A comprehensive study of the numerical properties of high-order residual-based dissipation terms for unsteady compressible flows leads to the design of well-behaved, low dissipative schemes of third-, fifth- and seventh-order accuracy. The dissipation and dispersion properties of the schemes are then evaluated theoreticaly, through Fourier space analysis, and numerically, through selected test cases including the inviscid Taylor-Green Vortex flow.

1. DynFluid
Fluid Dynamics Laboratory
Analysis of high-order residual-based
dissipation for unsteady
compressible flows
Karim GRIMICH
Joint Work with:
Paola CINNELLA
Alain LERAT
ICCFD-2804
1

2. Introduction
• Two usual ways for constructing high-accurate schemes on a structured grid:
 Extend the grid stencil
 use compact approximations (directional compact, DG, RBC …)
• Owing to their narrow stencil, compact approximations:
greatly reduce the error coefficient with respect to a non-compact
approximation of the same accuracy order
 improve treatment of irregular meshes
• Here we use centered compact approximation:
 With no filter,
 No artificial viscosity or limiter
• A Residual-Based Compact (RBC) scheme [Lerat, Corre, JCP 2001] can be
expressed only in terms of approximations of the exact residual (sum of all the
terms in the governing eqns), even for the numerical dissipation.
• Use of residual based approach enables genuinely multi dimensional dissipation.
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Fluid Dynamics Laboratory 2

3. Aims and motivations
• A queer feature is that the residual-based dissipation behaves as the discrete
residual, i.e. it is consistent with a null function
 When residual tends to zero high-order dissipation terms are left.
• Need for
identifying the actual dissipation operator through suitable expansions
 finding the necessary and sufficient conditions ensuring that this
operator is really dissipative for all flow conditions.
 studying the spectral properties of RBC schemes for a multidimensional
problem to get information on the evolution of Fourier modes supported
on a given grid.
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Fluid Dynamics Laboratory 3

4. Summary
I. High-order RBC schemes
II. The χ-criterion for RBC dissipation
III. Spectral properties of RBC schemes
IV. Numerical experiments
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Fluid Dynamics Laboratory 4

5. Summary
I. High-order RBC schemes
II. The χ-criterion for RBC dissipation
III. Spectral properties of RBC schemes
IV. Numerical experiments
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Fluid Dynamics Laboratory 5

6. High-order RBC schemes
Initial value problem
on a uniform mesh
~
~ ~
~
Mean and difference operators:
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7. High-order RBC schemes
Exact residual:
Residual-Based Compact schemes:
Third-order of accuracy using 3x3 points: RBC3
Fifth-order of accuracy using 5x5 points: RBC5
Seventh-order accuracy using 5x5 points: RBC7
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8. High-order RBC schemes
Spatial approximation of the main residual:
Pade approximation of first order derivatives
where
Applying to the all left hand side gives
Which truncation error is
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Fluid Dynamics Laboratory 8

9. High-order RBC schemes
Residual based dissipation operator:
Mid-point residuals using the same technique as for the main residual
Where and are  designed once for all [Lerat, Corre, JCP 2001]
 with no tuning parameter
 depending only on A=df/dw, B=dg/dw, and
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10. High-order RBC schemes
A peculiar feature of unsteady RBC schemes:
 A mass matrix appears due to difference operators applied to wt.
Operators applied to Operators applied to spatial
temporal derivatives derivatives
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Fluid Dynamics Laboratory 10

11. High-order RBC schemes
The residual-based operator is consistent with the first order and dissipative
operator [Lerat, Corre, JCP 2001]:
For an exact solution the residual is null
is consistent with some higher
order term
 To which high order term is it consistent with?
 Is this high-order term dissipative or not?
 How this high-order dissipation behaves?
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Fluid Dynamics Laboratory 11

12. Summary
I. High-order RBC schemes
II. The χ-criterion for RBC dissipation
III. Spectral properties of RBC schemes
IV. Numerical experiments
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Fluid Dynamics Laboratory 12

13. The χ-criterion for dissipation
Role of the x-discretization in r1 and the y-discretization in r2:
We define
With
Doing Taylor expansion of r1x around the point (j+1/2,k)
Similar result for
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14. The χ-criterion for dissipation
Fully discretized RBC dissipation:
The full dissipation of a RBCq scheme can be expressed as:
with:
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15. The χ-criterion for dissipation
The χ-criterion for dissipation:
In the linear case:
With
Since all derivatives in being even, its Fourier symbol is real.
Definition: The operator dq is said to be dissipative if
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Fluid Dynamics Laboratory 15

16. The χ-criterion for dissipation
The χ-criterion for dissipation [Lerat, Grimich, Cinnella, submitted to JCP 2012]:
Theorem: The operator dq is dissipative for any order q=2p-1, any pair (A,B)
and any functions Φ1 and Φ2 such that
if and only if
Criterion easily satisfied through a proper choice of Pade coefficients
 RBC3 and RBC7 presented in [Corre, Falissard, Lerat, C&F 2007]
do not satisfy the χ-criterion (weak instability for unsteady flows)
 RBC5 presented in [Corre, Falissard, Lerat, C&F 2007] satisfies
the χ-criterion
Analogous results proved in 3D
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Fluid Dynamics Laboratory 16

