Stability Analysis of the Muscl Method on General Unstructured Grids for Applications to Compressible Fluid Flow

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1 Stability Analysis of the Muscl Method on General Unstructured Grids for Applications to Compressible Fluid Flow F. Haider 1, B. Courbet 1, J.P. Croisille 2 1 Département de Simulation Numérique des Ecoulements et Aéroacoustique, Onera, Châtillon France 2 Laboratoire Mathématiques et Applications, Université de Metz, France Groupe de travail Méthodes Numériques 3 novembre 2008

2 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

3 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

4 The Software Package Cedre Overview of Cedre Overview of Cedre Software for numerical simulation of internal ows. Solver for compressible Navier-Stokes (RANS and LES). Multi-physics : gas dynamics, heat conduction, particles, radiation, reactive ows. Used by industry ( Eads, Snecma, Snpe ) and research organizations. Important Feature Discretization on general unstructured meshes : Important for industrial applications.

5 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

6 Spatial Discretization Basic Principle of the Muscl Finite-Volume Scheme x i 1 x i x i+1 x i+2 1 Reconstruct a piecewise linear solution from the cell averages. 2 Use it to compute upwinded uxes at the cell interfaces. 3 The uxes determine the dynamics of the cell averages.

7 Geometry of Unstructured Meshes x α k αβ x αβ h αβ n αβ k βα x β cell T α and face A αβ for convenience nαβ = kαβ = 0 if T α not a neighbor of T β fundamental relation β n αβ = 0

8 Spatial Discretization on Unstructured Meshes General Gradient Reconstruction on Polyhedral Meshes. u β Reconstruct a gradient σ α in each cell... h αγ σ α u αγ u γ σ α = β sαβ (uβ uα) u α k αγ... to compute second order accurate values at the cell interfaces uαγ = uα + kαγ σ α

9 Consistent Reconstruction Algebraic Condition for Second Order Reconstruction. Accuracy Condition Reconstruction must recover polynomials of degree one. This means that for all σ R d σ = β sαβ (hαβ σ) This equation with unknowns sαβ denes a family of admissible reconstructions.

10 General Solution of Consistent Reconstruction Consistency condition in matrix form Dene geometric matrix Hα with rows hαβ Dene reconstruction matrix Sα with columns sαβ Consistency condition becomes the linear matrix equation SαHα = I d General solution Sα = Ŝα + Λ α Bα Ŝα is a particular solution. Bα is a maximum rank solution of BαHα = 0.

11 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

12 Observation of Instabilities in 3D Applications Experience with Cedre Observation : Computations on unstructured grids are less accurate than those on cartesian grids. Hypothesis : Loss of accuracy is caused by slope limiters. Investigation : Perform same computations without limiters. Result : Computations break down on unstructured grids. Principal Questions Do these instabilities exist for the linear advection equation? Do these instabilities exist in the semi-discrete case?

13 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

14 Model Equation Examine the problem for the linear advection equation. t u (x, t) + c u (x, t) = 0 Spatial discretization transforms this into an ordinary dierential equation. du (t) dt = Ju (t) ; u = (u 1,..., u N ) Stability Condition u (t) C u (0) for all times t 0

15 Numerical Investigation of Spectra Apply spatial Muscl discretization to linear advection and compute spectra of the matrix J. 2D triangular mesh 3D tetrahedral mesh K40 K30 K20 K10 0 K10 50 K K30 K40 Curve 1

16 Conclusion Diagnosis for Semi-discrete Linear Advection Eigenvalues with positive real part in 3D on tetrahedral meshes : Spatial discretization is unstable. No instabilities for cartesian and deformed cartesian meshes. No instabilities for the rst order scheme. Conclusion Instabilities arise from the combination of linear reconstruction and unstructured meshes. It is necessary to examine the inuence of the slope reconstruction on stability.

17 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

18 Criterion to Select Reconstruction Problem A rigorous stability criterion is too dicult to obtain for the Muscl Scheme. Solution : Practical Criterion for Best Reconstruction Choose a consistent slope in each cell T α such that α T α du2 α dt becomes as small as possible. T α is the volume of cell T α.

