From Finite Volumes to Discontinuous Galerkin and Flux Reconstruction

Transcription

1 From Finite Volumes to Discontinuous Galerkin and Flux Reconstruction Sigrun Ortleb Department of Mathematics and Natural Sciences, University of Kassel GOFUN 2017 OpenFOAM user meeting March 21, 2017

2 Numerical simulation of fluid flow This includes flows of liquids and gases such as flow of air or flow of water src: NCTR src: NOAA Spreading of Tsunami waves Weather prediction src: NASA Goddard Flow through sea gates Requirements on numerical solvers CC BY 3.0 Flow around airplanes High accuracy of computation Detailed resolution of physical phenomena Stability and efficiency, robustness Compliance with physical laws (e.g. conservation) 1

3 Contents 1 The Finite Volume Method 2 The Discontinuous Galerkin Scheme 3 SBP Operators & Flux Reconstruction 4 Current High Performance DG / FR Schemes

4 Derivation of fluid equations Based on physical principles: conservation of quantities & balance of forces mathematical tools: Reynolds transport & Gauß divergence theorem Different formulations: Integral conservation law d u dx + F(u, u) n dσ = dt V V V s(u, x, t) dx Partial differential equation u t + F = s embodies the physical principles V n 2

5 Derivation of the continuity equation Based on Reynolds transport theorem d u(x, t) dx = dt V t rate of change in moving volume = u(x, t) V t t rate of change in fixed volume dx + u(x, t) v n dσ V t + convective transfer through surface (V t control volume of fluid particles) 3

6 Derivation of the continuity equation Based on Reynolds transport theorem d u(x, t) dx = dt V t rate of change in moving volume = u(x, t) V t t rate of change in fixed volume Physical principle: conservation of mass dx + u(x, t) v n dσ V t + convective transfer through surface (V t control volume of fluid particles) dm dt = d ρ ρ dx = dt V t V t t dx + ρv n dσ = 0 V t 3

7 Derivation of the continuity equation Based on Reynolds transport theorem d u(x, t) dx = dt V t rate of change in moving volume = u(x, t) V t t rate of change in fixed volume Physical principle: conservation of mass dx + u(x, t) v n dσ V t + convective transfer through surface (V t control volume of fluid particles) dm dt = d ρ ρ dx = dt V t V t t dx + ρv n dσ = 0 V t Divergence theorem yields [ ] ρ t + (ρv) dx = 0 ρ t + (ρv) = 0 V V t continuity equation 3

8 The compressible Navier-Stokes equations... are based on conservation of mass, momentum and energy u t + F = s [ ] u t + Finv + (A(u) u) = s inviscid & viscous fluxes Conservative variables u R 5, fluxes F R 3 5 and sources s R 5 ρ ρv 0 u = ρv, F = ρe ρv v + pi τ (ρe + p)v κ T τ v, s = ρg ρ(q + g v) convective fluxes, heat fluxes & surface forces p pressure, T temperature τ viscous stress tensor κ thermal conductivity body forces & heat sources g gravitational & electromag. forces q intern. heat sources simplified programming by representation in same generic form sufficient to develop discretization schemes for generic conservation law 4

9 General discretization techniques Finite differences / differential form approximation of nodal values and nodal derivatives easy to derive, efficient essentially limited to structured meshes Finite volumes / integral form approximation of cell means and integrals conservative by construction suitable for arbitrary meshes difficult to extend to higher order Finite elements / weak form weighted residual formulation quite flexible and general suitable for arbitrary meshes x i,j+1 x i 1,j x i,j x i+1,j x i,j 1 5

10 Finite volume schemes... based on the integral rather than the differential form Integral conservation enforced for small control volumes V i defined by computational mesh V = K i=1 V i Degrees of freedom: cell means u i (t) = 1 u(x, t) dx V i V i To be specified: concrete definition of control volumes type of approximation inside these cell-centered vs. vertex-centered possibly staggered for different variables numerical method for evaluation of integrals and fluxes 6

11 Why the integral form? Because this is the form directly obtained from physics. 1D scalar hyperbolic conservation law u (u) + f = 0 t x f (u) = 1 2 u2 u + u u = 0 t x (Burgers equation) 1, x < 0, init. cond.: u 0(x) = cos(πx), 0 x 1, 1, x > 1. PDE theory tells us: As long as the exact solution is smooth, it is constant along charactereristic curves 7

12 Why the integral form? Because this is the form directly obtained from physics. 1D scalar hyperbolic conservation law u (u) + f = 0 t x f (u) = 1 2 u2 u + u u = 0 t x (Burgers equation) 1, x < 0, init. cond.: u 0(x) = cos(πx), 0 x 1, 1, x > 1. PDE theory tells us: As long as the exact solution is smooth, it is constant along charactereristic curves Characteristic curves are straight lines and cross smooth solution breaks down integral form (time integrated) still holds 7

