Abstract
We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in sub-cells and treated by finite volume techniques Hence, our approach combines the good properties of the Discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations Due to the sub-cell approach the interior resolution of the Discontinuous Galerkin grid cell is nearly preserved and the number of degrees of freedom remains the same To fully benefit from the high order of the DG method we prefer rather large grid cells This is in contrast to the requirement of resolving small geometrical shapes or other local flow features With the possibilty to locally refine grids including non-conforming interfaces we close the gap between those two contrary requirements The overall implementation is designed to run very efficiently on massively parallel computers
Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Sub-cells Matthias Sonntag, Claus-Dieter Munz Institute of Aerodynamics and Gas Dynamics University of Stuttgart
SHARK-FV – May 23, 2016
Motivation I
high order methods
I
handle shocks, discontinuities, instabilities
I
very efficiently
I
on massive parallel systems 1.1 1 0.9 0.8
Density
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.3
Matthias Sonntag (IAG)
0.35
0.4
0.45
0.5 x
0.55
0.6
Shock Capturing for DG using FV Sub-cells
0.65
0.7
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Content
1
Discontinuous Galerkin Spectral Element Method
2
Finite Volume Sub-cell Shock Capturing
3
Mesh refinement with Mortar interfaces
4
Numerical Results
5
Conclusion & Outlook
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Content
1
Discontinuous Galerkin Spectral Element Method
2
Finite Volume Sub-cell Shock Capturing
3
Mesh refinement with Mortar interfaces
4
Numerical Results
5
Conclusion & Outlook
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Navier Stokes Equation
Conservation Law ut + ∇ · F (u, ∇u) = 0
in Ω
with the vector of conservative variables
ρ u = ρv ρE
and the convective and viscous fluxes F (u, ∇u) = F c (u) + F v (u, ∇u)
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Discretization & Weak Formulation I
hexahedral meshes
I
mapping to reference element J(ξ)ut + ∇ξ · F = 0
I
test function + integration hJut , φiE + h∇ · F, φiE = 0
I
partial integration
weak formulation hJut , φiE = −hf ∗ (u, ∇u, n), φi∂E + hF, ∇φiE where f ∗ (u, ∇u, n) is a numerical flux function (Riemann solver)
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Discontinuous Galerkin Spectral Element Method (DGSEM)
ui,1
ξ2 1
0 I
ξ1
ansatz functions
ˆ ij u
u−1,j
u=
N X
u ˆijk ψijk (ξ),
1
with ψijk = `i (ξ 1 )`j (ξ 2 )`k (ξ 3 )
i,j,k=0
where `i (ξ 1 ), . . . are the 1D Lagrange interpolation polynomials I
nodes of Gauss quadrature: {ξi }N i=0
I
collocation: points of interpolation = points of integration
I
Lagrange property `i (ξj ) = δij
I
test functions = ansatz functions
I
same approximation for fluxes: F =
Matthias Sonntag (IAG)
