GMRES acceleration of kinetic flux vector split (KFVS) Eulercodes

Abstract

The present work is concerned with the acceleration of Euler codes using the Generalized Minimal Residual (GMRES) algorithm. The codes taken up for acceleration are 2D cellcentred, timemarching, finitevolume explicit Euler codes based on the Kinetic Flux Vector Splitting (KFVS) scheme of Deshpande. The GMRES algorithm, originally developed by Saad and Schultz, is used to solve systems of linear algebraic equations involving nonsymmetric matrices. This is an iterative method involving projection onto the Krylov subspace. For symmetric matrices, other Krylovsubspace methods perform better than GMRES; however, for nonsymmetric matrices, GMRES has a clear advantage. This suitability has led to its application in CFD problems, where nonsymmetric matrices frequently arise after the linearization of nonlinear algebraic systems. One novel feature of this work is the analytical evaluation of the Jacobian-vector product (i.e., directional derivatives). The nonlinear GMRES method is essentially an inexact Newton method in which the Newton equations are solved approximately by GMRES at each outer iteration. To determine the search directions, GMRES requires Jacobian-vector products. In a Jacobianfree implementation, this product is approximated by a finitedifference quotient computed numerically, and the search directions (forming an orthonormal basis of the Krylov subspace generated by the initial residual and the system Jacobian) are computed using this numerical approximation. In contrast, this work develops subroutines to compute the Jacobian-vector product analytically and solves the minimization problem through QR factorization, without relying on external packages such as LINPACK or MINPACK. Analytical evaluation not only reduces the computational cost of generating search directions but also improves the orthogonality of the basis vectors compared to the approximate Jacobianfree approach. Better orthogonality, in turn, results in faster convergence. Another important contribution of this work is a novel preconditioning method. Preconditioning, which involves formulating an equivalent problem with the same solution but with a more tightly clustered eigenvalue spectrum, is crucial in CFD since it significantly accelerates the decay of residuals. Unlike conventional preconditioners, this work uses an extremely simple but highly effective method: the explicit timemarching KFVS code itself is used as a preconditioner for the system subsequently solved by GMRES. This leads to a substantial acceleration of the original CFD codes. As a result of the analytical Jacobian-vector product evaluation and the new preconditioning method, this work achieves significant acceleration of the explicit Euler codes tested-speedups as high as 7 in one case. This opens up the possibility of accelerating many explicit CFD codes currently used in aerospace organizations within the country and elsewhere, using GMRES.

Outline of the Thesis Chapter 1 Discusses various aspects of CFD, highlighting the need for convergenceacceleration methods-particularly in the Indian context. A brief review of existing acceleration techniques is provided, along with the motivation and objectives of the present work. The codes chosen for acceleration are also briefly described. Chapter 2 Explains the basics of the KFVS method, including the expressions for split fluxes for the cellcentred finitevolume formulation of the 2D Euler equations. Residual expressions in terms of conserved variables are derived; these are used in the analytical evaluation of directional derivatives. Sample results from the KFVS codes used in this study are also presented. Chapter 3 Covers the theoretical aspects of GMRES. Linear GMRES is described in detail, as it forms the basis of the nonlinear version. Newton’s method and inexact Newton methods for nonlinear systems are outlined. It is shown that nonlinear GMRES belongs to the class of inexact Newton methods. The GMRES algorithm for nonlinear problems is presented stepbystep. Chapter 4 Describes the method adopted to compute directional derivatives analytically, and presents the novel preconditioning method used to accelerate explicit codes. Differences between existing and proposed preconditioning approaches are highlighted. The preconditioned GMRES code and its memory requirements are briefly discussed. Chapter 5 Presents the results of numerical experiments. A systematic analysis of the results demonstrates that KFVSpreconditioned GMRES accelerates the explicit codes by a factor of four to seven.