13 Why the integral form? Because this is the form directly obtained from physics. 1D scalar hyperbolic conservation law u (u) + f = 0 t x f (u) = 1 2 u2 u + u u = 0 t x (Burgers equation) 1, x < 0, init. cond.: u 0(x) = cos(πx), 0 x 1, 1, x > 1. PDE theory tells us: As long as the exact solution is smooth, it is constant along charactereristic curves Characteristic curves are straight lines and cross smooth solution breaks down integral form (time integrated) still holds It is important to ensure correct shock speed 7

14 Discretization in conservative form In 1D, the FV scheme can be regarded as a FD scheme in conservative form u i (t) = 1 xi+1/2 u(x, t) dx x i x i 1/2 8

15 Discretization in conservative form In 1D, the FV scheme can be regarded as a FD scheme in conservative form u i (t) = 1 xi+1/2 u(x, t) dx x i x i 1/2 On a control volume V = [x i 1/2, x i+1/2 ], the exact solution fulfills d dt xi+1/2 x i 1/2 u dx + [f (u)] x i+1/2 x i 1/2 = 0 u n+1 i ui n discretized: x i t + f (u n i+1/2 ) f (un i 1/2 ) = 0 8

16 Discretization in conservative form In 1D, the FV scheme can be regarded as a FD scheme in conservative form u i (t) = 1 xi+1/2 u(x, t) dx x i x i 1/2 On a control volume V = [x i 1/2, x i+1/2 ], the exact solution fulfills d dt xi+1/2 x i 1/2 u dx + [f (u)] x i+1/2 x i 1/2 = 0 u n+1 i ui n discretized: x i t + f (u n i+1/2 ) f (un i 1/2 ) = 0 flux values f (u i±1/2 ) depending on unknown face quantities u i 1/2, u i+1/2 reconstruction necessary from available data..., u i 1, u i, u i+1,... Introduction of numerical flux functions f u n+1 ( i f (ui n, ui+1) n f (ui 1, n ui n ) ) = u n i x i t 8

17 Discretization in conservative form In 1D, the FV scheme can be regarded as a FD scheme in conservative form u i (t) = 1 xi+1/2 u(x, t) dx x i x i 1/2 On a control volume V = [x i 1/2, x i+1/2 ], the exact solution fulfills d dt xi+1/2 x i 1/2 u dx + [f (u)] x i+1/2 x i 1/2 = 0 u n+1 i ui n discretized: x i t + f (u n i+1/2 ) f (un i 1/2 ) = 0 flux values f (u i±1/2 ) depending on unknown face quantities u i 1/2, u i+1/2 reconstruction necessary from available data..., u i 1, u i, u i+1,... Introduction of numerical flux functions f The heart of FV schemes u n+1 i = u n i x i t ( f (u n i, u n i+1) f (u n i 1, u n i ) ) 8

18 Classical numerical flux functions linked to Riemann problems & characteristic directions 9

19 Classical numerical flux functions linked to Riemann problems & characteristic directions Upwind methods Scalar linear equation a > 0 ( u t + a u x = 0) u n+1 i = u n i x t (aun i au n i 1) (f (u i, u i+1 ) = au i ) Linear system of equations f (u i, u i+1 ) = A + u i + A u i+1 u n+1 i Nonlinear systems = u n i t ( A + (u n i u n i 1) + A (u n i+1 u n i ) ) x Flux vector splitting f(u) = f + (u) + f (u) f (u i, u i+1 ) = f + (u i ) + f (u i+1 ) Steger & Warming, van Leer, AUSM and variants 9

20 Classical approximate Riemann solvers for Euler equations Roe scheme exact solution to linear Riemann problem u t + A LR(u L, u R ) u x = 0 10

21 Classical approximate Riemann solvers for Euler equations Roe scheme exact solution to linear Riemann problem u t + A LR(u L, u R ) u x = 0 Properties of Roe matrix A LR A LR A(u) = Df(u) A LR (u, u) = A(u) A LR is diagonalizable f(u R ) f(u L ) = A LR (u R u L ) (mean value property) entropy-fix needed 10

22 Classical approximate Riemann solvers for Euler equations Roe scheme exact solution to linear Riemann problem u t + A LR(u L, u R ) u x = 0 Properties of Roe matrix A LR A LR A(u) = Df(u) A LR (u, u) = A(u) A LR is diagonalizable f(u R ) f(u L ) = A LR (u R u L ) (mean value property) entropy-fix needed HLL scheme Godunov-type scheme 10