PN
∀i, j
i,j,k=0
Fˆijk ψijk (ξ)
Shock Capturing for DG using FV Sub-cells
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Time Derivative Integral
∂ hJut , φiE = ∂dt
Z
1
Z
1
Z
1
J(ξ) −1
−1
−1
N X
! u ˆmno ψmno (ξ) ψijk (ξ)dξ 1 dξ 2 dξ 3
m,n,o=0
inserting Gauss quadrature + ansatz function:
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
SHARK-FV – May 23, 2016
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Time Derivative Integral
∂ hJut , φiE = ∂dt
Z
1
Z
1
Z
1
J(ξ) −1
−1
−1
N X
! u ˆmno ψmno (ξ) ψijk (ξ)dξ 1 dξ 2 dξ 3
m,n,o=0
inserting Gauss quadrature + ansatz function: N N X X ∂ = J(ξλµν ) u ˆmno `m (ξλ1 ) `n (ξµ2 ) `o (ξν3 ) ψijk (ξλµν )ωλ ωµ ων | {z } | {z } | {z } ∂dt m,n,o=0 λ,µ,ν=0 δmλ
Matthias Sonntag (IAG)
δnµ
Shock Capturing for DG using FV Sub-cells
δoν
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Time Derivative Integral
∂ hJut , φiE = ∂dt
Z
1
Z
1
Z
N X
1
J(ξ) −1
−1
−1
! u ˆmno ψmno (ξ) ψijk (ξ)dξ 1 dξ 2 dξ 3
m,n,o=0
inserting Gauss quadrature + ansatz function: N N X X ∂ = J(ξλµν ) u ˆmno `m (ξλ1 ) `n (ξµ2 ) `o (ξν3 ) ψijk (ξλµν )ωλ ωµ ων | {z } | {z } | {z } ∂dt m,n,o=0 λ,µ,ν=0 δmλ
=
∂ ∂dt
N X λ,µ,ν=0
δnµ
δoν
J(ξλµν )ˆ uλµν `i (ξλ1 ) `j (ξµ2 ) `k (ξν3 ) ωλ ωµ ων | {z } | {z } | {z }
Matthias Sonntag (IAG)
δiλ
δjµ
δkν
Shock Capturing for DG using FV Sub-cells
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Time Derivative Integral
∂ hJut , φiE = ∂dt
Z
1
Z
1
Z
N X
1
J(ξ) −1
−1
−1
! u ˆmno ψmno (ξ) ψijk (ξ)dξ 1 dξ 2 dξ 3
m,n,o=0
inserting Gauss quadrature + ansatz function: N N X X ∂ = J(ξλµν ) u ˆmno `m (ξλ1 ) `n (ξµ2 ) `o (ξν3 ) ψijk (ξλµν )ωλ ωµ ων | {z } | {z } | {z } ∂dt m,n,o=0 λ,µ,ν=0 δmλ
=
∂ ∂dt
N X λ,µ,ν=0
= J(ξijk )
δnµ
δoν
J(ξλµν )ˆ uλµν `i (ξλ1 ) `j (ξµ2 ) `k (ξν3 ) ωλ ωµ ων | {z } | {z } | {z } δiλ
δjµ
δkν
∂ˆ uijk ωi ωj ωk ∂dt
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Semidiscrete DG Formulation weak formulation hJut , φiE = −hf ∗ (u, ∇u, n), φi∂E + hF, ∇φiE " Jijk (ˆ uijk )t =
1 −fjk∗,ξ `ˆi (−1)
−
1 fjk∗,ξ `ˆi (1)
+
N X
# 1 Dˆiλ Fˆλjk
λ=0
" +
2 −fik∗,ξ `ˆj (−1)
−
2 fik∗,ξ `ˆj (1)
+
N X
# 2 Dˆjλ Fˆiλk
λ=0
" +
3 −fij∗,ξ `ˆk (−1)
−
3 fij∗,ξ `ˆk (1)
+
N X
# 3 Dˆkλ Fˆijλ
λ=0
I
only 1D operations due to tensor product structure Gauss integration weigths are hidden in ˆ· -terms
I
explicit Runge Kutta time integration
I
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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u
Algorithm
Extrapolate u → u−
1. MPI 2. other
SendS→M : u−
explicit Runge Kutta time integration
ReceiveM←S : u+ 1. MPI 2. other Lifting Volume Operator Lifting Surface Integral Lifting-Flux
Extrapolate ∇u → ∇u−
SendM→S : Lifting-Flux ReceiveS←M : Lifting-Flux 1. MPI 2. other
SendS→M : ∇u−
Volume Operator DG ReceiveM←S : ∇u+ Fluxes
1. MPI 2. other
Surface integral
1. other 2. MPI
SendM→S : Flux
ReceiveS←M : Flux
∂u ∂t Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Order Of Convergence (DG) poly. deg.
