Algebraic flux correction and its application to convection-dominated flow. Matthias Möller

Transcription

1 Algebraic flux correction and its application to convection-dominated flow Matthias Möller Institute of Applied Mathematics (LS III) University of Dortmund, Germany 8th Hirschegg Workshop on Conservation Laws 9-15 September 2007, Hirschegg 1

2 Overview 1 Algebraic flux correction Motivation Design principles Flux limiting 2 Nonlinear solution algorithm Choice of preconditioners Construction of Jacobians Numerical examples 3 Recent work Application to compressible flows Grid adaptivity 4 Conclusions 2

3 Motivation Scalar transport equation u t + (vu) = 0 High-order methods: wiggles Standard Galerkin FEM M c du dt + Ku = 0 Low-order methods: smearing Remedy: nonlinear combination of high- and low-order discretizations. 3

4 Algebraic flux correction, Kuz05a High-order scheme M c du dt + Ku = 0 K = {k ij}, k ij > 0, j i Apply artificial diffusion D = {d ij }, d ij = max{k ij, 0, k ji } = d ji Low-order scheme M l du dt + Lu = 0 L = K + D, l ij 0, j i Decompose the residual difference into internodal fluxes r H i r L i = f ij, j i f ij = [ m ij d dt + d ij] (uj u i ) = f ji Nonlinear high-resolution scheme containing limited antidiffusion M l du dt + Lu = f, f i = j i α ij f ij, 0 α ij 1 4

5 Algebraic flux correction, Kuz05a High-order scheme M c du dt + Ku = 0 K = {k ij}, k ij > 0, j i Apply artificial diffusion D = {d ij }, d ij = max{k ij, 0, k ji } = d ji Low-order scheme M l du dt + Lu = 0 L = K + D, l ij 0, j i Decompose the residual difference into internodal fluxes r H i r L i = f ij, j i f ij = [ m ij d dt + d ij] (uj u i ) = f ji Nonlinear high-resolution scheme containing limited antidiffusion M l du dt + Lu = f, f i = j i α ij f ij, 0 α ij 1 4

6 Algebraic flux correction, Kuz05a High-order scheme M c du dt + Ku = 0 K = {k ij}, k ij > 0, j i Apply artificial diffusion D = {d ij }, d ij = max{k ij, 0, k ji } = d ji Low-order scheme M l du dt + Lu = 0 L = K + D, l ij 0, j i Decompose the residual difference into internodal fluxes r H i r L i = f ij, j i f ij = [ m ij d dt + d ij] (uj u i ) = f ji Nonlinear high-resolution scheme containing limited antidiffusion M l du dt + Lu = f, f i = j i α ij f ij, 0 α ij 1 4

7 Algebraic flux correction, Kuz05a High-order scheme M c du dt + Ku = 0 K = {k ij}, k ij > 0, j i Apply artificial diffusion D = {d ij }, d ij = max{k ij, 0, k ji } = d ji Low-order scheme M l du dt + Lu = 0 L = K + D, l ij 0, j i Decompose the residual difference into internodal fluxes r H i r L i = f ij, j i f ij = [ m ij d dt + d ij] (uj u i ) = f ji Nonlinear high-resolution scheme containing limited antidiffusion M l du dt + Lu = f, f i = j i α ij f ij, 0 α ij 1 4

8 Flux limiting Nonlinear high-resolution scheme (lumped-mass version) M l du dt + Lu = f, f i = j i α ij f ij, f ij = d ij (u j u i ) = f ji α ij 0 f = 0 M l du dt + Lu = 0 α ij 1 f = Du M l du dt + Ku = 0 low-order scheme high-order scheme Goal: Determine α ij as close to 1 as possible such that (Lu f ) i = j i q ij (u j u i ), q ij 0 j i 5

9 Multidimensional flux correction 1. Compute the sums of positive/negative antidiffusive fluxes P + i = j i max{0, f ij }, P i = j i min{0, f ij } 2. Define the corresponding upper/lower bounds Q + i = j i q ij max{0, u j u i }, Q i = j i q ij min{0, u j u i } 3. Evaluate the nodal correction factors for positive/negative fluxes R + i = min{1, Q + i /P + i }, R i = min{1, Q i /P i } 4. Apply correction factors to the raw antidiffusive fluxes f i = j i α ij f ij a ij = { min{r + i, R j }, if f ij 0 min{r i, R + j }, otherwise 6