23 Classical approximate Riemann solvers for Euler equations Roe scheme exact solution to linear Riemann problem u t + A LR(u L, u R ) u x = 0 Properties of Roe matrix A LR A LR A(u) = Df(u) A LR (u, u) = A(u) A LR is diagonalizable f(u R ) f(u L ) = A LR (u R u L ) (mean value property) entropy-fix needed HLL scheme Godunov-type scheme approximates only one intermediate state based on integral conservation law 10

24 What about higher order schemes?

25 Challenges posed by hyperbolic conservation laws src: NOAA gust front CC-BY-SA-3.0 bow waves Computation of discontinuous solutions (shocks) Unphysical oscillations Oscillations of approximate solution Needs additional numerical dissipation 11

26 Difficulties regarding discontinuous solutions Example: Burgers equation u t + u u x = 0, u(x, 0) = sin(x) Exact solution 12

27 Difficulties regarding discontinuous solutions Example: Burgers equation u t + u u x = 0, u(x, 0) = sin(x) Exact solution Projection Goal: Controll oscillations 12

28 A first approach towards higher order FV schemes with linear reconstruction modify left and right states u n+1 i = u n i + x i t ( f (u n i, u n i+1) f (u n i 1, u n i ) ) = 0 13

29 A first approach towards higher order FV schemes with linear reconstruction modify left and right states u n+1 i = u n i + x i t ( f (u n i, u n i+1) f (u n i 1, u n i ) ) = 0 linear reconstruction within cells u(x) = u i + s(x x i ) preserve integral means how to compute slopes s? prevent creation of new max/min 13

30 A first approach towards higher order FV schemes with linear reconstruction modify left and right states u n+1 i = u n i + x i t ( f (u n i, u n i+1) f (u n i 1, u n i ) ) = 0 linear reconstruction within cells u(x) = u i + s(x x i ) preserve integral means how to compute slopes s? prevent creation of new max/min enforce TVD property (relates to properties of exact solution) i u n+1 i+1 u n+1 i i u n i+1 u n i sufficient condition { } x si x s i 0, 2 u i u i 1 u i+1 u i 13

31 A first approach towards higher order FV schemes with linear reconstruction modify left and right states u n+1 i = u n i + x i t ( f (u n i, u n i+1) f (u n i 1, u n i ) ) = 0 linear reconstruction within cells u(x) = u i + s(x x i ) preserve integral means how to compute slopes s? prevent creation of new max/min enforce TVD property (relates to properties of exact solution) i u n+1 i+1 u n+1 i i u n i+1 u n i sufficient condition { } x si x s i 0, 2 u i u i 1 u i+1 u i Typical example: s i = 1 x minmod(u i+1 u i, u i u i 1 ), minmod(a, b) = a a < b, ab > 0 b a b, ab > 0 0 otherwise 13

32 The ENO reconstruction ENO stands for essentially non-oscillatory higher order reconstruction via interpolation adaptively choose stencil avoid interpolation across shocks 14

33 The ENO reconstruction ENO stands for essentially non-oscillatory higher order reconstruction via interpolation adaptively choose stencil avoid interpolation across shocks ENO approach: successive increase of polynomial degree via Newton interpolation compare divided differences obtained by left or right extension 14

34 The ENO reconstruction ENO stands for essentially non-oscillatory higher order reconstruction via interpolation adaptively choose stencil avoid interpolation across shocks ENO approach: successive increase of polynomial degree via Newton interpolation compare divided differences obtained by left or right extension ENO properties: constructs only one reconstruction polynomial prone to round off errors 14

35 The WENO reconstruction WENO stands for weighted essentially non-oscillatory A priori: choice of main (central) stencil as well as secondary stencils Compute polynomial reconstruction on each stencil, conserve integral means Compute weights depending on oscillatory behaviour of reconstruction Evaluate weighted sum of reconstruction polynomials 15

36 The WENO reconstruction WENO stands for weighted essentially non-oscillatory A priori: choice of main (central) stencil as well as secondary stencils Compute polynomial reconstruction on each stencil, conserve integral means Compute weights depending on oscillatory behaviour of reconstruction Evaluate weighted sum of reconstruction polynomials WENO properties: The higher the discretization order the higher the number of required neighbor cells Unstructured grids: difficult to construct stencils High demand on resources (CPU time / memory requirement) 15

37 From Finite Volumes..... to Discontinuous Galerkin

38 Discontinuous Galerkin schemes discontinuous approximate solutions modern space discretization d u h Φ dx + F (u h dt, u+ h, n) Φ dσ F(u h ) Φ dx = q h Φ dx V i V i V i V i FV: Φ = Φ 0 DG: Φ = Φ 0, Φ 1,..., Φ N Closer link to given physical equations High accuracy & flexibility Compact domains of dependance Highly adapted to computations in parallel 16

