A Stable Spectral Difference Method for Triangles

A Stable Spectral Difference Method for Triangles Aravind Balan 1, Georg May 1, and Joachim Schöberl 2 1 AICES Graduate School, RWTH Aachen, Germany 2 Institute for Analysis and Scientific Computing, Vienna Technical University, Austria AIAA Aerospace Sciences Meeting January 4, 2011 Orlando, Florida Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 1 / 33

Outline 1 Background and Motivation 2 Spectral Difference(SD) Method 3 SD Method with Raviart-Thomas Elements 4 Stability Analysis 5 Results Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 2 / 33

Background and Motivation Spectral Difference (SD) high-order method for hyperbolic PDEs A quadrature free (pre-integrated) nodal Discontinuous Galerkin scheme Simple in formulation and implementation Found linearly unstable for triangles for order of accuracy > 2 Found stable with flux interpolation on Raviart-Thomas elements Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 3 / 33

SD Method for Triangles Hyperbolic conservation equation u( x,t) t + f (u) = 0 Transformation from reference element Φ : ξ x with J = x/ ξ (0,1) x1 x2 Φ :(ξ, η) (x,y) η (0,0) ξ (1,0) y x x3 Hyperbolic equation in reference domain ( u(ξ,t) t + 1 J ξ J J 1 f ) (u) Define Solution collocation nodes - ˆξ j Flux collocation nodes - ξ j = 0 Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 4 / 33

SD Method for Triangles Approximation of solution u h (ξ) = N m j=1 u jl j (ξ) l j P m l j (ˆξ k ) = δ jk u j = u h (ˆξj ) no. of degrees of freedom = N m = (m+1)(m+2) 2 Approximation of flux f h (ξ) = N m+1 j=1 f jˆlj (ξ) ˆlj P m+1 ˆlj ( ξ k ) = δ jk f j = f ) h ( ξj f j = { J J 1 f ( ξj ) ξj ˆT f num ξj ˆT f num n = h standard numerical flux no. of degrees of freedom = 2N m+1 Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 5 / 33

SD Method for Triangles Final form of the Spectral Difference scheme du (i) j dt + 1 N m+1 J (i) k=1 ) ξˆlk (i) (ˆξj f k = 0, j = 1,..., N m Linearly unstable for m 2 for triangles [Van den Abeele et al., 2008 ] Note - Each of the flux vectors need not be in P m+1 for the div to be in P m Raviart Thomas space Smallest space having div in P m Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 6 / 33

SD Method with Raviart-Thomas Elements Define Solution points - ˆξ j Flux points - ξ j, Directions - s j Approximation of solution u h (ξ) = N m j=1 u jl j (ξ) l j (ˆξ k ) = δ jk u j = u h (ˆξj ) Approximation of flux function in Raviart-Thomas (RT ) space f h (ξ) = N RT m j=1 f jψ j (ξ) ψ j ( ξ k ) s k = δ jk f j = f ) h ( ξj s j { ) J J 1 f ( ξj s j ξj f j = ˆT h ξj ˆT h standard numerical flux Nm RT = (m + 1) (m + 3) Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 7 / 33

SD Method with Raviart-Thomas Elements For a degree m, the RT space is defined as RT m = [P m ] 2 + (x, y) T P m. For m = 1, the monomials which form a basis in the RT space ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 x y 0 0 0 x 2 xy,,,,,,, 0 0 0 1 x y yx y 2 Less number of flux degrees of freedom compared to standard SD 2N m+1 N RT m = m + 3 Flux nodes distribution : m + 1 nodes on each edge and N RT m 3(m + 1) in the interior Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 8 / 33

SD Method with Raviart-Thomas Elements Final form of the new Spectral Difference scheme du (i) j dt + 1 Nm rt J (i) k=1 f (i) k ( ξ ψ ) ) k (ˆξj = 0, j = 1,..., N m Linearly stable for m = 1, 2, 3 in a simplified stability analysis Numerical experiments prove the viability of the scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 9 / 33

