Entropy-stable discontinuous Galerkin nite element method with streamline diusion and shock-capturing

Transcription

1 Entropy-stable discontinuous Galerkin nite element method with streamline diusion and shock-capturing Siddhartha Mishra Seminar for Applied Mathematics ETH Zurich

2 Goal Find a numerical scheme for conservation laws u t + i= which d f i (u) i = is arbitrarily high-order accurate is entropy stable satises a maimum principle (bound in L ) converges for scalar conservation laws is multidimensional is reasonably ecient

3 Avoid oscillations and too much diusion neither nor

4 Avoid oscillations and too much diusion II but rather

5 Outline Introduction 2 Method 3 Implementation 4 Results 5 Conclusions

6 Derivation of the entropy stable DG FEM Start with the conservation law u t + f (u) = Multiply with a test function w (smooth) u t w + f (u) w =

7 Derivation of the entropy stable DG FEM II t t n+ I n t n j /2 j+/2 T j Integrate over the elements N n= j J I n T j (u t w + f (u) w) ddt =

8 Derivation of the entropy stable DG FEM III Integrate by parts N n= j J + + ( I n (u w t + f (u) w ) ddt T j u(t n+ T j I n f n+ ) w(t T )d u(t+) n w(t+)d n j ( ) ( ) ) u w( j+/2 j+/2 I )dt f u + w( + n j /2 j /2 )dt =

9 Numerical ues Replace the u at the boundary by numerical ues that depend on states on both sides of the boundary N n= j J + + T j U I n F I n F ( I n (u w t + f (u) w ) ddt T j (u(t n+ ), u(t n+ + )) w(t n+ )d T j U ( ) u, j+/2 u+ w( j+/2 j+/2 )dt ( ) ) u, j /2 u+ w( + j /2 j /2 )dt = For the numerical u U we use the upwind u, ( U u(t ), n u(t+) ) n = u(t ) n ( u(t n ), u(t n +) ) w(t n +)d only this allows to do time stepping

10 Entropy stability Choose entropy function S(u) and an associated u Q(u) Want a discrete analogon of the entropy inequality S t + Q variable transformation: (entropy symmetrisation) u = u(v) where v = S u are the entropy variables. Discretize v instead of u. 2 entropy stable numerical u: ([Tad87]) F (a, b) = F (a, b) D (v(b) v(a)) 2 F : an entropy conservative u. It has to full a certain condition ( unique in scalar case). D: Diusion matri. Can be anything, as long as it is positive semidenite.

11 Complete description Choose the space of ansatz functions V (piecewise polynomials). Then the description is complete: Find v in V, such that N n= j J + + ( I n u(v(t n+ T j I n F I n F for all w in V. More compactly T j (u(v) w t + f (u(v)) w ) ddt n+ )) w(t T )d u(v(t )) n w(t+)d n j ) (u(v j+/2 ), u(v +j+/2 ) w( j+/2 )dt ) ) (u(v j /2 ), u(v +j+/2 ) w( + j /2 )dt = B DG (v, w) =

12 Properties This leads to entropy stability (use w = v, suitable boundary conditions): S N d = S d T j J j T j J j D(v + v ) j+/2,t, v + v 2 2 D(v + v ) j /2,t, v + v 2 2,n 2 S uu(θ n )(u u + ), u u + S d T j J j However, at discontinuities (shocks) this still leads to oscillations. That is why we introduce the streamline diusion / shock capturing.

13 Streamline Diusion ([JS87]; [JSH9]) Add the term B SD (v, w) = N n= j J I n SD (u v w t + f (u) v w ) D (u t + f (u) ) ddt T j B DG (v, w) + B SD (v, w) = with D SD = O( ) and D SD positive semidenite. Leads to additional diusion proportional to (u v v t + f (u) v v ) D(u t + f (u) ) = (u t + f (u) ) D SD (u t + f (u) )

14 Shock-capturing (Barth) Idea: at shocks the residual is big (is it?) add (homogeneous) diusion proportional to the residual B SC (v, w) = N n= j J I n T j ɛ n j (u t w t + u w ) ddt B DG (v, w) + B SD (v, w) + B SC (v, w) = (C α i R n + /2 β C ɛ n i,j b R n b,j j = I n (v T t j u v v t + v u v v ) ddt + R n i,j = I n (u t + f (u) ) u v (u t + f (u) )ddt T j ) parameter values: C i =, α =, C b =

15 Properties - Goals revisited formally arbitrarily high-order accurate entropy stable (without/with SD or SC) (global) maimum principle for the scalar case (using a logarithmic entropy) convergence for the scalar case? multidimensional reasonably ecient?

16 Implementation currently in MATLAB quite slow For each time interval we have to solve a non-linear system for the dofs associated to it. currently: mostly by a damped Newton method ( we have to compute the Jacobian) planned: Newton-Krylov method ( we have to compute only the multiplication with the Jacobian) preconditioner?

