AProofoftheStabilityoftheSpectral Difference Method For All Orders of Accuracy Antony Jameson 1 1 Thomas V. Jones Professor of Engineering Department of Aeronautics and Astronautics Stanford University International Conference on Advances in Scientific Computing in Memory of Professor David Gottlieb December 6-8, 2009, Brown University, Providence, RI
1 Introduction High Order Methods SD Applications SD Stability and Motivation 2 DG Stability Proof DG and Nodal DG Formulation Nodal DG Energy Stability 3 SD Stability Proof SD Formulation SD Reformulation Energy Stability of SD Scheme 4 Conclusion 5 Closing Remarks Dedication Acknowledgement
INTRODUCTION
High Order Methods Overview of Traditional Methods Traditional Methods 1 Structured Finite Difference and Finite Volume Methods simple formulation not geometrically flexible 2 Unstructured finite difference and finite volume methods inefficient but geometrically flexible limited to 2nd order accuracy
High Order Methods Overview of High Order Methods Modern High Order Methods 1 Discontinuous Galerkin Method geometrically flexible and locally conservative finite element weak formulations with coupled equations 2 Spectral Volume Method geometrically flexible and locally conservative finite volume direct formulations with uncoupled equations 3 Spectral Difference Method geometrically flexible and locally conservative finite difference direct formulations with uncoupled equations
SD Applications Various SD Applications Turbulence Transition Shock Capture Moving Deforming Mesh Adaptive Mesh Refinement
SD Applications Shock Capturing Use of artificial bulk viscosity with a dilatation sensor Artificial viscosity is switched on only in regions of strong negative dilatation (shocks) Smooth variation of artificial viscosity Use of adaptive mesh refinement in combination with artificial viscosity
SD Applications Turbulence Transition Test case: Transitional Flow over SD7003 airfoil at 4 degrees AOA and Re=60000 Code: 3D, parallel, unstructured solver for Navier-Stokes equations, NO sub-grid models Results: Good agreement with computational and experimental results Data Set Freestream Separation Transition Reattachment Turbulence x sep x tr /c x r /c TU-BS (Experiment) 0.08% 0.30 0.53 0.64 HFWT (Experiment) 0.1% 0.18 0.47 0.58 Visbal (ILES) 0 0.23 0.55 0.65 Uranga (ILES, DG) 0 0.23 0.51 0.60 Present ILES 0 0.23 0.53 0.64
Introduction DG Stability Proof SD Stability Proof Conclusion Closing Remarks SD Applications Euler Vortex Propagation and Deformable Mesh Blending vor: 5.0000E-04 6.1429E-03 1.1786E-02 1.7429E-02 2.3071E-02 2.8714E-02 3.4357E-02 vor: 5.0000E-04 4.0000E-02 5 5 0 0-5 -5-10 -5 0 5 10-10 6.1429E-03 1.1786E-02 1.7429E-02 2.3071E-02 2.8714E-02 3.4357E-02 4.0000E-02 High Order Temporal and Spatial Accuracy Demonstrated -5 0 5 10 Mesh Blending Technique with High Order Polynomial Numerical Results On Deformable Meshes Agree Well with Experiments Further Extension to FSI Problem Underway Plunging Airfoil and FSI on Deforming Mesh
SD Applications Adaptive Mesh Refinement Test Case: NACA0012 Airfoil, M=0.5, Steady Inviscid Flow Error Indicator 1: Unweighted Residual η k = R(Q H h ) Error Indicator 2: Entropy Adjoint η k = (V H h )T R(Q H h ) Two Mesh Adaptation Examples Illustrated
SD Stability and Motivation Spectral Difference Stability SD Stability Unstable for higher order triangular elements Sometimes weakly unstable in 1D depending on choices of flux collocation points
SD Stability and Motivation Motivation The aim is to prove: SD Stability in 1D The SD method for the 1D linear advection equation is stable for all orders of accuracy in an energy norm of Sobolev type. The norm is: Sobolev Norm u = (u 2 + cu (p)2 )dx where the coefficient c must be determined to cause a cancellation of terms (shown later).
SD Stability and Motivation STABILITY OF THE NODAL DISCONTINUOUS GALERKIN METHOD
Scalar Conservation Law Linear Advection Equation u t + u f (u) = x t + a u x = 0 we assume a > 0forrightrunningwave.
DG and Nodal DG Formulation Element Basis Function DG with Modal Basis NDG with Nodal Basis u h = nx u j Φ j j=1 u h = nx u j l j (x) j=1 where Φ is the modal basis where l is the Lagrange polynomial of degree p = n 1, and u j is the solution value at x j.