N=2
N=3
N=4
N=5
cells 16 32 64 128 256 16 32 64 128 256 16 32 64 128 256 8 16 32 64 128
Matthias Sonntag (IAG)
L2 error 8.14e-04 1.12e-04 1.49e-05 1.89e-06 2.38e-07 3.00e-05 1.80e-06 9.08e-08 6.36e-09 3.75e-10 1.74e-06 5.98e-08 2.27e-09 6.61e-11 2.20e-12 3.47e-06 4.48e-08 6.58e-10 1.02e-11 2.00e-13
L2 order 2.86 2.91 2.98 2.99 4.06 4.31 3.84 4.08 4.86 4.72 5.10 4.91 6.27 6.09 6.01 5.67
L∞ error 3.10e-03 4.34e-04 5.91e-05 7.51e-06 9.43e-07 1.15e-04 6.92e-06 3.65e-07 2.54e-08 1.53e-09 7.25e-06 2.50e-07 9.29e-09 2.75e-10 9.29e-12 1.36e-05 1.71e-07 2.52e-09 3.88e-11 6.89e-13
Shock Capturing for DG using FV Sub-cells
L∞ order 2.84 2.88 2.97 2.99 4.06 4.25 3.84 4.05 4.86 4.75 5.08 4.89 6.31 6.09 6.02 5.81
theor. order
3
4
5
6
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Content
1
Discontinuous Galerkin Spectral Element Method
2
Finite Volume Sub-cell Shock Capturing
3
Mesh refinement with Mortar interfaces
4
Numerical Results
5
Conclusion & Outlook
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
SHARK-FV – May 23, 2016
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Finite Volume Sub-cells DG
ui,1
ξ2 1
0
1 ξ1
u−1,j
ˆ ij u
I
= interface flux points (Riemann solver)
I
= interpolation/integration points
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Finite Volume Sub-cells DG
FV sub-cells
ui,1
ξ2 1
0
1 ξ1
u−1,j
ˆ ij u
I
= interface flux points (Riemann solver)
I
= interpolation/integration points
I I I
DG reference element is splitted into (N + 1)d sub-cells = FV sub-cell borders = inner flux points (Riemann solver)
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Finite Volume Method I
integration over sub-cell Z Z Jut dξ + κijk
I
i, j, k = 0, . . . , N
divergence theorem + numerical flux function Z Z Jut dξ + f ∗ (u, ∇u, n) dSξ = 0 κijk
I
∇ · F dξ = 0
κijk
∂κijk
choose nodal value u ˆijk as constant value of ijk-th FV sub-cell i ωj ωk h ∗,ξ1 ∗,ξ 1 Jijk (ˆ uijk )t = −fi− 1 ,j,k − fi+ 1 2 2 ,j,k ωi ωj ωk h i 2 ωi ωk ∗,ξ ∗,ξ 2 + −fi,j− − f 1 i,j+ 12 ,k 2 ,k ωi ωj ωk h i ωi ωj ∗,ξ 3 ∗,ξ 3 + −fi,j,k− 1 − f 1 i,j,k+ 2 2 ωi ωj ωk Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Coupling of DG and FV
P
I
interface fluxes computed on FV points
I
solution is interpolated to FV
I
fluxes projected back to DG conservative / free-stream preserving
I
I
even on curved meshes
⇒ coupling of DG and FV via numerical fluxes
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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2nd Order Reconstruction I
first order FV is very dissipative
I
improved by 2nd order reconstruction
I
total variation diminishing (TVD)
I
limiters: minmod, van Leer, Sweby, ...
I
computation of slopes:
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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u
Algorithm
Extrapolate u → u−
1. MPI 2. other
SendS→M : u− , Type
explicit Runge Kutta time integration
ReceiveM←S : u+ , Type 1. MPI 2. other Lifting Vol. Op. / Inner Reconstr. Lifting Surf Int. / Surf. Reconstr. 1. MPI Extrapolate ∇u → ∇u− 2. other Lifting-Flux
SendM→S : Lifting-Flux ReceiveS←M : Lifting-Flux SendS→M : ∇u−
Vol. Op. DG / FV inner faces ReceiveM←S : ∇u+ Fluxes
1. MPI 2. other
Surface integral
1. other 2. MPI
SendM→S : Flux
ReceiveS←M : Flux
∂u ∂t Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Indicators I
detect shocks
I
Persson indicator
I
Jameson indicator (3D sub-cell modification)
I
Ducros sensor (with Jameson to avoid turbulence detection)
I
upper and lower threshold indicator value upper threshold
lower threshold time DG
Matthias Sonntag (IAG)
FV
Shock Capturing for DG using FV Sub-cells
DG
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Timestep Restriction I
CFL condtion CFL =
I
∆t · λ ∆x
DG timestep: ∆t =
αRK (N) ∆xDG 2N + 1 λ
where: I I
I
αRK - constant of explicit time integration λ - max propagation velocity
FV timestep of sub-cell: ∆t = αRK (0) with
2 N+1
2 ∆xDG , N +1 λ
= size of FV sub-cell
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Order Of Convergence (FV sub-cells)
poly. deg.
