Arbitrary (Almost) Lagrangian-Eulerian Discontinuous Galerkin method - PowerPoint PPT Presentation

Arbitrary (Almost) Lagrangian-Eulerian Discontinuous Galerkin method for 1-D Euler equations Praveen Chandrashekar & Jayesh Badwaik 1 praveen@tifrbng.res.in Center for Applicable Mathematics Tata Institute of Fundamental Research Bangalore-560065, India http://cpraveen.github.io Hyp2016, Aachen, 1-5 August, 2016 Supported by Airbus Foundation Chair at TIFR-CAM, Bangalore http://math.tifrbng.res.in/airbus-chair 1now at Univ. of W¨ urzburg 1 / 28

Euler equations in 1-D ∂ u ∂t + ∂ f ( u ) = 0 ∂x     ρ ρv  , p + ρv 2 u = ρv f ( u ) =    E ρHv γ − 1 + 1 p 2 ρv 2 , E = H = ( E + p ) /ρ 2 / 28

KH instability using fixed mesh finite volume v 1 v 1 + V v 2 v 2 + V v 1 v 1 + V Figure 33. Kelvin Helmholtz instability test at time t = 2 . 0, computed with AREPO with a fixed mesh. In the three cases, different boost velocities along both the x - and y - directions have been applied. The fact that the results do not agree (and in particular not with the V = 0 result shown in the bottom of Figure 32) is direct evidence for a violation of Galilean invariance of the Eulerian approach. We note that we have obtained nearly identical results for this test when it is carried out with ATHENA instead of our code AREPO. From Volker Springel, https://arxiv.org/abs/0901.4107 3 / 28

Dissipation in upwind schemes Upwind scheme for u t + au x = 0 , modified PDE 2 | a | h (1 − ν ) ∂ 2 u ∂u ∂t + a∂u ∂x = 1 ν = | a | ∆ t ∂x 2 + O ( h 2 ) , h For Euler equations: | v − c | , | v | , | v | + c Roe flux, 100 cells, TVD limiter 1 . 4 DG(1), V=0 1 . 2 DG(1), V=10 Sod problem DG(1), V=100 1 . 0 � (1 . 000 , V , 1 . 0) x < 0 . 5 Density 0 . 8 ( ρ, v, p ) = 0 . 6 (0 . 125 , V , 0 . 1) x > 0 . 5 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 x 4 / 28

Mesh Moving cell C j ( t ) = ( x j − 1 2 ( t ) , x j + 1 2 ( t )) d 2 ( t ) = w n Mesh velocity : d tx j + 1 2 ( t ) = w j + 1 2 , t n < t < t n +1 j + 1 t x n +1 x n +1 C n +1 j − 1 j + 1 j 2 2 t n +1 dx dt = w n dx dt = w n j − 1 / 2 j +1 / 2 t n x n C n x n j − 1 j j + 1 2 2 x x j ( t ) = 1 2( x j − 1 2 ( t ) + x j + 1 2 ( t )) , h j ( t ) = x j + 1 2 ( t ) − x j − 1 2 ( t ) x j + 1 2 ( t ) − x x − x j − 1 2 ( t ) w n w n w ( x, t ) = 2 + 2 , ( x, t ) ∈ C j ( t ) × ( t n , t n +1 ) j − 1 j + 1 h j ( t ) h j ( t ) 5 / 28

Solution space Degree k piecewise polynomial solution k � u h ( x, t ) = u j,m ( t ) ϕ m ( x, t ) , x ∈ C j ( t ) m =0 Legendre basis functions: ξ = x − x j ( t ) 1 2 h j ( t ) C j − 1 C j C j − 1 √ ϕ m ( x, t ) = ˆ ϕ m ( ξ ) = 2 m + 1 P m ( ξ ) j − 1 j + 1 2 2 6 / 28

ALE-DG scheme ALE flux g ( u , w ) = f ( u ) − w u Numerical ALE flux g ( u − 2 ( t ) , u + ˆ 2 ( t ) := ˆ 2 ( u h ( t )) := ˆ 2 ( t ) , w j + 1 2 ( t )) g j + 1 g j + 1 j + 1 j + 1 l ’th moment � t n +1 � x j + 1 2 ( t ) g ( u h , w ) ∂ h n +1 u n +1 h n j u n = j,l + ∂xϕ l ( x, t )d x d t j j,l t n x j − 1 2 ( t ) � t n +1 2 ( t ) ϕ l ( x + 2 ( t ) ϕ l ( x − + [ˆ g j − 1 2 , t ) − ˆ g j + 1 2 , t )]d t j − 1 j + 1 t n Fully discrete scheme • Replace u h with a locally predicted solution U h • Quadrature in space and time 7 / 28