DG and Nodal DG Formulation Orthogonal Test Function The Galerkin formulations choose the same basis function to be the test function. The residual R h = u h t + x f (u h) is required to be orthogonal to the test function: Z xr x L R h Φ j dx = 0 or Z xr x L R h l j dx = 0
DG and Nodal DG Formulation Weak and Strong Forms Applying the technique of Integration-By-Parts to obtain the weak and strong forms of DG formulations: Weak Form Z xr x L uk h t Φ j dx Z xr x L f (uh k ) Φ j x dx + ˆf x R Φ j = 0 x L is required to be orthogonal to the test function: Strong Form Z xr x L uk h t Φ j dx + Φ j x L x f (uk h )dx +(ˆf f (uh k ))Φ x R j = 0 x L Z xr and similarly for Nodal DG formulation.
DG and Nodal DG Formulation Matrix Representation for DG Local Vectors u T =[u 1,...u n ] f = au Φ T =[Φ 1,...Φ n ] Mass and Stiffness Matrices M ij = S ij = Z xr x L Z xr x L Φ i Φ j dx Φ i Φ j dx Matrix Representation of the Weak Form M du dt S T f + ˆf x R Φ = 0 x L Matrix Representation of the Strong Form M du dt x R + Sf +(ˆf f )Φ = 0 x L
DG and Nodal DG Formulation Matrix Representations for Nodal DG Scheme Local Vectors u T =[u 1,...u n ] f = au l T =[l 1,...l n ] Mass and Stiffness Matrices M ij = S ij = Z 1 1 Z 1 1 l i l j dx l i l j dx Matrix Representation of the Weak Form M du dt S T f + ˆf l 1 1 = 0 Matrix Representation of the Strong Form M du dt + Sf +(ˆf f )l 1 1 = 0
DG and Nodal DG Formulation Matrix Representations for Nodal DG Scheme Multiplying by M 1,thestrongformofNodalDGformulation can be expressed as Matrix Representation of the Strong Form du dt + Df + M 1 l(ˆf f ) 1 1 = 0 where Differentiation Matrix D = M 1 S
Nodal DG Energy Stability Linear Advection Energy Estimate Multiplying the linear advection equation by u and integrating over x, Z b a u u t dx = a Z b a u u Z b dx = a x a ( u2 2 ) x dx Thus it satisfies the energy estimate d dt Z b a u 2 2 dx = 1 2 a(u2 a u 2 b )
Nodal DG Energy Stability Energy Estimate for the Discrete Solution Multiply the strong form by the local solution u T to obtain, u T M du dt + u T Sf + u T x R l(ˆf f ) = 0 x L Since M and S are pre-integrated exactly, this is equivalent to d dt Z xr x L Z u2 xr h 2 dx + a x L u u h h x dx + u x R h(ˆf au h ) = 0 x L Then integrate the middle term and combine it with the last term, d dt Z xr x L u2 h 2 dx = (u hˆf a u2 h 2 ) x R x L
Nodal DG Energy Stability Numerical Flux Element Interface Let u L and u R be values of u h on the left and right sides of a cell interface. For the numerical flux we now take ˆf = 1 2 a(u R + u L ) 1 2 α a (u R u L ), 0 α 1 where if α = 0wehaveacentralflux,andifα = 1wehavetheupwindflux.
Nodal DG Energy Stability Energy Estimate for the Discrete Solution On summing over all the elements: the L.H.S. of the energy estimate for the discrete DG solution derived previously yields: d dt Z xr x L u2 h 2 dx = d dt Z b a u 2 h 2 dx the R.H.S u 2 (u hˆf a h 2 ) has negative contribution at every element boundary, except the inflow boundary, which is strictly less than the boundary condition in the true solution, as shown in the next slide. x R x L
Nodal DG Energy Stability Energy Estimate for the Discrete Solution In the interior interface collecting the contributions elements on the left and right sides, there is a total negative contribution of u Rˆf a u 2 R 2 (u Lˆf a u2 L 2 ) = 1 2 a(u2 R u2 L ) 1 2 α a (u R u L ) 2 1 2 a(u2 R u2 L ) = 1 2 α a (u R u L ) 2 = negative contribution at every interior interface At the two end boundaries, Extrapolated upwind flux f b = au h at outflow boundary = negative contribution Use the true flux f a = au a at inflow boundary = positive but less than the true inflow boundary contribution au a u h 1 2 au2 h = 1 2 au2 a 1 2 a(u a u h ) 2 < 1 2 au2 a
Nodal DG Energy Stability Energy Stability of DG Energy Stability of DG This completes the proof that the DG scheme is energy stable for the linear advection equation.