N=2
N=3
N=4
cells 16 32 64 128 256 16 32 64 128 256 16 32 64 128 256
Matthias Sonntag (IAG)
no reconstruction L2 error L2 order 3.62e-02 1.89e-02 0.93 9.68e-03 0.97 4.89e-03 0.98 2.46e-03 0.99 2.87e-02 1.48e-02 0.95 7.55e-03 0.98 3.81e-03 0.99 1.91e-03 0.99 2.40e-02 1.24e-02 0.96 6.27e-03 0.98 3.16e-03 0.99 1.59e-03 0.99
minmod L2 error L2 order 9.32e-03 3.23e-03 1.53 1.04e-03 1.63 3.35e-04 1.64 1.07e-04 1.64 6.46e-03 2.14e-03 1.59 6.92e-04 1.63 2.22e-04 1.64 7.10e-05 1.64 4.81e-03 1.54e-03 1.64 5.09e-04 1.60 1.63e-04 1.64 5.26e-05 1.64
Shock Capturing for DG using FV Sub-cells
central L2 error L2 order 1.60e-03 3.83e-04 2.06 9.50e-05 2.01 2.37e-05 2.00 5.94e-06 2.00 1.25e-03 3.16e-04 1.99 7.92e-05 2.00 1.98e-05 2.00 4.97e-06 2.00 8.98e-04 2.30e-04 1.97 5.73e-05 2.00 1.43e-05 2.00 3.59e-06 2.00
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Remarks
I
same DOFs, same data structure I I I
straightforward implementation comparable performance (DG/FV sub-cells per element) conservative interpolation/projection between DG and FV sub-cells
I
2nd order reconstruction for FV sub-cells
I
slope limiters, TVD Indicators decide where to use FV sub-cells, where DG
I
I I I I
I
Persson indicator JST scheme like indicator Ducros sensort ...
no additional timestep restriction
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Content
1
Discontinuous Galerkin Spectral Element Method
2
Finite Volume Sub-cell Shock Capturing
3
Mesh refinement with Mortar interfaces
4
Numerical Results
5
Conclusion & Outlook
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Mesh refinement with Mortar interfaces Type 2 I
complex geometries ⇒ mesh refinement especially for hex-only-meshes
I
non-conforming interfaces
I
Mortar approach
I
3 different types in 3D
Type 1
Matthias Sonntag (IAG)
Type 3
Shock Capturing for DG using FV Sub-cells
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Mesh refinement with Mortar interfaces
I
Type 1 is combination of types 2 and 3
I
Interpolation-operators M 0 1 and M 0 2: M 0 1 : big → left half
I
M 0 2 : big → right half
Projection-operators M 1 0 and M 2 0: M 1 0 : left half → big
M 2 0 : right half → big
I
operators reduce to matrix-vector-product
I
use this operators in η or ξ direction
I
additional operators M * *FV for FV sub-cell shock capturing
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Content
1
Discontinuous Galerkin Spectral Element Method
2
Finite Volume Sub-cell Shock Capturing
3
Mesh refinement with Mortar interfaces
4
Numerical Results
5
Conclusion & Outlook
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Parallel performance (strong scaling)
speedup
10
4
103
pure DG pure FV half/half checker board ideal
6 elements per core 87 − 95%
102
102 I
polynomial degree N = 5
I
15 · 106 DOFs Matthias Sonntag (IAG)
103 cores
104
Shock Capturing for DG using FV Sub-cells
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Performance index (PID) wall-clock-time#cores #DOF#timesteps
I
PID =
I
24 to 12288 cores ⇒ 3072 to 6 elements per core 8
PID (µs/DOF)
7
6
5 pure DG pure FV half/half checker board
4
3
103
Matthias Sonntag (IAG)
104 Load (DOF/core) Shock Capturing for DG using FV Sub-cells
105
106 SHARK-FV – May 23, 2016
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Parallel performance (strong scaling)
102 101 100
102
8 PID (µs/DOF)
speedup
≈ 5.7 elements per core
103
7 6 5
cores I
realistic test case (N=8, 17582 cells)
I
strong scaling down to about 6 elements per core
I
cache effects ⇒ best performance
I
Hazel Hen of HLRS Stuttgart (Cray XC40) Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
104 105 Load (DOF/core)
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Forward Facing Step I I
Mach 3 flow over step in wind tunnel Mortar interfaces, refinement at 1. step ⇒ 6543 grid cells 2. additionally: top shear wave (red region) ⇒ 8343 grid cells
I
polynomial degree N = 5 ⇒ 2.3 × 105 DOF
I
computation by M. Hoffmann
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Forward Facing Step
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Forward Facing Step