19 Time Derivative of the Energy Function The time derivative of the energy can be written as a sum α T α du2 α dt = N {Θ α (u) + Φ α (u)} α=1 Θ α (u) = (c nαβ) + (uβ uα) 2 0 β Φ α (u) = 2 (c nαβ) + (uβ uα) kαβ sαγ (uγ uα) β γ

20 Local Criterion for Reconstruction Transform the Problem to Obtain Local Criterion Dene matrix Rα in cell T α with entries r (α) βγ = k αβ sαγ. Dene matrix Kα with rows kαβ then Rα = KαSα. The term Θ α (u) in the time derivative is always non-positive. The term Φ α (u) depends linearly on Rα = KαSα. Strategy to Select the Slope Choose Sα that minimizes KαSα under the constraint SαHα = I d.

21 Local Reconstruction Graphical Representation of the Local Reconstruction u β Rα = KαSα maps uβ uα to uαγ uα where h αγ u α u αγ k αγ u γ uαγ = uα + kαγ σ α If Rα = 0, then α T α du2 α dt 0 stability of the rst order upwind scheme!

22 Minimization Property of the Least Squares Reconstruction Minimization problem Find consistent reconstruction that minimizes KαSα : min { KαSα ; SαHα = I d } Solution exists for a large family of norms Least-squares slope solves min { KαSα ; SαHα = I d } for all unitarily invariant norms. This includes Spectral Norm, Frobenius Norm, Trace Norm. Least-squares reconstruction is optimal for the criterion.

23 Outline of the Proof I General form of reconstruction Sα = Ŝα + Λ α Bα. Least Squares Reconstruction Ŝα = ( Hα t 1 α) H t Hα. The matrix Ŝα satises BαŜ α t = 0, ŜαBα t = 0. This implies KαSαSα t K α t = KαŜαŜ α t K α t + KαΛ α BαBαΛ t t α K α t.

24 Outline of the proof II Denition A matrix norm. is called unitarily invariant if A = UAV for any matrix A and arbitrary unitary matrices U and V. Denition A norm g on R k is called a symmetric gauge function if it is an absolute and permutation invariant vector norm : g ( x 1,..., x k ) = g (x 1,..., x k ) and g (x) = g (Px) for any permutation matrix P.

25 Outline of the proof III Theorem Any unitarily invariant matrix norm A can be written as a symmetric gauge function of the vector of singular values of A A = g (ς 1 (A),..., ς k (A)). Theorem Let ς 1 (A) ς k (A) be the singular values of A and λ 1 (AA ) λ k (AA ) the eigenvalues of AA (They are the same as those of A A). Then ς i (A) = λ i (AA ), 1 i k.

26 Outline of the Proof IV Theorem (Weyl) Let P and Q be hermitian matrices. If Q is positive semi-denite, then λ i (P) λ i (P + Q), 1 i k. Weyl's Theorem shows ) ( ) λ i (KαŜαŜ α t K α t λ i KαŜαŜ α t K α t + KαΛ α BαBαΛ t t α K α t Therefore the singular values of KαSα satisfy ) ς i (KαSα), 1 i k. ς i (KαŜα

27 Outline of the proof V Theorem Any symmetric gauge function g is a monotone vector norm. Let x, y R k. x i y i, 1 i k, implies g(x 1,..., x k ) g(y 1,..., y k ). The combination of the theorems gives Theorem (Minimization property) Let. be any unitarily invariant matrix norm and Sα any reconstruction matrix Sα that is consistent, i.e. that is a solution of SαHα = I d. Then the least squares reconstruction matrix Ŝα satises KαŜα KαSα

28 Inuence of the Stencil Size on Stability A qualitative result for the least squares reconstruction Let Ŝα be the matrix of the least-squares reconstruction. What happens to the unitarily invariant matrix norms of if points are added to the stencil? KαŜα Theorem New reconstruction matrix Sα satises Kα Sα and even Kα Sα < KαŜα KαŜα under certain conditions. Conclusion : Larger stencils lead to more robust schemes.

