The Deflation Accelerated Schwarz Method for CFD C

The Deflation Accelerated Schwarz Method for CFD C. Vuik Delft University of Technology c. vuik@ewi. tudelft. nl http: //ta. twi. tudelft. nl/users/vuik/ J. Verkaik, B. D. Paarhuis, A. Twerda TNO Science and Industry ICCS congres, Atlanta, USA May 23, 2005 23/5/2005 1

Contents • • • Problem description Schwarz domain decomposition Deflation GCR Krylov subspace acceleration Numerical experiments Conclusions 23/5/2005 2

Problem description GTM-X: • • • CFD package TNO Science and Industry, The Netherlands simulation of glass melting furnaces incompressible Navier-Stokes equations, energy equation sophisticated physical models related to glass melting 23/5/2005 3

Problem description Incompressible Navier-Stokes equations: Discretisation: Finite Volume Method on “colocated” grid 23/5/2005 4

Problem description SIMPLE method: pressurecorrection system ( ) 23/5/2005 5

Schwarz domain decomposition Minimal overlap: Additive Schwarz: 23/5/2005 6

Schwarz domain decomposition GTM-X: • inaccurate solution to subdomain problems: 1 iteration SIP, SPTDMA or CG method • complex geometries • parallel computing • local grid refinement at subdomain level • solving different equations for different subdomains 23/5/2005 7

Deflation: basic idea Problem: convergence Schwarz method deteriorates for increasing number of subdomains Solution: “remove” smallest eigenvalues that slow down the Schwarz method 23/5/2005 8

Deflation: deflation vectors + 23/5/2005 9

Deflation: Neumann problem Property deflation method: systems with have to be solved by a direct method Problem: pressure-correction matrix is singular: has eigenvector for eigenvalue 0 singular Solution: adjust non-singular 23/5/2005 10

GCR Krylov acceleration Objective: efficient solution to Additive Schwarz: Property: slow convergence Krylov acceleration GCR Krylov method: • for general matrices (also singular) • approximates in Krylov space such that • is minimal • Gram-Schmidt orthonormalisation for search directions • optimisation of work and memory usage of Gram-Schmidt: restarting and truncating 23/5/2005 11

Numerical experiments 23/5/2005 12

Numerical experiments Buoyancy-driven cavity flow 23/5/2005 13

Numerical experiments Buoyancy-driven cavity flow: inner iterations 23/5/2005 14

Numerical experiments Buoyancy-driven cavity flow: outer iterations without deflation 23/5/2005 15

Numerical experiments Buoyancy-driven cavity flow: outer iterations with deflation 23/5/2005 16

Numerical experiments Buoyancy-driven cavity flow: outer iterations varying inner iterations 23/5/2005 17

Numerical experiments Glass tank model 23/5/2005 18

Numerical experiments Glass tank model: inner iterations 23/5/2005 19

Numerical experiments Glass tank model: outer iterations without deflation 23/5/2005 20

Numerical experiments Glass tank model: outer iterations with deflation 23/5/2005 21

Numerical experiments Glass tank model: outer iterations varying inner iterations 23/5/2005 22

Numerical experiments Heat conductivity flow h=30 Wm-2 K-1 T=303 K K = 1. 0 Wm-1 K-1 K = 100 Wm-1 K-1 Q=0 Wm-2 K = 0. 01 Wm-1 K-1 T=1773 K 23/5/2005 23

Numerical experiments Heat conductivity flow: inner iterations 23/5/2005 24

Conclusions • using linear deflation vectors seems most efficient • a large jump in the initial residual norm can be observed • higher convergence rates are obtained and wall-clock time can be gained • implementation in existing software packages can be done with relatively low effort • deflation can be applied to a wide range of problems 23/5/2005 25