Linear Stability Analysis Linear advection equation u( x, t) t + f (u) = 0, f(u) = (u c cosθ, u c sinθ), θ [0, π 2 ] Consider Cartesian mesh with each element formed by fusing two triangles j i SD formulation, using upwind fluxes t U ( (i,j) = ν AU (i,j) + BU (i 1,j) + CU (i,j 1)), Linear stability analysis (LSA) Fourier transformation : u û Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 10 / 33

Linear Stability Analysis SD discretization of the Fourier mode ûe i(kxx+kyy) dû dt = ν t Zû Z = ( A + Be iσ + Ce iκ) (σ, κ) = (k x h, k y h) Full stability = Stability of spacial discretization + time discretization Stability of spacial discretization eigensystem of Z Stable flux nodes Re(λ(Z)) 0 Optimal flux nodes M ax( λ(z) ) (Spectral Radius) is minimum Stability is independent of the position of solution nodes Stability is independent of the position of flux nodes on the edges Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 11 / 33

LSA - Spatial Discretization RT 1 1 interior flux point at centroid - stable RT 2 3 interior points each with two ortho directions form 6 flux nodes 3 interior points are varied as ξ i = ξ c + α(ξi e ξ c ), i = 1, 2, 3 α [0, 1] Stable 0.5 α 0.521, considering θ [0, π 2 ] Stable and optimal α = 0.5 [higher order quadrature nodes] Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 12 / 33

LSA - Spatial Discretization RT 3 6 interior points each with two ortho directions form 12 flux nodes 6 interior points are varied as ξ i = ξ c + α(ξ e i ξ c ), i = 1, 2, 3 α [0, 1] ξ i = ξ c + β(ξ e i ξ c ), i = 4, 5, 6 β [0, 1] Stable and optimal α = 0.725 nodes] β = 0.676 [higher order quadrature Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 13 / 33

LSA - Spatial Discretization Stability and optimality for RT 3 0.6 0.65 0.7 0.75 " 0.8 0.85 0.9 Stable Region 0.95 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1! Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 14 / 33

LSA - Spatial Discretization y y y Stable and optimal flux points 1 1.2 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.2 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x RT 1 0.4 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 x RT 2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 x RT 3 Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 15 / 33

LSA - Full Discretization Full discretization û n+1 = Gû n L 2 stability ρ(g) 1 Get allowable CFL number (= c t h ) Max CFL number for Shu-RK3 time discretization 0.45 0.4 0.35 RT 1 RT 2 RT 3 CFL number 0.3 0.25 0.2 0.15 0.1 0.05 0 5 10 15 20 25 30 35 40 45! Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 16 / 33

Results - 2D Linear Advection Equation Linear advection equation u( x, t) t + f (u) = 0, f(u) = (cx u, c y u) T Convergence study 1 2 RT1 RT2 RT3 3 Log(L Error) 4 5 6 7 8 1 1.2 1.4 1.6 1.8 2 Log(N) Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 17 / 33

Results - 2D Euler Equations Euler equations in 2D f(u) x + g(u) = 0 y ρ ρu u = ρu ρv f = ρu 2 + p ρuv g = E u (E + p) NACA0012-1440 mesh elements 2.5 ρv ρuv ρv 2 + p v (E + p), 2 1.5 1 Y 0.5 0-0.5-1 -1.5-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 X Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 18 / 33

Results - 2D Euler Equations Relaxation Technique Backward Euler / Damped Newton ( I t dr(u n ) ) U n = tr(u n ), du t Newton iteration (Quadratic convergence) Preconditioning - Incomplete LU factorization Linear system solver - Restarted GMRES algorithm Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 19 / 33

Results - 2D Euler Equations Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree Figure: Mach contours - RT 2 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 20 / 33

Results - 2D Euler Equations Test case 1 - Convergence of residual Log(Res) 0 2 4 6 8 10 RT 1 RT 2 RT 3 12 14 0 5 10 15 20 25 Iterations Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 21 / 33

Results - 2D Euler Equations Test case 1 -Comparison of 4th order DG and RT 3 schemes Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 22 / 33

Results - 2D Euler Equations Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree Figure: Mach contours - RT 1 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 23 / 33

Results - 2D Euler Equations Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree Figure: Mach contours - RT 2 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 24 / 33

Results - 2D Euler Equations Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree Figure: Mach contours - RT 3 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 25 / 33