Mesh velocity 2 = 1 w n 2[ v ( x − 2 , t n ) + v ( x + ˜ 2 , t n )] j + 1 j + 1 j + 1 A bit of smoothing 2 = 1 w n w n w n w n 3( ˜ 2 + ˜ 2 + ˜ 2 ) j + 1 j − 1 j + 1 j + 3 8 / 28

Predictor for k = 1 : Taylor expansion in space-time t t n +1 τ 1 = t n + 1 2 ∆ t n t n i − 1 i + 1 2 2 x j , t n ) + ( τ 1 − t n ) ∂ u h j ) ∂ u h U ( x q , τ 1 ) = u h ( x n ∂t ( x n j , t n ) + ( x q − x n ∂x ( x n j , t n ) Using conservation law, ∂ u ∂t = − ∂ f ∂x = − A ∂ u ∂x � ∂ u n U h ( x q , τ 1 ) = u n h ( x n A ( u n h ( x n ∂x ( x n h � j ) − ( τ 1 − t n ) j )) − w q I j ) 9 / 28

Predictor for k = 2 : Continuous expansion RK t t n +1 τ 2 τ 1 t n i − 1 i + 1 2 2 x d U m = − [ A ( U m ( t )) − w m ( t ) I ] ∂ ∂x U h ( x m , t ) =: K m ( t ) d t U m ( t n ) = u h ( x m , t n ) Using continuous expansion RK: for m = 0 , 1 , . . . , k n s � U m ( t ) = u h ( x m , t n ) + b s (( t − t n ) / ∆ t n ) K m,s , t ∈ [ t n , t n +1 ) s =1 where K m,s = K m ( t n + θ s ∆ t n ) , θ s ∆ t n is the stage time 10 / 28

Numerical flux for g ( u , w ) = f ( u ) − w u Rusanov flux g ( u l , u r , w ) = 1 2[ g ( u l , w ) + g ( u r , w )] − 1 ˆ 2 λ lr ( u r − u l ) λ lr = max {| v l − w | + c l , | v r − w | + c r } Roe flux g ( u l , u r , w ) = 1 2[ g ( u l , w ) + g ( u r , w )] − 1 ˆ 2 | A w | ( u r − u l ) A w = A w ( u l , u r ) satisfies g ( u r , w ) − g ( u l , w ) = A w ( u r − u l ) | A w | = R | Λ − wI | R − 1 q = 1 R , Λ evaluated at average state u (¯ q ) , ¯ 2 ( q l + q r ) , where q = √ ρ [1 , v, H ] ⊤ is the parameter vector introduced by Roe. 11 / 28

Limiting TVD and TVB limiters applied to characteristic variables 12 / 28

Positivity property First order scheme h n +1 u n +1 = h n u n g n g n ¯ j ¯ j − ∆ t n [ˆ 2 − ˆ 2 ] j j j + 1 j − 1 Restriction of time step to control change in cell size | w n 2 − w n 2 | ∆ t n ≤ βh n j , e.g. β = 0 . 1 j + 1 j − 1 Positivity of first order scheme The first order scheme with Rusanov flux is positivity preserving if the time step condition   (1 − 1 2 β ) h n βh n   j j ∆ t n ≤ ∆ t (1) := min 2 ) , n 1 | w n 2 − w n 2 ( λ n 2 + λ n 2 | j j − 1 j + 1 j + 1 j − 1   is satisfied. 13 / 28

Order of accuracy Initial condition ρ ( x, 0) = 1 + exp( − 10 x 2 ) , u ( x, 0) = 1 , p ( x, 0) = 1 Exact solution ρ ( x, t ) = ρ ( x − t, 0) , u ( x, t ) = 1 , p ( x, t ) = 1 The initial domain is [ − 5 , +5] and the final time is t = 1 units. 14 / 28