Nodal DG Energy Stability STABILITY OF THE SPECTRAL DIFFERENCE METHOD
SD Formulation SD Formulation - Solution Represent the SD scheme in terms of a reference element covering [-1,1]. Solution Points Distribution The discrete solution is locally represented by Lagrange interpolation on the solution collocation points x j as Solution Reconstruction u h (x) = nx u j l j (x) j=1 where for polynomials of degree p, n = p + 1.
SD Formulation SD Formulation - Flux At interior flux points f j = f (u h (ˆx j )); u h (ˆx j ) is interpolated from u h (x). Single valued numerical flux ˆf is used at boundaries f ( 1) and f (1). Flux Points Distribution Flux is represented by a polynomial of degree p + 1, Flux Reconstruction f h (x) = n+1 X j=1 f jˆlj (x) where ˆl j (x) are the Lagrange polynomials defined by the n + 1fluxcollocationpoints ˆx j,whichincludetheelementboundaries.
SD Formulation SD Formulation - Residual Flux Divergence Differentiate the flux polynomial at the solution collocation points to obtain Compute Residual du i dt = n+1 X j=1 f jˆl j (x i )
SD Reformulation Recast SD Formulation with Explicit Boundary Flux Interior Flux and Boundary Flux Correction f = au ˆf ( 1) =auh ( 1)+f CL, ˆf (1) =au h (1)+f CR f CL = ˆf ( 1) au h ( 1), f CR = ˆf (1) au h (1) Upon substitution, and noting u h (x) is a polynomial of degree p hence exactly represented by the sum, the flux becomes, Flux Reconstruction (Follow the Procedure Proposed by Huynh) f h (x) = n+1 X j=1 f jˆlj (x) = f h (x) =f CLˆl1 (x)+f CRˆln+1 (x)+a n+1 X j=1 = f h (x) =f CLˆl1 (x)+f CRˆln+1 (x)+au h (x) u h (ˆx j )ˆl j (x)
SD Reformulation Recast SD Scheme Rewrite the SD scheme as u h t = f h x = u h t = a u h x f CLˆl 1 f CRˆl n+1 Evaluating this at the solution points du i dt = a = a nx D ij u j f CLˆl 1 (x i ) f CRˆl n+1 (x i ) j=1 nx D ij u j f CLˆl 1 (x i ) for upwind numerical flux j=1 where D = M 1 S is the differentiation matrix associated with the solution collocation points, and is uniquely determined by the points location and thepolynomialdegreep.
SD Reformulation Matrix Form of SD Scheme The SD scheme can be converted to a form which resembles the nodal DG method by multiplying it by the mass matrix to produce SD Scheme in NDG Form X j M ij du j dt + a X j X S ij u j = f CL M ijˆl 1 (x j ) j Now since ˆl 1 ( 1) =1andˆl 1 (1) =0, Boundary Flux nx M ijˆl 1 (x j )= j=1 Z 1 1 = ˆl 1 l i 1 l i (x) 1 nx ˆl 1 (x j )l j (x)dx = j=1 Z 1 1 Z 1 1 l i (x)ˆl 1 (x)dx Z 1 l i (x)ˆl 1 (x)dx = l i ( 1) l i (x)ˆl 1 (x)dx 1
SD Reformulation SD Vs Nodal DG Upon substitution the resulting SD scheme becomes, SD Scheme X j M ij du j dt + a X j S ij u j = f CL l i ( 1)+ Z 1 1 «l i (x)ˆl 1 (x)dx which differs from the corresponding nodal DG equation, NDG Scheme X j M ij du j dt + a X j «S ij u j = f CL l i ( 1) only in the last term.
SD Reformulation SD Reformulation In order to make the last term in the SD scheme equal to the nodal DG scheme, we consider the following strategy by replacing the mass matrix M by a matrix Q > 0suchthatitwill eventually cause a cancellation of the desired terms. Choice of New Mass Matrix Q = M + C QD =(M + C)D = MD + CD = S = CD = 0 Thus each row of C must be orthogonal to every column of D.