Results - 2D Euler Equations Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree Figure: Mach contours - RT 2 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 26 / 33

Results - 2D Euler Equations Test case 2 - Convergence of Residual Log(Res) 0 2 4 6 8 10 RT 1 RT 2 RT 3 12 14 0 5 10 15 20 25 Iterations Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 27 / 33

Results - 2D Euler Equations Test case 2 -Comparison of 4th order DG and RT 3 Schemes Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 28 / 33

Results - 2D Euler Equations Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree Figure: Mach contours - RT 1 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 29 / 33

Results - 2D Euler Equations Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree Figure: Mach contours - RT 2 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 30 / 33

Results - 2D Euler Equations Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree Figure: Mach contours - RT 3 scheme Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 31 / 33

Summary and Outlook Difference between SD and new SD New SD formulation is found to be linearly stable Numerical results show the viability Needs to be extended to solve NS equations, also to simulate transonic flows Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 32 / 33

Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111, and by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA8655-08-1-3060, is gratefully acknowledged Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 33 / 33

LSA - Spatial Discretization Stability and optimality for RT 2 max ",#,$ Re(%) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9! max ",# Re($) 1 x 10 3 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95! 80 100 70 max, (Z) 80 60 40 20 60 50 40 60 0 40 20 0 0 0.2 0.4 0.6 0.8 30 1 20 10 Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 34 / 33

Runge-Kutta Time stepping The solution at (n + 1)-th iteration, U n+1, is obtained from U n as For stability analysis w (0) = U n, k 1 w (k) = α kl w (l) + tβ kl R (l) k = 1,..., p, l=0 w (0) = Û n, U n+1 = w (p), k 1 w (k) = α kl w (l) + νβ kl Zw (l) k = 1,..., p, (1) l=0 Û n+1 = w (p). If G (k) is the amplification matrix in the k-th intermediate step, then one obtains k 1 G (0) = I, G (k) = (α kl I + νβ kl Z) G (l) k = 1,..., p. l=0 Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 35 / 33

Runge-Kutta Time Stepping Shu-RK3 1 α = 3 1 4 4 1 2 3 0 3 1, β = 1 0 4 0 0 2 3. 5 stage 4th order SSP 1 0.4443704940 0.5556295059 α = 0.6201018513 0 0.3798981486 0.1780799541 0 0 0.8219200458, 0.0068332588 0 0.5172316720 0.1275983113 0.3483367577 0.3917522270 0 0.3684105926 β = 0 0 0.2518917742 0 0 0 0.5449747502. 0 0 0 0.0846041633 0.2260074831 Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 36 / 33

Backward-Euler Time Stepping Backward Euler Taylor series for R U n+1 U n t = R(U n+1 ), Rewrite R(U n+1 ) = R(U n ) + dr(u n ) t +... dt dr(u n ) dt t = dr(u n ) du du dt t dr(u n ) du (U n+1 U n ). If (U n+1 U n ) is denoted as U n, the implicit scheme is ( I t dr(u n ) ) U n = tr(u n ), du Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 37 / 33

Stability Analysis Numerical schemes should posess non-linear stability properties Eg. Total Variation Diminishing (TVD) N T T V (ū n ) = ū n i+1 ū n i, i=1 T V ( ū n+1) T V (ū n ). TVD property convergence A conservative numerical scheme can be made to satisfy TVD by using limiters Linearly unstable limiter will act on smooth regions affect order of accuracy Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 38 / 33

Finding Transfer Matrix Solution interpolation Solution at flux nodes Dubiner basis u h (ξ) = χ (ξ) = Dubiner basis at flux nodes N m j=1 u j l j (ξ) u h (ˆξj ) = u j u h ( ξk ) = N m j=1 χ j l j (ξ) ) χ ( ξk = N m j=1 N m j=1 u j l j ( ξk ) χ j l j ( ξk ) ) χ j = χ (ˆξj Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 39 / 33

Finding Differentiation Matrix The monomials in the RT space at solution nodes φ n (ˆξj ) = Nm RT k=1 a n,k ψk (ˆξj ) a n,k = φ n ( ξk ) s k Its divergence ξ φ n (ˆξj ) = Nm RT k=1 a n,k ( ξ ψ k ) (ˆξj ) Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 40 / 33