Order of accuracy: Rusanov flux k = 1 k = 2 k = 3 N Error Rate Error Rate Error Rate 100 4.370E-02 - 3.498E-03 - 3.883E-04 - 200 6.611E-03 2.725 4.766E-04 2.876 1.620E-05 4.583 400 1.332E-03 2.518 6.415E-05 2.885 9.376E-07 4.347 800 3.151E-04 2.372 8.246E-06 2.910 5.763E-08 4.239 1600 7.846E-05 2.280 1.031E-06 2.932 3.595E-09 4.180 Static mesh k = 1 k = 2 k = 3 N Error Rate Error Rate Error Rate 100 2.331E-02 - 3.979E-03 - 8.633E-04 - 200 6.139E-03 1.925 4.058E-04 3.294 1.185E-05 6.186 400 1.406E-03 2.0258 5.250E-05 3.122 7.079E-07 5.126 800 3.375E-04 2.0366 6.626E-06 3.077 4.340E-08 4.760 1600 8.278E-05 2.0344 8.304E-07 3.057 2.689E-09 4.573 Moving mesh 15 / 28

Randomized mesh velocity: HLLC flux 1 . 04 1 . 02 Example of randomized Mesh velocity velocity distribution for 1 . 00 smooth test case 0 . 98 0 . 96 − 10 − 5 0 5 10 x k = 1 k = 2 k = 3 N Error Rate Error Rate Error Rate 100 1.735E-02 - 1.798E-03 - 2.351E-04 - 200 4.179E-03 2.051 2.848E-04 2.676 1.416E-05 4.069 400 1.054E-03 2.035 4.301E-05 2.703 8.578E-07 4.041 800 2.615E-04 1.943 6.012E-06 2.838 5.476E-08 3.958 1600 7.279E-05 1.852 8.000E-07 2.909 3.505E-09 3.966 16 / 28

Single contact wave � (2 . 0 , 1 . 0 , 1 . 0) if x < 0 . 5 ( ρ, v, p ) = (1 . 0 , 1 . 0 , 1 . 0) if x > 0 . 5 Roe flux with 100 cells 2 . 0 2 . 0 1 . 8 1 . 8 1 . 6 1 . 6 Density Density 1 . 4 1 . 4 1 . 2 1 . 2 Exact Exact 1 . 0 1 . 0 DG(1) DG(1) 0 . 8 0 . 8 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 x x static mesh moving mesh 17 / 28

Sod test: x ∈ [0 , 1] , T = 0 . 2 � (1 . 000 , 0 . 0 , 1 . 0) if x < 0 . 5 ( ρ, v, p ) = (0 . 125 , 0 . 0 , 0 . 1) if x > 0 . 5 Roe flux, 100 cells and TVD limiter Exact Exact 1 . 0 1 . 0 DG(1) DG(1) 0 . 8 0 . 8 Density Density 0 . 6 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x x static mesh moving mesh 18 / 28

Sod test: x ∈ [0 , 1] , T = 0 . 2 � (1 . 000 , V , 1 . 0) if x < 0 . 5 ( ρ, v, p ) = (0 . 125 , V , 0 . 1) if x > 0 . 5 Roe flux, 100 cells and TVD limiter 1 . 4 1 . 4 DG(1), V=0 DG(1), V=0 1 . 2 DG(1), V=10 1 . 2 DG(1), V=10 DG(1), V=100 DG(1), V=100 1 . 0 1 . 0 0 . 8 0 . 8 Density Density 0 . 6 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 x x static mesh moving mesh 19 / 28

Shu-Osher problem: x ∈ [ − 5 , +5] , T = 1 . 8 � (3 . 857143 , 2 . 629369 , 10 . 333333) if x < − 4 ( ρ, v, p ) = (1 + 0 . 2 sin(5 x ) , 0 . 0 , 1 . 0) if x > − 4 Roe flux 5 . 0 5 . 0 4 . 5 4 . 5 4 . 0 4 . 0 3 . 5 3 . 5 Density Density 3 . 0 3 . 0 2 . 5 2 . 5 2 . 0 2 . 0 1 . 5 1 . 5 Exact Exact 1 . 0 1 . 0 DG(1) DG(1) 0 . 5 0 . 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 x x static mesh, 200 cells, M = 0 moving mesh, 200 cells, M = 0 20 / 28

Shu-Osher problem: x ∈ [ − 5 , +5] , T = 1 . 8 Roe flux 5 . 0 5 . 0 4 . 5 4 . 5 4 . 0 4 . 0 3 . 5 3 . 5 Density Density 3 . 0 3 . 0 2 . 5 2 . 5 2 . 0 2 . 0 1 . 5 1 . 5 Exact Exact 1 . 0 1 . 0 DG(1) DG(1) 0 . 5 0 . 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 x x static mesh, 200 cells, M = 100 static mesh, 300 cells, M = 100 21 / 28