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Shockless explosion combustion test case I
validation test case only
I
curved mesh
I
64388 grid cells
I
mortar interfaces
I
N = 3 ⇒ 4.1 × 106 DOFs
I
computation by N. Krais
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Shock Capturing for DG using FV Sub-cells
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Shockless explosion combustion test case
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Towards Air-Breathing Hypersonic Vehicles (M. Atak)
⇒ How effcient are DG methods for such class of problems? ⇒ This will be the first DNS of a compressible supersonic turbulent boundary layer with DG Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Shock Boundary Layer Interaction (M. Atak)
I
DNS of a shock wave interacting with a flat plate turbulent boundary layer at M = 2.67
I
shock intensity p2 /p1 = 1.3 yields a ”steady” state
I
structured hexahedra mesh with 1,125,000 grid cells
I
N = 5 ⇒ 243 × 106 DOF
I
indicator: Ducros with Jameson
I
up to 56,250 processors
I
more than 106 CPU hours
I
no mortar interfaces
I
computation & visualization by M. Atak
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Shock Boundary Layer Interaction (M. Atak)
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Content
1
Discontinuous Galerkin Spectral Element Method
2
Finite Volume Sub-cell Shock Capturing
3
Mesh refinement with Mortar interfaces
4
Numerical Results
5
Conclusion & Outlook
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Conclusion & Outlook Conclusion I
Discontinuous Galkerkin Spectral Element Method
I
Shock Capturing with Finite Volume Sub-cells
I
Mortar interfaces for mesh refinement
I
Numerical examples
Outlook I
adaptiv mesh refinement (AMR)
I
dynamic load balancing
References I
M. Sonntag, C.-D. Munz: Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Sub-cells, Journal of Scientific Computing, in Review.
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Thank you for your attention! Visit I I
flexi-project.org nrg.iag.uni-stuttgart.de
Sod Shock Tube
I
left ρ 1.0 Initial conditions: v 0 p 0.125 Mesh: 10 DG elements
I
N = 11
I
Jameson indicator
I
IJameson =
right 0.125 0 0.1
N X pmin,ijk − 2pijk + pmax,ijk ωi ωj ωk , pmin,ijk + 2pijk + pmax,ijk Vol(Element)
i,j,k=0
where pmin,ijk is the minimal pressure of the surrounding nodes I
tend = 0.2
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Sod Shock Tube
Exact solution DG part of solution FV part of solution
Density
1
0.5
0
0.2
0.4
0.6
0.8
1
x
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Strong Shock-Vortex Interaction I
Mach Shock: 7.0
I
Mach Vortex: 1.7
I
Grid: h = 1/100
I
N=5
I I
FV distribution: equidistant Indicator: Jameson (with Persson on FV → DG)
I
Riemann: LF (HLLE at BC)
I
Limiter: MinMod
I
CFL: 0.5
I
Initial condition: numerical converged stationary 1D-shock
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Shock Capturing for DG using FV Sub-cells
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Strong Shock-Vortex Interaction
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Strong Shock-Vortex Interaction
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Double Mach Reflection
I
send shock wave diagonally into a wall
I
shock speed: Mach 10
I
Grid: h = 1/120 (480 x 120 cells)
I
polynomial degree N = 5 ⇒ 2 × 106 DOF
I
Indicator: Jameson (with Persson on FV → DG)
I I
Riemann: HLLE Limiter: MinMod
I
no mortar interfaces
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Double Mach Reflection
Matthias Sonntag (IAG)
Shock Capturing for DG using FV Sub-cells
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Double Mach Reflection 1
22.5
0.75
20
0.5 16
0.25 0 0
0.5
1.0
1.5
2.0
2.5
3.0 3.2
12
1 0.75
8
0.5 4
0.25 0
1.5 0
0.5
Matthias Sonntag (IAG)
1.0
1.5
2.0
Shock Capturing for DG using FV Sub-cells
2.5
3.0 3.2 SHARK-FV – May 23, 2016
47 / 39
Double Mach Reflection ρ 22.3
0.59
20
16
0.4
12
0.2
8
4 0 2.15 Matthias Sonntag (IAG)
1.4 2.35
2.55
Shock Capturing for DG using FV Sub-cells
2.75
2.86 SHARK-FV – May 23, 2016
48 / 39
