Using exact Jacobians in an implicit Newton–Krylov method

Bramkamp, F.; Bücker, H. Martin; Rasch, Arno

doi:10.1016/j.compfluid.2005.10.003

Cited by 15 publications

(5 citation statements)

References 34 publications

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“…where ∆ q n = q n+1 − q n , in which the superscripts n and n + 1 denote the current and the next time steps, respectively. ∂R n ∂ q is the Jacobian matrix of the DG spatial discretization, which is computed analytically (except for the dissipative part of the numerical flux using divided differencing [34]) in this paper.…”

Section: Implicit Lu-sgs Schemementioning

confidence: 99%

A High-Order Discontinuous Galerkin Method for Solving Preconditioned Euler Equations

et al. 2022

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A high-order discontinuous Galerkin (DG) method is presented for solving the preconditioned Euler equations with an explicit or implicit time marching scheme. A detailed description is given of a practical implementation of a precondition matrix of the type of Weiss and Smith and of the DG spatial discretization scheme employed, with particular emphasis on the artificial viscosity-based shock capturing techniques. The curved boundary treatment is proposed through adopting a NURBS surface equipped with a radial basis function interpolation to propagate the boundary displacement to the interior of the mesh. The resulting methods are verified by simulating flows over two-dimensional airfoils, such as symmetric NACA0012 or asymmetric RAE2822, and over three-dimensional bodies, such as an academic hemispherical headform or aerodynamic ONERA M6 wing. Numerical results show that the present method functions for both transonic and nearly incompressible flow simulations, and the proposed treatment of curved boundaries, play an important role in improving the accuracy of the obtained solutions, which are in good agreement with available experimental data or other numerical solutions reported in literature.

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Section: Implicit Lu-sgs Schemementioning

confidence: 99%

A High-Order Discontinuous Galerkin Method for Solving Preconditioned Euler Equations

et al. 2022

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“…The NK algorithm has long been used for solving the RANS equations for external aerodynamic analysis in two and three dimensions [21][22][23][24][25][26]. These early works mainly dealt with relatively benign conditions where the flow is mostly attached.…”

Section: Introductionmentioning

confidence: 99%

Newton–Krylov Solver for Robust Turbomachinery Aerodynamic Analysis

Mohanamuraly

Wang³

et al. 2020

AIAA Journal

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Steady computational fluid dynamics solvers based on the Reynolds-averaged Navier-Stokes equations are the primary workhorse for turbomachinery aerodynamic analysis due to their good engineering accuracy at a low computational cost. However, even state-of-the-art steady solvers suffer from convergence slowdown or failure when applied to challenging off-design conditions. This severely limits the reliable nonlinear and linearized turbomachinery aerodynamic analysis over a wide operating range. To alleviate the convergence difficulties, a nonlinear flow solver using the Newton-Krylov method is developed. This is the first time the Newton-Krylov

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“…This method can accelerate convergence by more than an order of magnitude relative to an explicit solver [50]. AD was reported to improve the performance and robustness of a finite volume scheme by using a set of techniques for computing accurate derivatives of non-linear functions compared to an approximate one [51]. Also, the vertex-elimination, automatic differentiation approach was used to calculate the Jacobians of functions as efficient as hand-coded Jacobian [52].…”

Section: Introductionmentioning

confidence: 99%

“…However, there is no analytical Jacobian, to the best of the authors’ knowledge, derived or used for fully transformed Navier-Stokes equations on staggered grids. In fact, the derivation of an analytical Jacobian for advanced schemes, such as upwind methods, can become very complex as recognized by several researchers [60, 51, 61]. …”

Section: Introductionmentioning

confidence: 99%

A Newton–Krylov method with an approximate analytical Jacobian for implicit solution of Navier–Stokes equations on staggered overset-curvilinear grids with immersed boundaries

Asgharzadeh

Borazjani

2017

Journal of Computational Physics

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The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42 – 74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal Jacobian when the stretching factor was increased, respectively. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80–90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

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Using exact Jacobians in an implicit Newton–Krylov method

Cited by 15 publications

References 34 publications

A High-Order Discontinuous Galerkin Method for Solving Preconditioned Euler Equations

A High-Order Discontinuous Galerkin Method for Solving Preconditioned Euler Equations

Newton–Krylov Solver for Robust Turbomachinery Aerodynamic Analysis

A Newton–Krylov method with an approximate analytical Jacobian for implicit solution of Navier–Stokes equations on staggered overset-curvilinear grids with immersed boundaries

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