Discontinuous Galerkin methods Lecture 2

Transcription

1 y y RMMC 2008 Discontinuous Galerkin methods Lecture 2 1 Jan S Hesthaven Brown University Jan.Hesthaven@Brown.edu y y Scattering about a vertical cylinder in a finite-width channel cal model based on the linear Padé (2,2) ro tational velocity version we set up a test-0.25 for 0.01 open-channel flow. We seek to model the scattering of an incident wave field which is prop agating toward a bottom-mounted rigid cylin der positioned in the middle of a finite-width x x y x 8.4 Scattering about a vertical cylinder in a finitewidth channel A numerical study is carried out to both test the consistency of the imposed boundary conditions and investigate how the geometric representation of the the domain may 0.25 affect the computed solution. In a numeri- channel Due to the symmetry, the solution is mathe E E E E E E E E E E E E E E E x Figure 8.12: Scattering x of waves about a Darmstadt International Workshop, October 2004 p.79 cylinder in a finite-width channel.

2 A brief overview of what s to come Lecture 1: Introduction, Motivation, History Lecture 2: Basic elements of DG-FEM Lecture 3: Linear systems and some theory Lecture 4: A bit more theory and discrete stability Lecture 5: Attention to implementations Lecture 6: Nonlinear problems and properties Lecture 7: Problems with discontinuities and shocks Lecture 8: Higher order/global problems

3 Lecture 2 Lets briefly recall the issues A bit of notation A second look at the scheme Returning to the example Looking forward

4 Let us recall Local flexibility to achieve high-order and geometric flexibility in the spirit of FEM Explicit scheme and problem control in the spirit of FVM... but many answers remain unanswered How do we achieve high-order accuracy? How do we choose the numerical flux? Is the scheme stable? What is the price?

5 A bit of notation It is the multi-element component of FEM/FVM which gives the geometric flexibility Ω Ω h = K k=1 D k, D k-1 D k D k+1 x 1 l =L x k-1 x K r =xk l x k r =xk+1 l r =R and the solution is approximated as Local norms K u(x, t) u h (x, t) = u k h(x, t), k=1 (u, v) D k = uv dx, u 2 D =(u, u) k D, k D k n inner product and norm

6 A bit of notation Global norms (u, v) Ω,h = K (u, v) D k, u 2 Ω,h =(u, u) Ω,h. k=1 To deal with the solutions across an interface, we define The jump: The average: u [[u]] = ˆn u + ˆn + u +, {{u}} = u + u +, 2 u + = local/interior solution = neighbor/exterior solution outward pointing normal

7 The basics of DG-FEM To understand the role of the different choices, let us consider an example u t + f(u) =0, x [L, R] =Ω, x as ( )=. This is subject to the ap First recall d dt u h 2 Ω,h = a ( u 2 (R) u 2 (L) ), t x f(u) =au. by multiplication of Eq. (2.1) by ( Leading to suitable boundary conditions as u(l, t) =g(t) if a 0, u(r, t) =g(t) if a and energy conservation when nonoverlapping elements u(r) =u(l)

8 The basics of DG-FEM Let us now assume that the local solution is x D k : u k h(x, t) = N p n=1 Where we have Some local modal basis A local nodal basis û k n(t)ψ n (x) = N p i=1 u k h(x k i,t)l k i (x). have introduced two complementary expressions for the lo l k i (x) ψ n (x)

9 The basics of DG-FEM Let us now assume that the local solution is x D k : u k h(x, t) = We form the local residual h and form the local residual x D k : R h (x, t) = uk h t + f k h x and require this to vanish locally in a Galerkin sense N p n=1 Where we have Some local modal basis A local nodal basis û k n(t)ψ n (x) = D k R h (x, t)l k j (x) dx =0, N p i=1 u k h(x k i,t)l k i (x). have introduced two complementary expressions for the lo l k i (x) ψ n (x) ack to the finite element scheme, we

10 The first DG schemes The lack of solution uniqueness at the interface is addressed as in FVM by a numerical flux f = f (u h,u+ h ), it is consistent [i.e u k h dl k j D k t lk j fh k dx glk j dx = [ f l k j g Gauss theorem once again, [ ] [ ] with the corresponding strong form being orem once again, D k R h (x, t)l k j (x) dx = [ (f k h f )l k j ] x k+1 x k. ] x k+1 x k, Naturally, the choice of the flux is important!

11 The basics of DG-FEM To simplify the notation, introduce Weak: M k ij = D k l k i (x)l k j (x) dx, S k ij = Dk l ki (x) dlk j dx dx. to obtain the two basic forms of DG-FEM M k duk h dt Strong: (S k ) T f k h M k g k h = f (x k+1 )l k (x k+1 )+f (x k )l k (x k ) M k duk h dt + S k f k h M k g k h =(f k h (x k+1 ) f (x k+1 ))l k (x k+1 ) (f k h (x k ) f (x k ))l k (x k ).