39 High resolution of DG scheme Double Mach reflection: Shock hitting fixed wall u u + Inflow Outflow Fixed wall Initial shock Excellent shock resolution & Detailed representation of fine structures Density distribution: FV vs. DG scheme FV scheme DG scheme (N = 4) 17

40 The triangular grid DG scheme Hyperbolic conservation law in 2D u(x, t) + F(u(x, t)) = 0, (x, t) Ω R+ t Initial conditions: Boundary conditions: u(x, 0) = u 0 (x) inflow/outflow, reflecting walls 18

41 The triangular grid DG scheme Hyperbolic conservation law in 2D u(x, t) + F(u(x, t)) = 0, (x, t) Ω R+ t Initial conditions: Boundary conditions: u(x, 0) = u 0 (x) inflow/outflow, reflecting walls Approximation u h,n (x, t): piecewise polynomial in x, degree N T h = {τ 1, τ 2,..., τ #T h} Triangulation Ω u h,n (, t) 18

42 The triangular grid DG scheme Multiplication by test functions Φ P N (τ i ), Integration over τ i d u Φ dx + F(u) Φ dx = 0 dt τ i τ i Use divergence theorem d u Φ dx + F(u) n Φ dσ F(u) Φ dx = 0 dt τ i τ i τ i u h,n H(u h,n, u+ h,n, n) F(u h,n) Nutze orthogonale Polynombasis { Φ 1, Φ 2,..., Φ q(n) } von P N (τ i ) u h,n τi (x, t) = q(n) k=1 û i k (t)φ k(x), q(n) = (N + 1)(N + 2)/2 19

43 The triangular grid DG scheme Multiplication by test functions Φ P N (τ i ), Integration over τ i d u Φ dx + F(u) Φ dx = 0 dt τ i τ i Use divergence theorem d u Φ dx + F(u) n Φ dσ F(u) Φ dx = 0 dt τ i τ i τ i u h,n F (u h,n, u+ h,n, n) F(u h,n) 20

44 The triangular grid DG scheme Multiplication by test functions Φ P N (τ i ), Integration over τ i d u Φ dx + F(u) Φ dx = 0 dt τ i τ i Use divergence theorem d u Φ dx + F(u) n Φ dσ F(u) Φ dx = 0 dt τ i τ i τ i u h,n F (u h,n, u+ h,n, n) F(u h,n) Use orthogonal polynomial basis { Φ 1, Φ 2,..., Φ q(n) } of P N (τ i ) u h,n τi (x, t) = q(n) k=1 û i k (t)φ k(x), q(n) = (N + 1)(N + 2)/2 20

45 With respect to orthogonal basis Time Evolution of coefficients ( d dt ûi k = F (u h,n, u+ h,n, n)φ kdσ + τ i Quadrature rules ) F(u h,n ) Φ k dx / Φ k 2 L 2 τ i 21

46 With respect to orthogonal basis Time Evolution of coefficients ( d dt ûi k = F (u h,n, u+ h,n, n)φ kdσ + τ i Quadrature rules ) F(u h,n ) Φ k dx / Φ k 2 L 2 τ i System of ODEs for coefficients û i k d ) dt Û(t) = L h,n (Û(t), t, Û = [ û i ] k k=1,...,q(n), i=1,...,#t h 21

47 With respect to orthogonal basis Time Evolution of coefficients ( d dt ûi k = F (u h,n, u+ h,n, n)φ kdσ + τ i Quadrature rules ) F(u h,n ) Φ k dx / Φ k 2 L 2 τ i System of ODEs for coefficients û i k d ) dt Û(t) = L h,n (Û(t), t, Û = [ û i ] k k=1,...,q(n), i=1,...,#t h e.g. Runge Kutta time integration Cockburn and Shu ( , 1998) 21

48 With respect to orthogonal basis Time Evolution of coefficients ( d dt ûi k = F (u h,n, u+ h,n, n)φ kdσ + τ i Quadrature rules ) F(u h,n ) Φ k dx / Φ k 2 L 2 τ i System of ODEs for coefficients û i k d ) dt Û(t) = L h,n (Û(t), t, Û = [ û i ] k k=1,...,q(n), i=1,...,#t h e.g. Runge Kutta time integration Cockburn and Shu ( , 1998) Allows easy incorporation of modal filters Meister, Ortleb, Sonar 12 21

