LU-SGS preconditioned Newton-Krylov solver applied to industrial relevant test cases
Arpit Aggarwal, Ralf Hartmann · elib (German Aerospace Center) · 2019
DLR-TAU is a compressible flow solver based on finite volume methods on unstructured grids.Over the last years, this CFD code has matured and is now regularly employed on industrial applications.The working horse in TAU is the Lower Upper Symmetric Gauss-Seidel (LU-SGS) scheme [2] which is used as (single grid) solver or as smoother in an agglomeration multigrid.Including inviscid flux Jacobians, eigenvalues and an (implicit) local time step size this LU-SGS relaxation scheme was shown to be stronger than the traditionally used explicit Runge-Kutta iteration schemes.Nevertheless, these explicit and slightly implicit relaxation schemes are too weak to solve some of the more complex flow problems causing the flow solver to stall or even to diverge.Given our previous experience on implicit solvers developed in other flow solvers (cf.[4] in the context of Discontinuous Galerkin methods and [3] in the context of FV methods), we now present the development of a new fully implicit solver in the DLR-TAU code.It is based on the Backward Euler method which replaces the LU-SGS iteration as single grid solver or multigrid smoother.Applied to nonlinear steady state problems, this implicit time integration method iterates in pseudo-time and recovers Newton's method when the CFL number and thus the local time-step size tends to infinity.Each Backward Euler step involves a linear problem which is solved using the preconditioned Generalized Minimum Residual (GMRes) method.This Krylov solver is right preconditioned with a single forward and backward sweep of the LU-SGS scheme.By increasing the number of LU-SGS sweeps within one GMRes iteration step, the condition number of the linear problems can be further reduced making the iterated LU-SGS scheme a stronger preconditioner for the GMRes method allowing the use of fewer Krylov iterations and vectors (with reduced memory requirements).Each GMRes iteration requires multiplication of the system matrix with a vector.This matrix includes the mass matrix divided by the local time step size and the (complete) Jacobian matrix which is given by the derivative of the residual vector with respect to conservative variables.This matrix-vector multiplication is implemented matrix-free by approximating it with a (one-sided or symmetric) difference quotient of residual vector evaluations.Also the LU-SGS preconditioner is matrix-free such that the resulting overall LU-SGS preconditioned Newton-Krylov method is realized matrix-free.The related memory requirements are significantly lower than using an assembled (and stored) system matrix and/or matrix-based preconditioners like an Incomplete Lower-Upper (ILU) decomposition of the system matrix.Newton's method provides a very fast (ideally quadratic) convergence once the solution iterate is sufficiently close to the solution.Starting with any initial solution (e.g. with free-stream quantities), one of the main challenges is to find a solution iterate which is in the region of attraction of Newton's method.Four (so-called globalization) techniques are implemented to address this issue: Full multigrid is used as the primary solver for the computations.The solution process is started on an agglomerated coarse grid level.This coarse grid solution is then interpolated to the next finer grid level where it serves as a start solution of the implicit solver.On all but the coarsest grid level a V-cycle is applied. The Backward Euler can be viewed as a relaxation of Newton's method. The Switched Evolution Relaxation (SER) method is employed to start the solution with a small CFL number CFL init , and subsequently increase it according to CFL new = CFL init *(||R freestream || L2 /||R current || L2 ) α , where α is 0.4 for 3D turbulent cases.