TAU Solver Improvement [Implicit methods] Richard Dwight Megadesign 23-24 May 2007 Folie 1 > Vortrag > Autor
Outline Motivation (convergence acceleration to steady state, fast unsteady) Implicit methods for stationary problems Characterization Investigation of specific methods Time-accurate schemes: Semi-Implicit Runge-Kutta (SIRK) methods (with Ursula Mayer). Conclusions Folie 2 > Vortrag > Autor
Folie 3 > Vortrag > Autor Begin with the semi-discrete problem with residual R: Use the backward-euler scheme in time: Taylor expansion of non-linear term: Yields, after rearranging the scheme: Which is a linear system in ΔW: 0 ) ( ) ( ) ( 2 = Δ + + Δ + Δ t O W R W W W R t n i n j j n i δ ij R W A = Δ ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) ( 2 1 t O t dt dw W W R W R t O t dt W dr W R W R i N j j j n i n i n i n i n i Δ + Δ + = Δ + Δ + = + Derivation of an Implicit Method 0 ) ( = + W R dt dw i i 0 ) ( 1 1 = + Δ + + n n n R W t W W
Characterization of Implicit Methods Parameter space Not to scale!!! Ω Δt + R W ΔW n = R( W n ) Newton method: Exact, complete Jacobian. Exact solution of linear system. Fewer iterations Folie 4 > Vortrag > Autor
Specific Implicit Operators Newton Method - RAE2822 Case 09 Folie 5 > Vortrag > Autor
Specific Implicit Operators Newton Method (RAE2822, Case 09) Start-up/stability probs. Folie 6 > Vortrag > Autor
Specific Implicit Operators LU-SGS - Jacobian approximation F (W L,W R,n ij )= 1 2 (F (W L; n ij )+F (W R ; n ij )) 1 2 D(W L,W R ; n ij ), R i = = 1 2 j N (i) j N (i) F (W i,w j ; n ij ) { F (Wi ) n ij } + 1 2 0 j N (i) F (W ; n) = F (W ) n, and { F (Wj ) n ij } 1 2 j N (i) n ij =0 j N (i) D(W i,w j ; n ij ) R i W i = j N (i) D(W i,w j ; n ij ) W i Folie 7 > Vortrag > Autor
Specific Implicit Operators LU-SGS Folie 8 > Vortrag > Autor
Specific Implicit Operators Matrix-Free Newton-Krylov Finite difference Jacobian. Solution with preconditioned Krylov. Folie 9 > Vortrag > Autor
Specific Implicit Operators Line Implicit Tri-diagonal Jacobian. Direct banded solver. Jacobian + solver fit perfectly. Folie 10 > Vortrag > Autor
Specific Implicit Operators Something Else??? Jacobian??? Linear solution??? Folie 11 > Vortrag > Autor
Specific Implicit Operators "Better" LU-SGSs "Better" Jacobian or "Better" linear solver Folie 12 > Vortrag > Autor
Specific Implicit Operators SGS and Jacobi versions of LU-SGS Same Jacobian as LU-SGS Multiple symmetric Gauss-Seidel sweeps, SGS(n) or Multiple Jacobi iterations, Jacobi(n) Folie 13 > Vortrag > Autor
Specific Implicit Operators SGS and Jacobi versions of LU-SGS Folie 14 > Vortrag > Autor
Specific Implicit Operators Existence of "Holy Grail" Jameson's "Solution in 5 multigrid cycles"??? Swanson's "Optimal multigrid"??? (not an implicit method). Folie 15 > Vortrag > Autor
Specific Implicit Operators Newton Method vs Approximate Newton Method Jacobian based on 1 st -order flux Solution: Krylov, GS, etc. Folie 16 > Vortrag > Autor
Specific Implicit Operators FOKI - Carefully choosen approximate Newton method Folie 17 > Vortrag > Autor
Specific Implicit Operators FOKI - Carefully choosen approximate Newton method Folie 18 > Vortrag > Autor
Implicit Methods Conclusions Wide range of methods investigated. No "holy grail" found. Lots of implicit schemes are "much of a muchness". Compromise method LU-SGS. Some promising more accurate methods exist (FOKI); stabilization techniques needed. Work continues. Folie 19 > Vortrag > Autor
Time Accurate Schemes: Semi-Implicit Runge-Kutta (with Ursula Mayer) Folie 20 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) Explicit scheme Δt limited by CFL condition. Folie 21 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Semi-Implicit scheme (N. Nikitin) N. Nikitin, Third-Order-Accurate Semi-Implicit Runge-Kutta Scheme for Incompressible Navier-Stokes Equations, International Journal for Numerical Methods in Fluids 51, pp. 221-233, 2006. Folie 22 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Semi-Implicit scheme (N. Nikitin) => Third-order accuracy retained!!! Folie 23 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Shock Tube Problem Folie 24 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Shock Tube Problem Folie 25 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Supersonic Step Problem Folie 26 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Supersonic Step Problem Folie 27 > Vortrag > Autor
Semi-Implicit Runge-Kutta (SIRK) in TAU Conclusion The new method allows for the application of arbetrary implicit operators in 3 rd and 4 th order time accurate schemes. Factor of 10 reduction in CPU time over dual-time for supersonic step. Unconditional stability achieved without an inner iteration. Unconditional stability for Navier-Stokes requires a higher quality implicit operator. Folie 28 > Vortrag > Autor
Thank you for your attention Folie 29 > Vortrag > Autor