Blending Dual Time Stepping and Newton‐Krylov Methods for Unsteady Flows

Abstract

We explore the idea of using nonlinear schemes as preconditioners in Newton-Krylov schemes for unsteady flow computations. Analysis shows that left preconditioning changes the Newton scheme in a non equivalent way, leading to a stall in Newton convergence, whereas right preconditioning leads to a sound method. Finite Volume SchemeWe consider a general finite volume discretization on a nonmoving grid, which is represented by the grid function R(w), which acts on the vector of all conserved variables w:The diagonal matrix V represents the volume of the cells of the grid. As an exemplary implicit time integrator we use BDF-2 which results for a fixed timestep ∆t in the nonlinear equation system for the unknown w = w n+1which we use to define the function F(w). Dual Time steppingThe dual time stepping scheme solves the equation system (1) by adding a pseudo time derivative and computing the steady state of the following equation system: ∂w ∂t * + F(w) = 0. This is done using the nonlinear multigrid method for the computation of steady flows of Jameson et al [1]. For Euler flows, this needs only three to five multigrid steps per time step, whereas for Navier-Stokes flows, this is significantly slower and sometimes more than a hundred steps are needed for convergence. Newton-Krylov-MethodThe numerical solution of system (1) can also be done using Newton's method. One Newton step is given by:We solve this linear equation system with system matrix A = (∂w )| w (k) using matrix free Krylov subspace methods. Since Krylov subspace methods never need the matrix A explicitely, but only matrix-vector products, we circumvent the expensive computation of the Jacobian to obtain a matrix-free method. This is done by approximating all matrix vector products by finite difference approximations of directional derivatives, using a suitable epsilon:As reported by several authors, GMRES-like methods that have an optimality property are more suitable for this approach than methods like BiCGSTAB with short recurrences. We will use GMRES and GCR.