17 Wave equation (smooth initial data) h t + cm = m t + ch = relative L error deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC number of cells

18 Burgers' equation u t + (u 2 /2) = relative L error 2 deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC deg=4, no SD/SC deg=4, SD deg=4, SD+SC number of cells

19 Burgers' equation deg = 2, N = no SD/SC SD SD+SC eact u

20 Burgers' equation SD + SC, N = 8.2 u deg= deg= deg=2 deg=3 deg=4 eact

21 ρ u Euler equations - Sod shock tube no SD/SC SD SD+SC eact.8.6 no SD/SC SD SD+SC eact no SD/SC SD SD+SC eact p.6.4 N = 8, deg =

22 Euler equations - Sod shock tube N = 8, deg = no SD/SC SD SD+SC eact ρ

23 Euler equations - Sod shock tube N = 8, SD+SC deg= deg= deg=2 deg=3 eact ρ

24 ρ u Euler equations - La shock tube no SD/SC SD SD+SC eact no SD/SC SD SD+SC eact p no SD/SC SD SD+SC eact N = 8, deg = 2 5 5

25 Euler equations - La shock tube N = 8, deg = no SD/SC SD SD+SC eact ρ

26 Euler equations - shock-entropy wave interaction N = 32, SD+SC ρ deg= deg= deg=2 reference.5 5 5

27 Introduction Method Implementation Results Conclusions Euler equations in 2D - Sod shock tube no SD/SC SD.5.5 ρ ρ y 2 5 SD+SC y ρ.5 Nc = 288, deg = y 2 5

28 Euler equations in 2D - Sod shock tube SD+SC, N c = 288, deg = 2.5 ρ.5 2 y 2 5 5

29 Introduction Method Implementation Results Conclusions Euler equations in 2D - Sod shock tube deg = deg =.5.8 ρ ρ y 2 deg = 2 5 y ρ Nc = 288, SD+SC y 2 5

30 Conclusions we get an entropy-stable DG FE method by: discretizing entropy variables using entropy stable numerical ues the solution is quite oscillatory at discontinuities by streamline diusion and shock-capturing we get a much less oscillatory solution, but it is quite diusive at contact discontinuities parameters and eact form of shock-capturing? convergence proofs? implementation: ecient solution of the non-linear systems?

31 Bibliography Claes Johnson and Anders Szepessy. On the convergence of a nite element method for a nonlinear hyperbolic conservation law. Mathematics of Computation, 49(8):427444, 987. Claes Johnson, Andres Szepessy, and Peter Hansbo. On the convergence of shock-capturing streamline diusion nite element methods for hyperbolic conservation laws. Mathematics of computation, 54(89):729, 99. Eitan Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws. i. Mathematics of Computation, 49(79):93, 987.

32 Appendi SC - boundary residual Linear Advection Euler Sod shock tube Euler La shock tube Linear Advection in 2D Burgers'/Advection in 2D Wave in 2D Euler Sod shock tube in 2D

33 SC - boundary residual R n = b,j ( ( ) ( u n + u n u u n v + u ) n d T j ) ) + (F (u j+/2, u+j+/2 ) f (u j+/2 ) u (F (u j+/2, u+j+/2 ) f (u j+/2 ) dt v I n ) ) ) + (F (u j /2, u+j /2 ) f (u+j /2 ) u (F (u j /2, u+j /2 ) f (u+j /2 ) 2 dt v I n

34 Linear advection u t + au = relative L error deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC deg=4, no SD/SC deg=4, SD deg=4, SD+SC 2 3 number of cells

35 Linear advection equation in 2D u t + au + bu y = relative L error 2 3 deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC 4 2 h

36 Burgers'/Advection equation in 2D u t + au + (u 2 /2) y = relative L error deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC 2 2 h

37 u u Burgers'/Advection equation in 2D deg = deg = deg = 2 y y N c = 84, SD+SC

38 Burgers'/Advection equation in 2D deg = 2, N c = 84, SD+SC

39 Burgers'/Advection equation in 2D no SD/SC SD SD+SC N c = 84, deg = 2

40 Burgers'/Advection equation in 2D SD+SC, N c = 84, deg = 2

41 Wave equation in 2D h t + cm + cn y = m t + ch = n t + ch y = relative L error deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC 5 2 h

42 ρ u Euler equations - Sod shock tube deg= deg= deg=2 deg=3 eact.8.6 deg= deg= deg=2 deg=3 eact deg= deg= deg=2 deg=3 eact p.6.4 N = 8, SD+SC.2 5 5

43 Euler equations - Sod shock tube N = 8, SD+SC deg= deg= deg=2 deg=3 eact ρ

44 ρ u Euler equations - La shock tube deg= deg= deg=2 deg=3 eact deg= deg= deg=2 deg=3 eact p deg= deg= deg=2 deg=3 eact N = 8, SD+SC.5 5 5

45 Euler equations - La shock tube N = 8, SD+SC deg= deg= deg=2 deg=3 eact ρ

46 Euler equations in 2D - Sod shock tube relative L error deg= deg=, no SD/SC deg=, SD deg=, SD+SC deg=2, no SD/SC deg=2, SD deg=2, SD+SC deg=3, no SD/SC deg=3, SD deg=3, SD+SC 2 h