49 Damping strategies for DG Modify approximate solution Modify equation Limiters (H)WENO- Reconstruction Explicit dissipation u + F(u) = ɛ u t Use neighboring data at shocks Often N = 1 Cockburn,Shu 89 Krivodonova 07 Use stencil data Weighted interpol. polynomials Qiu,Shu 04/05 In conservation law or discretization Time step O(h 2 ) Jaffre,Johnson,Szepessy 95 Persson,Peraire 06 Feistauer,Kučera 07 22

50 Damping for spectral methods 1D periodic case: t u + x f (u) = 0, x [ π, π] Fourier method: t u N + x P Nf (u N ) = 0 u N (x, t) = û k (t)e ikx k N Modal Filtering Modify coefficients at times t n un(x,t σ n ) = ( ) k σ ûk n e ikx N k N σ : [0, 1] [0, 1] Gottlieb, Lustman, Orzag 81 23

51 Damping for spectral methods 1D periodic case: t u + x f (u) = 0, x [ π, π] Fourier method: t u N + x P Nf (u N ) = 0 u N (x, t) = û k (t)e ikx k N Modal Filtering Modify coefficients at times t n un(x,t σ n ) = ( ) k σ ûk n e ikx N k N Spectral viscosity Add special viscosity term [ p+1 p p u ] N ɛ N ( 1) x p Q N x p Gottlieb, Lustman, Orzag 81 Tadmor 89 24

52 Damping for spectral methods 1D periodic case: t u + x f (u) = 0, x [ π, π] Fourier method: t u N + x P Nf (u N ) = 0 u N (x, t) = û k (t)e ikx k N Modal Filtering Modify coefficients at times t n un(x,t σ n ) = ( ) k σ ûk n e ikx N k N Simple implementation Link Spectral viscosity Add special viscosity term [ p+1 p p u ] N ɛ N ( 1) x p Q N x p Convergence theory, Parameter choice 25

53 Shock-density interaction (Shu-Osher test case) Initial conditions (ρ, v 1, v 2, p) = { ( , , 0, ) if x < 4 ( sin(5x), 0, 0, 1) if x 4 DG with modal filtering approximate density solution (t = 1.8) Discretization: N = 5, K =

54 Further classical test cases forward facing step density N = 5, K = 6496 double Mach reflection density N = 4, K = shock-vortex interaction pressure N = 7, K =

55 Finite Volumes & Discontinuous Galerkin..... and beyond?

56 Contents 1 The Finite Volume Method 2 The Discontinuous Galerkin Scheme 3 SBP Operators & Flux Reconstruction 4 Current High Performance DG / FR Schemes

57 Conservative schemes in 1D Finite Volume Method Discontinuous Galerkin Method Cockburn/Shu 89 Spectral Difference Method Kopriva/Kolias 96, Liu et al. 06, Wang et al. 07 Flux Reconstruction Method Huynh 11 VCJH Energy Stable FR Method Vincent et al. 10 SBP-SAT schemes Originally: Kreiss/Scherer 74 28

58 The 1D DG scheme Scalar hyperbolic conservation law in 1D u(x, t) + f (u(x, t)) = 0, t > 0, x Ω = [α, β] t x 29

59 The 1D DG scheme Scalar hyperbolic conservation law in 1D u(x, t) + f (u(x, t)) = 0, t > 0, x Ω = [α, β] t x Subdivision of Ω in Ω = Ω i = [x i, x i+1 ] i i approximation of u on Ω i uh i N+1 (x, t) = u i l (t)φi l (x) k=1 u i = (u1 i,..., ui p+1 )T with basis functions Φi l solution vector The DG scheme in strong form Ω i uh i f t Φi h kdx+ Ωi i x Φi kdx = [fi 1,i f h i (x i )]Φ i k(x i ) [fi,i+1 f h i (x i+1 )]Φ i k(x i+1 ) 29

60 The 1D DG scheme Scalar hyperbolic conservation law in 1D u(x, t) + f (u(x, t)) = 0, t > 0, x Ω = [α, β] t x Subdivision of Ω in Ω = Ω i = [x i, x i+1 ] i i approximation of u on Ω i uh i N+1 (x, t) = u i l (t)φi l (x) k=1 u i = (u1 i,..., ui p+1 )T with basis functions Φi l solution vector The DG scheme in strong form Ω i uh i f t Φi h kdx+ Ωi i x Φi kdx = [fi 1,i f h i (x i )]Φ i k(x i ) [fi,i+1 f h i (x i+1 )]Φ i k(x i+1 ) equivalent to M i dui dt + S i f i = [(f h f )Φ i ] x i+1 x i M i kl = Ω i Φ i k Φi l dx S i kl = Ω i Φ i k x Φi l dx Φ i = (Φ i 1,..., Φi p+1 )T 29