17. Summary
I. High-order RBC schemes
II. The χ-criterion for RBC dissipation
III. Spectral properties of RBC schemes
IV. Numerical experiments
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Fluid Dynamics Laboratory 17

18. Spectral properties of RBC
RBC schemes spatial discretization expresses as
and can be rewritten as
Where is the operator applied to wt and the operator applied to w.
In a more compact way:
with
along with initial condition these ordinary differential equations define a Cauchy
problem that is studied in the following.
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Fluid Dynamics Laboratory 18

19. Spectral properties of RBC
We study the linear problem
Its Fourier transform is
(1)
Where is the velocity vector and is a 2D wave
vector.
We introduce
which is the reduced wave number aligned aligned with advection direction
where and are the CFL numbers
and is the reduced wave number vector
Thus (1) rewrites:
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Fluid Dynamics Laboratory 19

20. Spectral properties of RBC
Taking the Fourier transform of the spatial discretization of RBC schemes
(2)
Introducing the modified reduced wave number along the advection direction
(2) re-writes
We define the error with respect to the exact wave number
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21. Spectral properties of RBC
 Cauchy stability of RBC schemes
 Dispersion properties of RBC schemes
 Dissipation properties of RBC schemes
If RBC schemes were infinitely accurate we would have
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22. Spectral properties of RBC
 Cauchy stability of RBC schemes
unstable
unstable
stable
stable
RBC7 with χ≠0
RBC7 with χ=0
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23. Spectral properties of RBC
Advection along a mesh direction
 Dispersion and dissipation properties of RBC schemes (example of RBC7)
1D cut 1D cut
Dispersion Dissipation
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24. Spectral properties of RBC
Advection along a mesh direction
 1D cut
Dispersion Dissipation
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Fluid Dynamics Laboratory 24

25. Spectral properties of RBC
Advection along the mesh diagonal
 Dispersion and dissipation properties of RBC schemes (example of RBC7)
Dispersion Dissipation
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26. Spectral properties of RBC
Resolvability
: cutoff reduced wavelength for an
error lower than 10-3
: minimum number of points per
wavelength for an error lower than 10-3
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27. Summary
I. High-order RBC schemes
II. The χ-criterion for RBC dissipation
III. Spectral properties of RBC schemes
IV. Numerical experiments
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Fluid Dynamics Laboratory 27

28. Numerical experiments
The discretization has to be completed by selecting a time
approximation method
Here, we choose the 2nd-order accurate, A-stable Gear scheme
Stability region includes the complex stable half-plane
• Coupled with a dissipative scheme, this results in an unconditionally
stable discretization
Associated nonlinear system solved via a dual-time stepping technique
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29. Advection of a sine wave
Validation of the χ-criterion for dissipation:
χ≠0
χ≠0
25x25 mesh λ
χ=0
χ=0
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Fluid Dynamics Laboratory 29

30. Advection of a sine wave
Resolvability of RBC schemes:
6 points per wavelength 8 points per wavelength 16 points per wavelength
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Fluid Dynamics Laboratory 30

31. Inviscid Taylor-Green vortex
Representative model to within the inertial range
Periodic boundary conditions +
t=0 t=3 t=5
RBC5 scheme
1283 mesh
Q-criterion colored by kinetic energy M=0.3 (Shu et al, J.Sci.Comput. 2005)
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Fluid Dynamics Laboratory 31

32. Inviscid Taylor-Green vortex
Time evolution of the total kinetic energy
643 mesh 1283 mesh
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Fluid Dynamics Laboratory 32

33. Inviscid Taylor-Green vortex
RBC5 scheme
Physics well captured: vortex stretching
1283 mesh
vorticity contours
t=3 t=5
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Fluid Dynamics Laboratory 33

34. Conclusion
• RBC schemes unsteady high-order dissipation has been identified
 the χ-criterion for dissipation has been demonstrated ensuring that this
operator is really dissipative for all flow conditions.
 the spectral properties of RBC schemes for a multidimensional problem
gave information on the evolution of Fourier modes supported on a given
grid: stability, dissipation, dispersion.
• Numerical experiments confirmed
the validity of the χ-criterion.
 the low dissipative and dispersive behavior of high-order RBC
schemes
 the high resolvability of high-order RBC schemes
• Perspectives
 LES, DES simulations
 design a suitable high-order time integration technique
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Fluid Dynamics Laboratory 34

35. Thank you for your attention
Any question?
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Fluid Dynamics Laboratory 35

36. References
Main references on previous works on RBC schemes:
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Fluid Dynamics Laboratory 36

37. Converging cylindrical shock
Converging cylindrical shock [Chisnell, JFM 1957]  analytical estimation of the
pressure behind a moving
cylindrical shock
No filter
No Artificial Viscosity
No Tuning parameter
Shock strength becomes infinite when it reaches the axis
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Fluid Dynamics Laboratory 37