12 The basics of DG-FEM Let us consider the strong form ( ) M k d ( dt uk h + S k ( ) au k ) [ ] x k h = l k (x)(au k h (au h ) r ). ( ) x k l ced the nodal versions of the local operators ( ) N p N p u T h M k u h = u k h(x k D k i )l k i (x) u k h(x k j )l k j (x) dx = u k h 2 D ; k i=1 j=1 First note that Then also note that hat is, it recovers the local energy. Furthermore, consider u T h S k u h = D k N p i=1 = 1 2 [(uk h) 2 ] xk r x k l u k h(x k i )l k i (x). N p j=1 u k h(x k j ) dlk j dx dx = D k u k h(x)(u k h(x)) dx

13 The basics of DG-FEM Combining this, we achieve from d dt M k d dt uk h + S k ( au k) [ = h l k (x)(au k h (au h ) ) ced the nodal versions of the local operators ( ) the local energy estimate u k h 2 D k h = a[(uk h) 2 ] xk r [ x k l +2 [ u k h(au k h (au) ) ] x k r x k l as for the original equation, require that which should behave in such a way that ] x k r x k l.. K k=1 d dt u k h 2 D k = d dt u h 2 Ω,h 0. Stability

14 The basics of DG-FEM Let us consider the contribution from one interface each of which looks like ˆn (au 2 h(x ) 2u h (x )(au) (x ) ) + ˆn + (au 2 h(x + ) 2u h (x + )(au) (x + ) ) 0 at every interface. Here ( ) refers to the left (e.g., ), and right side We have the freedom ) to ( choose numerical flux to guarantee this. Consider f α = 1 ) ( f =(au) = {{au}} + a 1 α 2 α = 0, - upwind flux pwind α = 1, fl- central/average flux α = 0, w [[u]]. This yields this the gives local a contribution term from each interface a (1 α)[[u h ]] 2, 0, for 0 α 1

15 The basics of DG-FEM What about the boundary conditions? a>0:u(l, t) =g(t) Approach #1: f L = ag(t),f R = au k h (xk r ) Approach #2: f L = au k h (x1 l )+2g(t),f R = au k h (xk r )

16 The basics of DG-FEM Approach #1: K 1 d dt u h 2 Ω,h = a (1 α) [[u k h(x k r)]] 2 (1 α)a(u 1 h(x 1 l )) 2 a(u K h (x K r )) 2. For 0 Approach #2: k=1 1 we have global stability. tflow. This yields a global energy estimate as K 1 d dt u h 2 Ω,h = a (1 α) [[u k h(x k r)]] 2 a(u 1 h(x 1 l )) 2 a(u K h (x K r )) 2. k=1 gain, we recover stability for 0 a 0, 0 α 1 1. A particularly interesting case Stability Energy dissipation for α 1 Energy conservation for α =1,u(L) =u(r)

17 The basics of DG-FEM What did we learn from this? Stability is enforced through the flux choice. No restrictions on the local basis, e.g., it need not be polynomial, and is chosen to provide accuracy. The numerical solution is discontinuous between elements. Boundary conditions and interface conditions are imposed weakly. All operators are local. Due to the weak interface based coupling, there are no restrictions on element size and local approximation.

18 Back to the example Let us consider a simple example u u 2π =0, x [0, 2π], t x u(x, 0) = sin(lx), l = 2π λ, ons and initial condition as N\ K Convergence rate 1 4.0E E E E E E E E E E E E E E E E E E E E E E E The error clearly behaves as u u h Ω,h Ch N+1. proximation that gives the

19 Back to the example What about time dependence Final time (T) π 10π 100π 1000π 2000π (N,K)=(2,4) 4.3E E E-01 >1 >1 (N,K)=(4,2) 3.3E E E E E-01 (N,K)=(4,4) 3.1E E E E E-03 The error behaves as u u h Ω,h C(T )h N+1 (c 1 + c 2 T )h N+1,

20 Back to the example u(x,2π) x/2π u(x,2π) x/2π Central flux Upwind flux

21 Back to the example What about cost? N\K dominates and destroys the expected convergence rate. Higher order is cheaper Time C(T )K(N + 1) 2, N\K Convergence rate with the final time in a linear 1 4.0E E E E E E E E E E E E E E E E E E E E E E E

22 A few remarks We know now It is the flux that gives stability It is the local basis that gives accuracy. The scheme is very (VERY) flexible.