10 Multidimensional flux correction 1. Compute the sums of positive/negative antidiffusive fluxes P + i = j i max{0, f ij }, P i = j i min{0, f ij } 2. Define the corresponding upper/lower bounds Q + i = j i q ij max{0, u j u i }, Q i = j i q ij min{0, u j u i } 3. Evaluate the nodal correction factors for positive/negative fluxes R + i = min{1, Q + i /P + i }, R i = min{1, Q i /P i } 4. Apply correction factors to the raw antidiffusive fluxes f i = j i α ij f ij a ij = { min{r + i, R j }, if f ij 0 min{r i, R + j }, otherwise 6

11 Multidimensional flux correction 1. Compute the sums of positive/negative antidiffusive fluxes P + i = j i max{0, f ij }, P i = j i min{0, f ij } 2. Define the corresponding upper/lower bounds Q + i = j i q ij max{0, u j u i }, Q i = j i q ij min{0, u j u i } 3. Evaluate the nodal correction factors for positive/negative fluxes R + i = min{1, Q + i /P + i }, R i = min{1, Q i /P i } 4. Apply correction factors to the raw antidiffusive fluxes f i = j i α ij f ij a ij = { min{r + i, R j }, if f ij 0 min{r i, R + j }, otherwise 6

12 Multidimensional flux correction 1. Compute the sums of positive/negative antidiffusive fluxes P + i = j i max{0, f ij }, P i = j i min{0, f ij } 2. Define the corresponding upper/lower bounds Q + i = j i q ij max{0, u j u i }, Q i = j i q ij min{0, u j u i } 3. Evaluate the nodal correction factors for positive/negative fluxes R + i = min{1, Q + i /P + i }, R i = min{1, Q i /P i } 4. Apply correction factors to the raw antidiffusive fluxes f i = j i α ij f ij a ij = { min{r + i, R j }, if f ij 0 min{r i, R + j }, otherwise 6

13 Nonlinear solution algorithm Nonlinear algebraic system for 0 < θ 1 N(u) := Lu f M l u n+1 u n t + θn(u n+1 ) + (1 θ)n(u n ) = 0 Fixed-point iteration u (0) := u n u (m+1) = u (m) C 1 F (u (m) ), m = 0, 1,... until F (u (m+1) ) tol or F (u (m+1) ) tol F (u n ). Practical implementation C u (m+1) = F (u (m) ) m = 0, 1,... u (m+1) = u (m) u (m+1) u (0) = u n 7

14 Nonlinear solution algorithm Nonlinear algebraic system for 0 < θ 1 N(u) := Lu f F (u n+1 u n+1 u n ) := M l + θn(u n+1 ) + (1 θ)n(u n ) = 0 t Fixed-point iteration u (0) := u n u (m+1) = u (m) C 1 F (u (m) ), m = 0, 1,... until F (u (m+1) ) tol or F (u (m+1) ) tol F (u n ). Practical implementation C u (m+1) = F (u (m) ) m = 0, 1,... u (m+1) = u (m) u (m+1) u (0) = u n 7

15 Choice of preconditioner Linearized system C u (m+1) = F (u (m) ) Diagonal mass matrix C = M l / t cheap, but only usable for very small time steps Low-order operator C = M l / t + θ L fixed-point defect correction scheme may converge slowly no update needed if the velocity field remains unchanged Jacobian operator C = M l / t + θ N(u) u nonlinear operator N(u) is constructed discretely flux limiter may not be globally differentiable Idea: approximate by divided differences and see what happens 8

16 Jacobian matrix, Moe07a Nodal approximation of Jacobian J = {j ik } D k [N i ] := N i(u + he k ) N i (u he k ) 2h j ik = D k [N i ] + O(h 2 ) D k [N i ] = D k [(Lu f ) i ] = D k [(Ku + Du f ) i ] = D k [ j k ij u j + j i (1 α ij )f ij ] = k ik + j D k [k ij ]u j + j i D k [(1 α ij )f ij ] Averaged coefficient kik := k ik(u + he k ) + k ik (u he k ) 2 9