61 SBP schemes Generalized definition of 1D SBP scheme Del Rey Fernández et al. 14 M symmetric positive definite D := M 1 S approximates x S + S T = B with (x µ ) T Bx ν = (x i+1 ) µ+ν (x i ) µ+ν SBP mimics integration by parts 30

62 SBP schemes Generalized definition of 1D SBP scheme Del Rey Fernández et al. 14 M symmetric positive definite D := M 1 S approximates x S + S T = B with (x µ ) T Bx ν = (x i+1 ) µ+ν (x i ) µ+ν SBP mimics integration by parts fulfilled by DG scheme M du dt + Sf = [(f h f )Φ] x i+1 x i M kl = Ω i Φ k Φ l dx S kl = Ω i Φ k x Φ ldx B kl = [Φ k Φ l ] x i+1 x i Gauss-Lobatto (GLL) and Gauss-Legendre (GL) DG schemes: B GLL = diag{ 1, 0,..., 0, 1}, B GL,N=1 = diag{ 3, 3}, B GL,N=2 = 1 1 ξ 2 0 ξ 3 ξ 3 1 ξ 2 ξ ξ 3 ξ 3 ξ ξ 3 ξ 3, ξ =

63 Advantages of SBP schemes Provable linear stability Energy stability w.r.t. 1 2 u 2 M 31

64 Advantages of SBP schemes Provable linear stability Energy stability w.r.t. 1 2 u 2 M Relation to quadrature formulae Corresponding QF preserve certain properties of functional e.g. discrete divergence theorem Hicken/Zingg 13 31

65 Advantages of SBP schemes Provable linear stability Energy stability w.r.t. 1 2 u 2 M Relation to quadrature formulae Corresponding QF preserve certain properties of functional e.g. discrete divergence theorem Hicken/Zingg 13 Correct discretization of split form conservation laws split forms (e.g. skew-symmetric) Better control of oscillations preservation of secondary quantities, e.g. kinetic energy 31

66 Advantages of SBP schemes Provable linear stability Energy stability w.r.t. 1 2 u 2 M Relation to quadrature formulae Corresponding QF preserve certain properties of functional e.g. discrete divergence theorem Hicken/Zingg 13 Correct discretization of split form conservation laws split forms (e.g. skew-symmetric) Better control of oscillations preservation of secondary quantities, e.g. kinetic energy possible lack of discrete conservation 31

67 Advantages of SBP schemes Provable linear stability Energy stability w.r.t. 1 2 u 2 M Relation to quadrature formulae Corresponding QF preserve certain properties of functional e.g. discrete divergence theorem Hicken/Zingg 13 Correct discretization of split form conservation laws split forms (e.g. skew-symmetric) Better control of oscillations preservation of secondary quantities, e.g. kinetic energy possible lack of discrete conservation SBP schemes: equivalent telescoping form Fisher et al. 12 if convergent, then weak solution (Lax-Wendroff) 31

68 Observation Discontinuous Galerkin Energy Stable Flux Reconstruction (VCJH) 32

69 Observation Discontinuous Galerkin Energy Stable Flux Reconstruction (VCJH) These methods meet as SBP schemes! 32

70 Spectral difference and flux reconstruction schemes The SD scheme [Wang et al. 07] SD u f + t x = 0 Construction of f SD f SD f h u h 33

71 Spectral difference and flux reconstruction schemes The SD scheme [Wang et al. 07] SD u f + t x = 0 Construction of f SD Generalized by FR scheme [Huynh 11] f SD f FR = fh i + [fi 1,i fh i (x i )] g L + [fi,i+1 fh i (x i+1 )] g R }{{}}{{} f CL { f CR where g L, g R P N+1 gl (x i ) = 1 g with R (x i ) = 0 g L (x i+1 ) = 0 g R (x i+1 ) = 1 f h u h DG for g L, g R right & left Radau polynomials 33

72 Spectral difference and flux reconstruction schemes The SD scheme [Wang et al. 07] SD u f + t x = 0 Construction of f SD Generalized by FR scheme [Huynh 11] f SD f FR = fh i + [fi 1,i fh i (x i )] g L + [fi,i+1 fh i (x i+1 )] g R }{{}}{{} f CL { f CR where g L, g R P N+1 gl (x i ) = 1 g with R (x i ) = 0 g L (x i+1 ) = 0 g R (x i+1 ) = 1 in matrix-vector form [Allaneau/Jameson 11] f h u h DG for g L, g R right & left Radau polynomials expand u h, f h and g L, g R in same basis {Φ k } multiply by M with M kl = Ω i Φ k Φ l dx M du dt + S f = f CLM g L f CRM g R 33