SD Reformulation Choice of Matrix Q In order to find a row vector which is orthogonal to every column of D, consider p th difference operator d T nx j=1 d j R p (x j )=R (p) p for any polynomial of degree p. Then further operation by the differentiation matrix D gives X X d i i j D ij R p (x j )=R (p+1) p = 0 Thus we find the desired matrix C that is orthogonal to D where c is an arbitrary parameter. Q = M + C = M + cdd T, with C = cdd T
SD Reformulation SD Scheme using Mass Matrix Q SD Scheme using Mass Matrix Q X j Q ij du j dt + a X j X X S ij u j = f CL M ijˆl 1 (x j ) f CL C ijˆl 1 (x j ) j j Extra Term X X f CL C ijˆl 1 (x j )= c f CL d i j j (p+1) d jˆl 1 (x j )= c f CL ˆl l (p) 1 i SD Scheme using Mass Matrix Q X j Q ij du j dt + a X j S ij u j = f CL l i ( 1)+ Z 1 1 «l i (x)ˆl (p+1) 1 (x)dx c ˆl 1 l (p) i
SD Reformulation Choice of Cancellation Constant Now if we can choose c so that the last two terms on the right cancel, we can attain an energy estimate with the norm u T Qu replacing u T Mu in each element. Required Cancellation f CL Z 1 1 l i (x)ˆl 1 (x)dx = f CL c ˆl (p+1) 1 l (p) i
SD Reformulation Choice of Cancellation Constant Choose the interior flux collocation points as the zeros of the Legendre polynomial L p (x) of degree p. Then L.H.S Z 1 1 ˆl1 (x)l i (x)dx =( 1)p+1 1 2 Z 1 1 xl p (x)l i (x)dx, where ˆl 1 (x) =( 1) p 1 2 (1 x)l p(x) Only the leading term in xl i (x) contributes to the integral. Let L.H.S L p (x) =c p x p +... = 1 3 5... (2p 1) x p +... p! and xl i (x) =pa i x p +... where a i is the leading coefficient in l i (x).
SD Reformulation Choice of Cancellation Constant The left hand side of the extra term becomes: L.H.S Z 1 1 xl p (x)l i (x)dx = 2p 2p + 1 a i c p The right hand side of the extra term can be expressed as R.H.S and since l i (x) =a i x p +... ˆl (p+1) =( 1) p+1 1 2 (p + 1)!c p l i (p) = p!a i
SD Reformulation Choice of Cancellation Constant Thus the desired cancellation is obtained by setting Cancellation Constant c = 2p 2p + 1 1 c 2 p 1 p!(p + 1)! > 0 In the case that the interface flux is not fully upwind, a similar calculation shows that the convection from the right boundary is correspondingly reduced, so that finally SD Scheme Cast in Nodal DG Form X j Q ij du j dt + a X j S ij u j = f CL l i ( 1) f CR l i (1)
Energy Stability of SD Scheme Energy Stability of the SD Scheme Since u h is a polynomial of degree p, i d iu i = u (p) h,andin each element X X u i Q ij u j = i j Z xr xl (uh 2 + cu(p)2 )dx h Now the same argument that was used to prove the energy stability of the nodal DG scheme establishes the energy stability of the SD scheme with the norm Z b a (uh 2 + cu(p)2 )dx h for the case of solution polynomials of degree p, provided that the interior flux collocation points are the zeros of L p (x).
Energy Stability of SD Scheme Behavior of the Cancellation Constant It can be seen that c decreases very rapidly with increasing p, as is illustrated by the following table of values of c. cvsp p c 1 1 3 2 4 135 3 1 1050
Energy Stability of SD Scheme CONCLUSION
Conclusion The result is consistent with the conclusion of Van den Abeele, Lacor and Wang that the stability of the spectral difference method depends only on the location of the flux collocation points. While it establishes the stability of the SD scheme when the interior flux collocation points are the zeros of the Legendre polynomial L p (x), itdoesnotprecludethestabilityofthesd scheme with other choices of the flux collocation points, possibly in a different norm. However, extensive calculations for the second, third and fourth order cases (not included here) have indicated that the conditions for the cancellation of the last two terms of the boundary correction can only be satisfied by choosing the interior flux collocation points as the zeros of L p (x).
Conclusion As the order of accuracy is increased the norm in which the SD method has been shown to be stable asymptotically approaches the usual energy norm. In light of the theorem that all norms are equivalent in a finite dimensional space, the proof also establishes the stability of the SD schemes in any norm. Whether a similar proof of stability for all orders of accuracy can be established for the multidimensional case with either tensor product or simplex elements remains a subject for future research.
ACKNOWLEDGEMENT AND DEDICATION
Dedication Dedication Dedication This paper is dedicated to the memory of David Gottlieb, a man of extraordinary honor, integrity and generosity, whose seminal contributions throughout his career have laid the foundation for the emergence of practically useful higher order methods in scientific computing.
Acknowledgement Acknowledgement Acknowledgement The author is indebted to both the National Science Foundation and the Air Force Office of Scientific Research for their support of the research that led to this paper through grants 0708071 and FA9550-07-1-0195.