17 Jacobian matrix, Moe07a Nodal approximation of Jacobian J = {j ik } D k [N i ] := N i(u + he k ) N i (u he k ) 2h j ik = D k [N i ] + O(h 2 ) D k [N i ] = D k [(Lu f ) i ] = D k [(Ku + Du f ) i ] = D k [ j k ij u j + j i (1 α ij )f ij ] = k ik + j D k [k ij ]u j + j i D k [(1 α ij )f ij ] Averaged coefficient kik := k ik(u + he k ) + k ik (u he k ) 2 9

18 Jacobian matrix, Moe07a Nodal approximation of Jacobian J = {j ik } D k [N i ] := N i(u + he k ) N i (u he k ) 2h j ik = D k [N i ] + O(h 2 ) D k [N i ] = D k [(Lu f ) i ] = D k [(Ku + Du f ) i ] = D k [ j k ij u j + j i (1 α ij )f ij ] = k ik + j D k [k ij ]u j + j i D k [(1 α ij )f ij ] Averaged coefficient kik := k ik(u + he k ) + k ik (u he k ) 2 9

19 Sparsity pattern, Moe07a Jacobian coefficients j ik = k ik + j D k[k ij ]u j + j i D k[(1 α ij )f ij ] There exists an edge ij iff the FE basis functions have overlapping supports G := K Structure of the Jacobian Remarks z kk 0, g ii = 1, g ij = 1 ij Z := J z ik 0 ik z jk 0 ik ij Same structure is used for edge-oriented stabilization techniques Sparsity pattern can be generated by multiplication: Z = G 2 10

20 Sparsity pattern, Moe07a Jacobian coefficients j ik = k ik + j D k[k ij ]u j + j i D k[(1 α ij )f ij ] There exists an edge ij iff the FE basis functions have overlapping supports G := K g ii = 1, g ij = 1 ij u he k k Structure of the Jacobian Z := J z kk 0, z ik 0 ik z jk 0 ik ij Remarks Same structure is used for edge-oriented stabilization techniques Sparsity pattern can be generated by multiplication: Z = G 2 10

21 Sparsity pattern, Moe07a Jacobian coefficients j ik = k ik + j D k[k ij ]u j + j i D k[(1 α ij )f ij ] There exists an edge ij iff the FE basis functions have overlapping supports G := K Structure of the Jacobian Remarks z kk 0, g ii = 1, g ij = 1 ij Z := J z ik 0 ik z jk 0 ik ij Same structure is used for edge-oriented stabilization techniques Sparsity pattern can be generated by multiplication: Z = G 2 10

22 Sparsity pattern, Moe07a Jacobian coefficients j ik = k ik + j D k[k ij ]u j + j i D k[(1 α ij )f ij ] There exists an edge ij iff the FE basis functions have overlapping supports G := K Structure of the Jacobian Remarks z kk 0, g ii = 1, g ij = 1 ij Z := J z ik 0 ik z jk 0 ik ij Same structure is used for edge-oriented stabilization techniques Sparsity pattern can be generated by multiplication: Z = G 2 10

23 Sparsity pattern, Moe07a Jacobian coefficients j ik = k ik + j D k[k ij ]u j + j i D k[(1 α ij )f ij ] There exists an edge ij iff the FE basis functions have overlapping supports G := K Structure of the Jacobian Remarks z kk 0, g ii = 1, g ij = 1 ij Z := J z ik 0 ik z jk 0 ik ij Same structure is used for edge-oriented stabilization techniques Sparsity pattern can be generated by multiplication: Z = G 2 10

24 Example: Stationary convection-diffusion Convection-diffusion equation v u d u = 0 v = (cos 10, sin 10 ) FEM-TVD: d = 10 3 Initial guess in Ω = (0, 1) 2 u u (x, y) = Boundary conditions { 1 x if y otherwise Q 1 -elements u(x, 0) = 0, u(1, y) = 0, { u 1 if y 0.5, (x, 1) = 0, u(0, y) = y 0 otherwise. 11

25 Example: τ u + v u d u = 0 d = 10 3, τ = 1.0, BiCGSTAB+ILU, rel lin 10 3 norm of nonlinear residual 10 0 Newton level 6 Newton level Newton level 8 Defcor level 6 Defcor level Defcor level number of nonlinear iterations NVT CPU NN NL Newton s method defect correction Newton s method vs. defect correction total CPU time reduces by factor 2-3 reduction of nonlinear iterations by factor 4-16