29 Outline of the Proof I In cell T α, let the matrix Hα have rows {hαβ 1,..., hαβ n }. Add l new cells {T γ1,..., T γl } to the reconstruction stencil. Dene matrix Ȟα with rows {hαγ 1,..., hαγ l }. New geometric matrix is given by H α t = [ ] Hα t Ȟ α t. The matrix Hα satises ( H t α Hα) 1 = ( H t α H α + Ȟ t αȟα) 1.

30 Outline of the proof II Least squares reconstruction matrices Sα = ( H t α Hα) 1 H t α, Ŝα = ( H t α H α) 1 H t α. The matrix Sα satises Kα Sα S t α K t α = Kα ( H t α Hα) 1 K t α. The Sherman-Morrison-Woodbury identity shows ( Kα +Kα Hα t 1 ( α) H t Kα = Kα Hα t H α + ȞαȞα) t 1 t Kα + ( ( Hα α) t 1 ( H Ȟα t I + Ȟα Hα t 1 1 α) H Ȟα) t Ȟα ( H t α H α) 1 K t α.

31 Outline of the Proof III Weyl's Theorem shows λ i (Kα Sα S t α K t α ) ( ) λ i KαŜαŜ α t K α t Therefore the singular values of Kα Sα satisfy ) ( ) ς i KαŜα, 1 i k. ς i (Kα Sα This implies for any unitarily invariant matrix norm. Kα Sα KαŜα

32 Outline 1 Context The Software Package Cedre Spatial Discretization on General Polyhedral Meshes 2 Stability of Unstructured Muscl Schemes Instabilities in Three-dimensional Applications Preliminary Investigation Reconstruction and Stability on Unstructured Grids Applications to Gas Dynamics

33 Practical Conclusions for Cedre Practical Conclusions Theoretical results suggest (but no complete proof yet!) It is not reasonable to expect any consistent reconstruction to give better stability than the Least Squares Method. Use of larger stencils than rst neighborhood is recommended in three dimensions. Numerical Study for Linear Advection There are grids where the Least Squares Method is stable but alternative methods are not. In the case of piecewise linear reconstruction the second neighborhood turns out to be sucient for stability on tetrahedral meshes.

34 Practical Implementation Practical Solution for Solver Cedre Cedre handles large grids by parallel computing and grid partitioning. Large reconstruction stencils are not easy to implement. Practical Solution for Consistent and Stable Reconstruction Compute least-squares slope on rst neighborhood. Take weighted average of slopes over rst neighborhood. Numerical computations of spectra show that this method is stable. This method is consistent.

With the new scheme, the simulation of a three-dimensional subsonic ow

35 Application to Compressible Gas Dynamics Three-dimensional Flow Over a Deep Cavity. With the new scheme, the simulation of a three-dimensional subsonic ow over a deep cavity is possible without slope limiters on tetrahedral grids. Entropy Distribution Pressure Signal in the Cavity

36 Application to Compressible Gas Dynamics Three-dimensional Flow Over a Deep Cavity. Comparison of the spectra of a pressure signal in the upstream wall of the cavity. The limiter used is the best limiter available in Cedre: expensive!! Result without limiters Result with limiters SPL SPL f f

Application to Compressible Gas Dynamics Three-dimensional Jet Noise Computations Inuence of slope limiters on a jet noise computation on tetrahedral grids.

37 Application to Compressible Gas Dynamics Three-dimensional Jet Noise Computations Inuence of slope limiters on a jet noise computation on tetrahedral grids. Blue line : new scheme - limiters are active only inside the nozzle. Red and green line : new and old scheme with slope limiters active on the entire domain. Sensor Positions Pressure Signal at Sensor 15

38 Summary Without limiters, spatial Muscl discretization of linear advection leads to instabilities on unstructured three-dimensional grids containing tetrahedra and prisms. Instabilities can be eliminated by the least-squares method and larger stencils than the rst neighborhood. Outlook The criterion can be used for higher order reconstructions.

39 Appendix For Further Reading I F. Haider, J.P. Croisille, B. Courbet Stability Analysis of the Cell Centered Finite-Volume Muscl Method on Unstructured Grids. Submitted (2008)

Benchmark 3D: the VAG scheme

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