23 A few remarks We know now It is the flux that gives stability It is the local basis that gives accuracy. The scheme is very (VERY) flexible. BUT - we have doubled the number of degrees of freedom along the interfaces In 1D not a big deal -- penalty is N+1 N

24 Does it generalize? Let us first consider the scalar conservation law u t + f(u) x =0, x [L, R], where the e initial boundary conditions conditions are u(l, t) =g 1 (t) when f u (u(l, t)) 0, u(r, t) =g 2 (t) when f u (u(r, t)) 0. tly as for the linear case discussed previousl Assume as usual that x D k : u k h(x, t) = N p n=1 û k n(t)ψ n (x) = N p i=1 u k h(x k i,t)l k i (x). have introduced two complementary expressions for the lo

25 Does it generalize? From this we directly recover ( ) u k h t φk h fh k (u k h) dφk h dx = ( dx ) D k D k ˆn f φ k h dx, strong form and the ( corresponding ) strong form D k ( u k h t + f h k(uk h ) ) φ k h dx = x D k ˆn (f k h (u k h) f ) φ k h dx, Here we have the general local test functions φ k h Vk h. x D k : φ k h(x) = N p ˆφ k nψ n (x), n=1 tions as

26 Does it generalize? This yields exactly local unknowns N p D k N p equations for the and the corresponding strong form D k Here we have also introduced ( u k h t ψ n fh k (u k h) dψ ) n ( dx ) = dx ( )) ( u k h t + f h k(uk h ) ) ψ n dx = x D k ˆn f ψ n dx, ( ( ) ) D k ˆn (f k h (u k h) f ) ψ n dx, the weak and strong semi-discrete form, respectively. In both c x D k : f k h (u k h)= N p n=1 ˆf k nψ n (x) = N p i=1 The details of this will return! f h (x k i )l k i (x).

27 Does it generalize? The only thing that remains unknown is the flux f = f (u h,u+ h ) We rely on the hugely successful theory of finite volume monotone schemes

28 Does it generalize? The only thing that remains unknown is the flux f = f (u h,u+ h ) We rely on the hugely successful theory of finite volume monotone schemes The flux is consistent The flux is monotone f(u h )=f (u h,u h ) increasing in a f (a, b){ decreasing in b

29 Does it generalize? There are many choices for this The Lax-Friedrich flux where the global LF flux is given by and the local LF flux is obtained for NOTE: for f LF (a, b) = f(a)+f(b) 2 choice. The definition of C C + C 2 ˆn (a b), allows for some v max f u(s) inf u h (x) s sup u h (x) f = au max f u(s). min(a,b) s max(a,b) we recover the flux seen earlier

30 Does it generalize?... but the FV literature is filled with alternatives Exact Riemann solvers Godunov fluxes Engquist-Osher fluxes Approximate Riemann fluxes (Roe, van Leer, HLLC etc)

31 Does it generalize?... but the FV literature is filled with alternatives Exact Riemann solvers Godunov fluxes Engquist-Osher fluxes Approximate Riemann fluxes (Roe, van Leer, HLLC etc) To keep things simple we shall mainly focus on the LF flux which generally works very well, but is also the most dissipative flux.

32 Does it generalize? Let us now consider the system of conservation laws u t + f(u) =0, x [L, R], x omponent is introduced to o s u =[u 1 (x, t),...,u m (x, t)] T m m. where the boundary conditions are B L u(l, t) =g 1 (t) at x = L, B R u(r, t) =g 2 (t) at x = R, B The only essential difference is that C in the LF flux depends on the eigenvalues of f u = f u.

33 Does it generalize? The last frontier -- multidimensional problem No essential difference - the weak form ( u k ( ) ( h t φk h f k h(u k h) φ k ) h dx = D k The strong form D k ˆn f φ k h dx, the strong form ( ) ( ) ) u k ( h + f k D k t h(u k h) φ k h dx = ˆn f k h(u k h) f ) φ k D k h dx. with the LF-flux f = {{f h (u h )}} + C 2 [[u h]]. C = max u ( λ ˆn f ), u eigenvalue of the matrix.

34 Lets summarize We already know a lot about the basic DG-FEM Stability is provided by carefully choosing the numerical flux. Accuracy appear to be given by the local solution representation. We can utilize major advances on monotone schemes to design fluxes. The scheme generalizes with very few changes to very general problems -- multidimensional systems of conservation laws.

35 Lets summarize We already know a lot about the basic DG-FEM Stability is provided by carefully choosing the numerical flux. Accuracy appear to be given by the local solution representation. We can utilize major advances on monotone schemes to design fluxes. The scheme generalizes with very few changes to very general problems -- multidimensional systems of conservation laws. At least in principle -- but what can we actually prove?