26 Example: τ u + v u d u = 0 d = 10 3, τ = 10.0, BiCGSTAB+ILU, rel lin 10 3 norm of nonlinear residual 10 0 Newton level 6 Newton level Newton level 8 Defcor level 6 Defcor level Defcor level number of nonlinear iterations NVT CPU NN NL Newton s method defect correction Newton s method vs. defect correction total CPU time reduces by factor 2-3 reduction of nonlinear iterations by factor 4-16

27 Example: τ u + v u d u = 0 d = 10 4, τ = 1.0, BiCGSTAB+ILU, rel lin 10 3 norm of nonlinear residual 10 0 Newton level 6 Newton level Newton level 8 Defcor level 6 Defcor level Defcor level number of nonlinear iterations NVT CPU NN NL Newton s method defect correction Newton s method vs. defect correction total CPU time reduces by factor 2-3 reduction of nonlinear iterations by factor 4-16

28 Example: τ u + v u d u = 0 d = 10 4, τ = 10.0, BiCGSTAB+ILU, rel lin 10 3 norm of nonlinear residual 10 0 Newton level 6 Newton level Newton level 8 Defcor level 6 Defcor level Defcor level number of nonlinear iterations NVT CPU NN NL Newton s method defect correction Newton s method vs. defect correction total CPU time reduces by factor 2-3 reduction of nonlinear iterations by factor

29 Example: τ u + v u d u = 0 d = 10 4, τ = 10.0, BiCGSTAB+ILU, rel lin 10 3 norm of nonlinear residual 10 0 Newton level 6 Newton level Newton level 8 Defcor level 6 Defcor level Defcor level number of nonlinear iterations NVT CPU NN NL Newton s method defect correction Newton s method vs. defect correction total CPU time reduces by factor 2-3 reduction of nonlinear iterations by factor

30 Example: Burgers equation in space-time Inviscid Burgers equation ( ) u 2 x + t u = 0 2 Discrete upwind: 32,768 P 1 -elements Space-time domain Ω = (0, 1) (0, 0.5) Boundary conditions 1 if 0 x < 0.4 t = if 0.4 x 8 t = 0 u(x, t) = 0 if 0.8 < x 1 t = 0 or x = 0 0 t 0.5 u u h 1 = e 2 u u h 2 = e 2 13

31 Example: Burgers equation in space-time Inviscid Burgers equation ( ) u 2 x + t u = 0 2 FEM-TVD: 32,768 P 1 -elements Space-time domain Ω = (0, 1) (0, 0.5) Boundary conditions 1 if 0 x < 0.4 t = if 0.4 x 8 t = 0 u(x, t) = 0 if 0.8 < x 1 t = 0 or x = 0 0 t 0.5 u u h 1 = e 3 u u h 2 = e 2 13

32 Example: τ u + x ( 1 2 u2 ) + t u = 0 Discrete upwind: 1 nit./ τ, BiCGSTAB+ILU, rel lin 10 3 τ = 0.1 Newton defect correction error NVT CPU NN NL CPU NN NL u u h 1 u u h e e e e e e-2 τ = 1.0 Newton defect correction error NVT CPU NN NL CPU NN NL u u h 1 u u h e e e e e e-2 Discrete Jacobian operator yields robust Newton iteration 14

33 Example: τ u + x ( 1 2 u2 ) + t u = 0 FEM-TVD: up to 10 nit./ τ n, τ n [0.01, 1.0], rel nonl 10 3 BiCGSTAB+ILU, rel lin 10 3 Newton defect correction CPU NSTEP NN NL CPU NSTEP NN NL error u u h e e e-3 u u h e e e-2 Newton s method vs. defect correction total CPU time reduces by factor 3 reduction of outer iterations by factor

34 Example: Swirling flow, Leveque Swirling flow in Ω = (0, 1) 2 u t + (vu) = 0 Time-dependent velocity t (0, T ) v x = sin 2 (πx) sin(2πy)g(t) v y = sin 2 (πy) sin(2πx)g(t) g(t) = cos(πt/t ), T = 1.5 Time step control: PID t [10 3, 10 1 ], un+1 u n u n+1 0.5% 16

35 Example: Swirling flow, contd. Newton defect correction NVT NSTEP CPU NN NL CPU NN NL (70) (77) (66) nodes nodes nodes adaptive time step error reduction 17 CPU improved by factor 4.5; reduction of iterations by factor 7

