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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

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Anisotropic Element-Local Implicit Time Stepping for High Order Flux Reconstruction

Semih Akkurt 1 Freddie Witherden 2 Peter Vincent 1

1Imperial College London 2Texas A&M

June 18, 2020

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Overview

Introduction Motivation Literature Review

Numerical Formulation

Test Cases 2D Cylinder SD7003 Wing

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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.

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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.

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Line Implicit Applications in the context of FVM

Figure: Application of line implicit for FVM. D. Mavriplis, 1999.

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Line Implicit in the context of DG

Figure: Application of line implicit across elements in a DG type method. P.O. Persson, 2013.

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Local Line Implicit Coupling

dU dt = R(U) (1) for implicit formulation Un+1 − Un ∆t = R(Un+1) (2) Un+1 is an unknown, approximate it by using Taylor series expansion at x0 = Un; R(Un+1) = R(Un) + ∂R ∂U (Un+1 − Un) (3) J = ∂R

∂U , and the final form is

1 ∆t I − J

(Un+1 − Un) = R(Un)

(4)

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Local Line Implicit Coupling

take the final form 1 ∆t I − J

(Un+1 − Un) = R(Un)

(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.

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Diagonal block jacobian matrix

Figure: Quad element; p=2 Figure: non-zero structure of the local matrix for the Euler equation

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Construction of the Jacobian Matrix

Analytic formulation is possible but complicated Numerical differentiation is more expensive but easier to implement ∂R ∂Ui = R(Un

i + ǫ) − R(Un i )

ǫ (6) Solution points that do not share a stencil can be perturbed simultaneously. Number of RHS calls: number of colors×nvar×(p+1)ndim.

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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.

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Proof of Concept - Different p order in x and y directions

Figure: px = 2; py = 4.

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Asymmetric Polynomial Order

Table: Maximum time-step when the flux Jacobian set to zero using a DIRK scheme.

DOF 2402 ℘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

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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 2402 ℘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

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Test Case - 2D Cylinder

Figure: Cylinder @Re=200.

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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

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Test Case - SD7003 wing

Figure: SD7003 @Re=40000.

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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

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Future Work

Enabling p-multigrid Adaptive time stepping Convergence acceleration

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The End

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