coefficients, and smoothing parameters are investigated. Numerical results in comparison with other classic preconditioning methods show a reduction of the iteration number required to compute the steady-state and unsteady solution. Results show about 55 ∼ 77 and 40 ∼ 72% reduction in iteration number for steady and unsteady flows simulation, respectively.

An accurate and efficient gridless method based on implicit, fast, and constrained weights optimization schemes for compressible flows

An accurate and efficient gridless method is presented for calculation of inviscid compressible flows in subsonic to supersonic flows. The Taylor series least squares is used for discretization of spatial derivatives at each point. Fast gridless method using the second- and fourth-order artificial dissipation terms is applied for solving the Euler equations. Two methods are proposed for increasing the accuracy; implementing the constrained weights optimization and using high-order Taylor series expansion on the fast gridless method. The explicit and implicit dual-time schemes are applied for temporal discretization. To speed up convergence, local time stepping and residual smoothing techniques are used. The capability and accuracy of the methods are compared with other gridless methods, finite volume method and AGARD data for some test cases in subsonic, transonic and supersonic flows. Results show that the use of the constraint weights optimization increases the accuracy especially with fewer point distributions in comparison with conventional gridless methods and it is more efficient than other gridless methods.

Reconsideration of Residual Smoothing Technique for Improving the Accuracy of Approximate Solutions of CGS-type Iterative Methods

Reconsideration of Residual Smoothing Technique for Improving the Accuracy of Approximate Solutions of CGS-type Iterative Methods

A quasi-minimal residual variant of IDRstab using the residual smoothing technique

Quasi-Minimal Residual (QMR)-type methods have been developed for improving the oscillations in the residual norms of the Bi-Conjugate Gradient (Bi-CG) method and the hybrid Bi-CG methods. The IDRstab method, which combines the Induced Dimension Reduction (IDR)(s) method with higher-order stabilizing polynomials, is often more effective than IDR(s) and the hybrid Bi-CG methods, but the residual norms oscillate. In this paper, we design a QMR variant of IDRstab by using a residual smoothing technique in order to obtain smoother convergence behavior. Numerical experiments show the effectiveness of our QMR variant of IDRstab.

A multidimensional characteristic-based method for making incompressible flow calculations on unstructured grids

This paper presents a new multidimensional characteristic-based scheme (MCB) for numerical simulation of incompressible flows on unstructured grids in conjunction with the artificial compressibility method. The original MCB scheme has been proposed by the authors already and this research is the extension of the aforementioned scheme to unstructured grids. The significant difference between MCB and the conventional characteristic-based scheme (CB) is the multidimensional nature of the scheme, which allows information to propagate in any direction instead of only normal to the cell interface. In addition, local time stepping and the residual smoothing technique have been used for convergence acceleration. The accuracy and utility of the proposed scheme have been studied by means of numerical tests for different Reynolds numbers and the results obtained using the new scheme are in good agreement with the standard benchmark solution in the literature.

GMRES implementations and residual smoothing techniques for solving ill-posed linear systems

There are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations.

Quasi-minimal residual smoothing technique for the IDR($s$) method

The IDR( s) proposed by Sonneveld and Gijzen is an efficient method for solving large nonsymmetric linear systems. In this paper, QMRIDR( s), a new variant of the IDR( s) method is presented. In this method, the irregular convergence behavior of IDR( s) is remedied and both the fast and smooth convergence behaviors are expected. Numerical experiments are reported to show the performance of our method.

Convergence Acceleration for Solving the Compressible Navier-Stokes Equations

A fully implicit operator for implicit smoothing of residuals in the framework of explicit Runge-Kutta time stepping and multigrid acceleration is developed. To avoid memory overhead associated with storing the complete flux Jacobians, the Jacobians are expressed in terms of Mach number to enable economic computation of all flux Jacobians during iteration. The implicit operator allows increasing Courant-Friedrichs-Lewy numbers of the basic explicit scheme to the order of 100, and it properly addresses the stiffness in the discrete equations associated with highly stretched meshes. The proposed method was applied to different cases of viscous, turbulent airfoil flow, and convergence rates ranging from 0.8 to 0.87 were achieved. Comparing the present scheme to well-tuned state-of-the-art methods using common implicit residual smoothing techniques, CPU time is better than halved.

Interpolating functions of matrices on zeros of quasi-kernel polynomials

The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kernel polynomials are considered. Theoretical results as well as numerical experiments illustrate the effectiveness of our approach. Copyright © 2004 John Wiley & Sons, Ltd.

Residual Smoothing Techniques: Do They Improve the Limiting Accuracy of Iterative Solvers?

Many iterative methods for solving linear systems, in particular the biconjugate gradient (BiCG) method and its “squared” version CGS (or BiCGS), produce often residuals whose norms decrease far from monotonously, but fluctuate rather strongly. Large intermediate residuals are known to reduce the ultimately attainable accuracy of the method, unless special measures are taken to counteract this effect. One measure that has been suggested is residual smoothing: by application of simple recurrences, the iterates x n and the corresponding residuals r n :≡ b − Ax n are replaced by smoothed iterates y n and corresponding residuals s n :≡ b− Ay n. We address the question whether the smoothed residuals can ultimately become markedly smaller than the primary ones. To investigate this, we present a roundoff error analysis of the smoothing algorithms. It shows that the ultimately attainable accuracy of the smoothed iterates, measured in the norm of the corresponding residuals, is, in general, not higher than that of the primary iterates. Nevertheless, smoothing can be used to produce certain residuals, most notably those of the minimum residual method, with higher attainable accuracy than by other frequently used algorithms.

