Anisotropic Element-Local Implicit Time Stepping for High Order Flux Reconstruction Semih Akkurt 1 Freddie Witherden 2 Peter Vincent 1 1 Imperial College London 2 Texas A&M June 18, 2020 Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 1 / 20
Overview Introduction 1 Motivation Literature Review Numerical Formulation 2 Test Cases 3 2D Cylinder SD7003 Wing Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 2 / 20
Motivation There is a maximum limit for time step in explicit solvers Memory requirement is high in implicit solvers Coupling is maximal in wall normal direction Figure: S. Langer, 2013. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 3 / 20
Line Implicit Applications in the context of FVM RANS solvers with high aspect ratio boundary layer elements The minimum distance is in the wall normal direction Figure: Application of line implicit for FVM. P. Eliasson, 2009. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 4 / 20
Line Implicit Applications in the context of FVM Figure: Application of line implicit for FVM. D. Mavriplis, 1999. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 5 / 20
Line Implicit in the context of DG Figure: Application of line implicit across elements in a DG type method. P.O. Persson, 2013. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 6 / 20
Local Line Implicit Coupling d U dt = R (U) (1) for implicit formulation U n +1 − U n = R (U n +1 ) (2) ∆ t U n +1 is an unknown, approximate it by using Taylor series expansion at x 0 = U n ; R (U n +1 ) = R (U n ) + ∂ R ∂ U (U n +1 − U n ) (3) J = ∂ R ∂ U , and the final form is � 1 � (U n +1 − U n ) = R (U n ) (4) ∆ t I − J Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 7 / 20
Local Line Implicit Coupling take the final form � 1 � (U n +1 − U n ) = R (U n ) ∆ t I − J (5) and simplify Jacobian by ignoring all the contributions except the ones from wall normal direction. This results an element local matrix for each element. Element local matrix has some zero blocks, which allows further simplifications in the implementation. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 8 / 20
Diagonal block jacobian matrix Figure: Quad element; p=2 Figure: non-zero structure of the local matrix for the Euler equation Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 9 / 20
Construction of the Jacobian Matrix Analytic formulation is possible but complicated Numerical differentiation is more expensive but easier to implement = R ( U n i + ǫ ) − R ( U n i ) ∂ R (6) ∂ U i ǫ Solution points that do not share a stencil can be perturbed simultaneously. Number of RHS calls: number of colors × n var × ( p +1) n dim . Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 10 / 20
Proof of Concept One directional coupling removes CFL restriction in that particular direction. A test code where the polynomial order can be set differently in different directions is used. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 11 / 20
Proof of Concept - Different p order in x and y directions Figure: p x = 2; p y = 4. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 12 / 20
Asymmetric Polynomial Order Table: Maximum time-step when the flux Jacobian set to zero using a DIRK scheme. DOF 240 2 ℘ y 0 ℘ y 1 ℘ y 2 ℘ y 3 ℘ y 4 ℘ x 0 0.0380 ℘ x 1 0.0170 ℘ x 2 0.0110 0.0080 0.0065 ℘ x 3 0.0080 0.0070 0.0055 ℘ x 4 0.0065 0.0055 0.0045 Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 13 / 20
Asymmetric Polynomial Order with Local Line Implicit Table: Maximum time-step with a flux Jacobian in x direction only using the same DIRK scheme. DOF 240 2 ℘ y 0 ℘ y 1 ℘ y 2 ℘ y 3 ℘ y 4 ℘ x 0 0.0698 0.0370 0.0213 0.0138 0.0092 ℘ x 1 0.0746 0.0347 0.0219 0.0134 0.0089 ℘ x 2 0.0743 0.0328 0.0210 0.0129 0.0088 ℘ x 3 0.0728 0.0317 0.0208 0.0128 0.0087 ℘ x 4 0.0710 0.0311 0.0205 0.0128 0.0087 Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 14 / 20
Test Case - 2D Cylinder Figure: Cylinder @Re=200. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 15 / 20
2D Cylinder case performance comparison Table: Speed up factors with different methods for 2D Cylinder. ∆ t /∆ τ Line Coupling Wall Time(m) SUF BDF2 Euler 8e-5 Off 118.5 1.00 BDF2 Euler 1.2e-3 On 13.0 9.12 STD RK45 2e-4 Off 38.3 3.09 Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 16 / 20
Test Case - SD7003 wing Figure: SD7003 @Re=40000. Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 17 / 20
SD7003 case performance comparison Table: Speed up factors with different methods for SD7003. ∆ t /∆ τ Line Coupling Wall Time(h) SUF BDF2 Euler 1.75e-5 Off 249.5 1.00 BDF2 Euler 7e-5 On 85.0 2.94 STD RK45 4.4e-5 Off 96.0 2.60 Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 18 / 20
Future Work Enabling p-multigrid Adaptive time stepping Convergence acceleration Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 19 / 20
The End Semih Akkurt (Imperial College) Local Line Implicit FR June 18, 2020 20 / 20
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