73 Spectral difference and flux reconstruction schemes The SD scheme [Wang et al. 07] SD u f + t x = 0 Construction of f SD Generalized by FR scheme [Huynh 11] f SD f FR = fh i + [fi 1,i fh i (x i )] g L + [fi,i+1 fh i (x i+1 )] g R }{{}}{{} f CL { f CR where g L, g R P N+1 gl (x i ) = 1 g with R (x i ) = 0 g L (x i+1 ) = 0 g R (x i+1 ) = 1 in matrix-vector form [Allaneau/Jameson 11] f h u h DG for g L, g R right & left Radau polynomials expand u h, f h and g L, g R in same basis {Φ k } multiply by M with M kl = Ω i Φ k Φ l dx M du dt + S f = f CLM g f CRM g L R [ du ] dt + D f = f CLg f CRg, D = L R M 1 S 33

74 Relation of FR framework to DG scheme Reformulate M du dt + Sf = f CLM g L f CRM g R f CL = f i 1,i f i h (x i ) f CR = f i,i+1 f i h (x i+1) 34

75 Relation of FR framework to DG scheme Reformulate f CL = f i 1,i f i h (x i ) M du dt + Sf = f CLM g L f CRM g f CR = fi,i+1 f h i (x i+1) R As M g = g Φdx = [gφ] x i+1 x i gφ dx ( g is representation of g ) Ω i Ω i 34

76 Relation of FR framework to DG scheme Reformulate f CL = f i 1,i f i h (x i ) M du dt + Sf = f CLM g L f CRM g f CR = fi,i+1 f h i (x i+1) R As M g = g Φdx = [gφ] x i+1 x i gφ dx ( g is representation of g ) Ω i Ω i we have (as g R (x i ) = g L (x i+1 ) = 0) RHS FR = f CL Φ(x i ) f CR Φ(x i+1 ) }{{} RHS DG + (f CL g L + f CR g R ) Φ dx Ω } i {{} Deviation from DG 34

77 Relation of FR framework to DG scheme Reformulate f CL = f i 1,i f i h (x i ) M du dt + Sf = f CLM g L f CRM g f CR = fi,i+1 f h i (x i+1) R As M g = g Φdx = [gφ] x i+1 x i gφ dx ( g is representation of g ) Ω i Ω i we have (as g R (x i ) = g L (x i+1 ) = 0) RHS FR = f CL Φ(x i ) f CR Φ(x i+1 ) }{{} RHS DG + Two equivalent formulations for FR scheme: M du dt + S f = RHS DG + (f CL g L + f CR g R ) Φ dx Ω i du dt + D f = f CL g L f CR g R (f CL g L + f CR g R ) Φ dx Ω } i {{} Deviation from DG 34

78 VCJH schemes: energy stable FR Derivation of energy stable FR schemes: M du dt + S f = RHS DG + (f CL g L + f CR g R ) Φ dx Ω i K du dt + K Df = f CLK g f CRK g, K pos. semidef. with K D = 0 L R 35

79 VCJH schemes: energy stable FR Derivation of energy stable FR schemes: M du dt + S f = RHS DG + (f CL g L + f CR g R ) Φ dx Ω i K du dt + K Df = f CLK g f CRK g, K pos. semidef. with K D = 0 L R Summing up yields ( ) ( ) ( ) du M + K dt +Sf = RHS DG +f CL g L Φ dx K g + f CR g R Φ dx K g L R Ω i Ω i 35

80 VCJH schemes: energy stable FR Derivation of energy stable FR schemes: M du dt + S f = RHS DG + (f CL g L + f CR g R ) Φ dx Ω i K du dt + K Df = f CLK g f CRK g, K pos. semidef. with K D = 0 L R Summing up yields ( ) ( ) ( ) du M + K dt +Sf = RHS DG +f CL g L Φ dx K g + f CR g R Φ dx K g L R Ω i Ω i VCJH schemes [Vincent/Castonguay/Jameson 10] Choose g L, g R such that red terms vanish for suitable K 35

81 VCJH schemes: energy stable FR Derivation of energy stable FR schemes: M du dt + S f = RHS DG + (f CL g L + f CR g R ) Φ dx Ω i K du dt + K Df = f CLK g f CRK g, K pos. semidef. with K D = 0 L R Summing up yields ( ) ( ) ( ) du M + K dt +Sf = RHS DG +f CL g L Φ dx K g + f CR g R Φ dx K g L R Ω i Ω i VCJH schemes [Vincent/Castonguay/Jameson 10] Choose g L, g R such that red terms vanish for suitable K Similar to DG: M M + K (modified mass matrix) filtered DG scheme [Allaneau/Jameson 11] 35