Acceleration and stabilization properties of minimal residual smoothing technique in multigrid

We analyze the standard multigrid method accelerated by a minimal residual smoothing (MRS) technique. We show that MRS can accelerate the convergence of the slow residual components, thus accelerates the overall multigrid convergence. We prove that, under certain hypotheses, MRS stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration. The analysis is customarily conducted on the two-level method.

Two-grid analysis of minimal residual smoothing as a multigrid acceleration technique

We analyze the two-level method accelerated by a minimal residual smoothing (MRS) technique. The two-grid analysis is sufficient for our purpose because our MRS acceleration scheme is only applied on the finest level of the multigrid method. We prove that the MRS acceleration scheme is a semi-iterative method with respect to the underlying two-level iteration and that the MRS accelerated two-level method is a polynomial acceleration of first order. We explain why MRS may not effectively accelerate standard multigrid method for solving Poisson-like problems. The iteration matrices for the MRS accelerated coarse-grid-correction operator and the MRS accelerated two-level operator are obtained. We give bounds for the residual reduction rates of the accelerated two-level method. Numerical experiments are employed to support the analytical results.

Comparison of Krylov-type methods for complex linear systems applied to high-voltage problems

Krylov-subspace methods for complex-valued systems are still a present research topic in numerical mathematics. Due to the lack of theoretical results systematic numerical experiments are very important in order to find a sufficiently robust solution method. An important problem in high-voltage power plants are arc-overs on moist or contaminated insulators. A quantitative knowledge of the electric behaviour of layers of water droplets is necessary in order to understand the resulting ageing processes. Discretization of the electro-quasistatic problem leads to large sparse complex systems of linear equations. Comparisons between different Krylov-subspace methods, also in combination with residual smoothing techniques, and parameter studies are presented in this paper.

Multigrid with inexact minimal residual smoothing acceleration

We introduce some inexact versions of the minimal residual smoothing (IMRS) technique to accelerate the standard multigrid convergence. These are modified versions of the minimal residual smoothing (MRS) technique introduced in a recent paper (Zhang, to appear). The IMRS acceleration schemes minimize the residual norm of the multigrid iterate in a subspace and reduce the cost of the standard MRS acceleration by about 40% for two-dimensional problems and frequently achieve even faster convergence. Numerical experiments are employed to compare the performance of the exact and the inexact minimal residual smoothing schemes.

Minimal residual smoothing in multi-level iterative method

A minimal residual smoothing (MRS) technique is employed to accelerate the convergence of the multi-level iterative method by smoothing the residuals of the original iterative sequence. The sequence with smoothed residuals is re-introduced into the multi-level iterative process. The new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi-level method and the sequence generated by MRS technique. The cost of this acceleration scheme is independent of the original operator and in many cases is negligible. The emphasis of this paper is on the practical implementation of MRS acceleration techniques in the multi-level method. The discussions are focused on the two-level method because the acceleration scheme is only applied on the finest level of the multi-level method. Numerical experiments using the proposed MRS acceleration scheme to accelerate both the two-level and multi-level methods are conducted to show the efficiency and the cost-effectiveness of this acceleration scheme.

Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number

A fourth-order compact finite difference scheme is employed with the multigrid algorithm to obtain highly accurate numerical solution of the convection-diffusion equation with very high Reynolds number and variable coefficients. The multigrid solution process is accelerated by a minimal residual smoothing (MRS) technique. Numerical experiments are employed to show that the proposed multigrid solver is stable and yields accurate solution for high Reynolds number problems. We also show that the MRS acceleration procedure is efficient and the acceleration cost is negligible. © 1997 John Wiley & Sons, Inc.

Residual Smoothing Techniques for Iterative Methods

An iterative method for solving a linear system $Ax = b$ produces iterates $\{ x_k \} $ with associated residual norms that, in general, need not decrease “smoothly” to zero. “Residual smoothing” techniques are considered that generate a second sequence $\{ y_k \} $ via a simple relation $y_k = (1 - \eta _k )y_{k - 1} + \eta _k x_k $. The authors first review and comment on a technique of this form introduced by Schonauer and Weiss that results in $\{ y_k \} $ with monotone decreasing residual norms; this is referred to as minimal residual smoothing Certain relationships between the residuals and residual norms of the biconjugate gradient (BCG) and quasi-minimal residual (QMR) methods are then noted, from which it follows that QMR can be obtained from BCG by a technique of this form; this technique is extended to generally applicable quasi-minimal residual smoothing. The practical performance of these techniques is illustrated in a number of numerical experiments.

Solutions of the Euler equations for transonic and supersonic aircraft

A method for numerically computing steady incompressible flow fields about fighter-type aircraft at transonic and supersonic speeds is described. Discretization of the physical space around the aircraft is accomplished with a single-block H-O type mesh. The compressible Euler equations are discretized on this mesh using a fully conservative finite-volume method and are integrated to steady state via a five-stage Runge-Kutta scheme. The use of a maximum local time step, together with a residual smoothing technique and enthalpy damping, speeds up the solution process. Additional acceleration of convergence toward the steady state is provided by cycling the entire numerical scheme through coarser meshes in a multigrid sequence. The method is efficient and reliable. Examples of flow fields computed on a variety of aircraft at subsonic, transonic, and supersonic speeds are included.