36 Example: Swirling flow, contd. Newton defect correction NVT NSTEP CPU NN NL CPU NN NL (70) (77) (66) nodes nodes nodes norm of nonlinear residual adaptive time step number of nonlinear iterations 17 CPU improved by factor 4.5; reduction of iterations by factor 7

37 Outlook: Compressible flows, Kuz05b Divergence Form U t + 3 d=1 F d x d = 0 A d = F d U Quasi-linear form U t + 3 U d=1 Ad x d = 0 FEM discretization [ M du ] C dt = i j i a ij(u j u i ) a ij = r ij Λ ij r 1 ij Transformation to local characteristic variables makes it possible to perform upwinding and algebraic flux correction as in the scalar case. M = 3, α = 0, density Mach number 18

38 Grid adaption, Moe triangles triangles triangles triangles Density distribution on the final mesh cutline at y = 0.6 grid1 grid2 grid3 grid

39 Time-dependent grid adaptivity 20

40 Time-dependent grid adaptivity 20

41 Time-dependent grid adaptivity 20

42 Time-dependent grid adaptivity 20

43 Time-dependent grid adaptivity 20

44 Time-dependent grid adaptivity 20

45 Time-dependent grid adaptivity 20

46 Conclusions and Outlook nonlinear high-resolution schemes can be constructed by adding artificial diffusion + limited antidiffusion implicit flux correction schemes can be combined with algebraic Newton methods to speed up convergence Jacobian matrix is approximated by divided differences and assembled efficiently edge-by-edge in a black-box fashion the sparsity pattern of the Jacobian needs to be extended which can be accomplished by standard matrix multiplication Further research: compressible Euler/Navier-Stokes equations time-dependent grid adaptation + goal oriented error estimation 21

47 References Kuz05a D. Kuzmin, M.M. Algebraic flux correction I. Scalar conservation laws. In: Flux-Corrected Transport, Principles, Algorithms, and Applications, D. Kuzmin, R. Löhner, S. Turek (eds.). Springer: Germany, 2005; Kuz05b D. Kuzmin, M.M. Algebraic flux correction II. Compressible Euler equations. In: Flux-Corrected Transport, Principles, Algorithms, and Applications, D. Kuzmin, R. Löhner, S. Turek (eds.). Springer: Germany, 2005; Moe06 M.M., D. Kuzmin. Adaptive mesh refinement for high-resolution finite element schemes. IJNMF 52, 2006; Moe07a M.M. Efficient solution techniques for implicit finite element schemes with flux limiters. IJNMF (in press). Moe07b M.M, D. Kuzmin, D. Kourounis. Implicit FEM-FCT algorithms and discrete Newton methods for transient problems. Tech. Rep. 340, 2007, University of Dortmund. 22

48 PID time step control Normalize relative changes e n = un u n 1 tol u n Reject time step if e n > 1 t n := β t n, 0 < β < 1 Compute new time step t n+1 = ( en 1 e n e n Parameters (Valli, Carey, Coutinho) ) kp ( ) ki ( 1 e 2 ) kd n 1 t n e n e n 2 k P = 0.075, k I = 0.175, k D = 0.01 Back 23

49 Semi-implicit FCT limiter, Part I 1. Initialization P ± i Q ± i R ± i 0 2.Compute positivity-preserving intermediate low-order solution ũ = u n + (1 θ) tm 1 l Lu n 3. Evaluate raw antidiffusive flux f n ij P ± i := P ± i + max min {0, f n = td n ij (un i ij }, P ± j := P ± j + max min 4. Update admissible increments for both nodes Q ± i uj n ) and update {0, f n ij } := max min {Q± i, ũ j ũ i }, Q ± j := max min {Q± j, ũ i ũ j } 5. Limite the raw antidiffusive fluxes f n ij R ± i := m i Q ± i /P ± i f ij := { min{r + i, R j }fij n, min{r i, R + j }fij n, if F ij n > 0, otherwise. 24

50 Semi-implicit FCT limiter, Part II 1. Evaluate the target flux f ij = [m ij + θ td (m) ij ](u (m) i u (m) j ) [m ij (1 θ) tdij n ](ui n uj n ) 2.Constrain each flux by means of f ij { f ij = min{f ij, max{0, f ij }}, if f ij > 0, max{f ij, min{0, f ij }}, otherwise. 3. Insert the limited antidiffusive flux into the right-hand side b(m) i := b (m) i + f ij, b(m) j := b (m) j f ij Back 25