82 VCJH schemes: energy stable FR Derivation of energy stable FR schemes: M du dt + S f = RHS DG + (f CL g L + f CR g R ) Φ dx Ω i K du dt + K Df = f CLK g f CRK g, K pos. semidef. with K D = 0 L R Summing up yields ( ) ( ) ( ) du M + K dt +Sf = RHS DG +f CL g L Φ dx K g + f CR g R Φ dx K g L R Ω i Ω i VCJH schemes [Vincent/Castonguay/Jameson 10] Choose g L, g R such that red terms vanish for suitable K Similar to DG: M M + K (modified mass matrix) filtered DG scheme [Allaneau/Jameson 11] Fulfills SBP property! [D = M 1 S = (M + K) 1 S] 35

83 Comparison of low order DGSBP schemes & Use of kinetic energy preservation and skew-symmetric forms

84 Smooth solutions to 1D Navier-Stokes equations Non-linear acoustic pressure wave ρ(x, 0) = 1, v(x, 0) = 1, p(x, 0) = sin(2πx), x [0, 1] periodic BC, viscosity µ = 0.002, Prandtl number Pr = 0.72 Gauss-Legendre vs. Gauss-Lobatto nodes [N = 1 on 80 cells, KEP flux, T = 20; reference: N = 3 on 500 cells] 1.01 DG Gauss Legendre DG Gauss Lobatto Reference Solution DG Gauss Legendre DG Gauss Lobatto Reference Solution Pressure Kinetic Energy x x Higher accuracy of Gauss-Legendre DG scheme. 36

85 2D decaying homogeneous turbulence Computed on cartesian grid discretizing Ω = [0, 2π] 2, periodic b.c. T = 1 T = 2 T = 5 T = 10 T = 0: Initial energy spectrum given in Fourier space by ( ) [ 2s+1 ( E(k) = as 1 k exp s + 1 ) ( ) ] 2 k 2 k p k p 2 k p for wave number k = k 2 x + k 2 y (Parameters k p = 12, a s = ) 37

86 Comparison standard DG vs. DG-KEP scheme (I) Energy spectrum T = 10 Gauss nodes, N = 1 Re= E(k) E(k) DG standard 40 cells DG skew 40 cells Reference t=10 Initial energy spectrum k DG standard 80 cells DG skew 80 cells Reference t=10 Initial energy spectrum k cells cells SBP operators allow for conservative discretization of fluid equations in skew-symmetric form. 38

87 Comparison standard DG vs. DG-KEP scheme (II) Re= E(k) E(k) DG standard 80 cells DG skew 80 cells Reference t= DG standard 160 cells DG skew 160 cells Reference t=10 Initial energy spectrum Initial energy spectrum k k cells cells Better representation of energy spectrum for KEP scheme. Specifically for in underresolved case. 39

88 Current successful implementations of DG and FR schemes

89 The FLEXI Project DG space discretization explicit time stepping massive scalability F. Hindenlang, G. J. Gassner, C. Altmann, A. Beck, M. Staudenmaier, C. Munz, Explicit discontinuous Galerkin methods for unsteady problems, Computers & fluids 61, pp ,

90 Flux reconstrution with Python: PyFR high order flux reconstruction parallel platforms team named a 2016 finalist for Gordon Bell Prize F. D. Witherden, A. M. Farrington, P. E. Vincent, PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach, Computer Physics Communications 185, pp ,

91 Spectral/hp Element Framework: Nektar++ CG / DG operators hierarchical and nodal expansion bases temporal and spatial adaption, MPI parallel communication C. D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. De Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, B. Jordi, H. Xu, Y. Mohamied, C. Eskilsson, B. Nelson, P. Vos, C. Biotto, R.M. Kirby, S.J. Sherwin, Nektar++: An open-source spectral/ element framework, Computer Physics Communications 192, pp ,

92 DG for OpenFOAM? 4/b/b05 1/index.de.jsp developement within DG framework BoSSS (Bounded Support Spectral Solver) to be successively implemented in OpenFOAM level set method for multiphase flow high order DG method arbitrarily high accuracy at phase interfaces with cut-cell method N. Müller, S. Krämer-Eis, F. Kummer, M. Oberlack, A high-order discontinuous Galerkin method for compressible flows with immersed boundaries, Int. J. Numer. Meth. Engng. 110, pp. 3 30,

93 Summary 1 The Finite Volume Method 2 The Discontinuous Galerkin Scheme 3 SBP Operators & Flux Reconstruction 4 Current High Performance DG / FR Schemes 44

94 Thank you for your attention!