A posteriori analysis of discontinuous Galerkin schemes for systems - PowerPoint PPT Presentation

A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws Jan Giesselmann joint work with Ch. Makridakis (Univ. of Sussex), T. Pryer (Univ. of Reading) Sino-German Symposium on Modern Numerical Methods for Compressible Fluid Flows and Related Problems May 21 - 27 2014 1 / 22

Outline 1. Introduction 2. Relative entropy 3. Semi-discrete DG schemes 4. Reconstruction approach 5. Error estimate 6. Numerical examples 2 / 22

Introduction We study the approximation of (systems of) hyperbolic conservation laws in one space dimension u t + f ( u ) x = 0 in (0 , ∞ ) × I , (CLAW) where u : (0 , ∞ ) × I → U ⊂ R n and f ∈ C 2 ( U , R n ) , by semi-discrete discontinuous Galerkin schemes. Initial conditions: u (0 , · ) = u 0 . Periodic boundary conditions. 3 / 22

Introduction We study the approximation of (systems of) hyperbolic conservation laws in one space dimension u t + f ( u ) x = 0 in (0 , ∞ ) × I , (CLAW) where u : (0 , ∞ ) × I → U ⊂ R n and f ∈ C 2 ( U , R n ) , by semi-discrete discontinuous Galerkin schemes. Initial conditions: u (0 , · ) = u 0 . Periodic boundary conditions. Entropy/ entropy flux pair : η : U → R , q : U → R such that ∇ η D f = ∇ q . ( ∗ ) Definition: Entropy solution A weak solution of (CLAW) is called an entropy solution with respect to the entropy/ entropy flux pair ( η, q ) provided it satisfies η ( u ) t + q ( u ) x ≤ 0 in the sense of distributions. 3 / 22

Known results A posteriori error estimates: Ohlberger, Kröner ’00: Upwind finite volume schemes for scalar conservation laws. Hartmann, Houston ’02: Space-time DG schemes for systems of conservation laws. Jovanovic, Rohde ’05: Finite volume schemes for Friedrichs systems. Dedner, Ohlberger, Makridakis ’07: Runge-Kutta DG schemes for scalar conservation laws. A priori error estimates for DG schemes: Zhang, Shu ’04, ’10: For RKDG2, RKDG3 schemes the spatial error is O ( h p + 1 2 ) . It is O ( h p +1 ) for upwind flux. 4 / 22

Aim Construct numerical solution u h t + f ( u h ) x = R h with residual R h . Construct explicit estimator E = E ( u h ) such that � u − u h � L ∞ (0 , T ; L 2 ( I )) ≤ E ( u h ) . E ( u h ) should be of the same order as the error. 5 / 22

Remarks on entropies In the scalar case every function η : U → R is an entropy. In the scalar case entropy solutions (satisfying the entropy inequality for all convex entropies) are unique. For systems of hyperbolic conservation laws there is usually only one (physically motivated) entropy/entropy flux pair. (For systems) entropy solutions need not be unique. In most cases the entropy is convex. But there are important cases where it is not. In the sequel we consider a system endowed with a strictly convex entropy. 6 / 22

Relative entropy In which way does the entropy inequality ensure stability? Relative entropy & relative entropy flux; (Dafermos ’79, Di Perna ’79): Let ( η, q ) be an entropy/entropy flux pair. η ( u | v ) := η ( u ) − η ( v ) − ∇ η ( v )( u − v ) q ( u | v ) := q ( u ) − q ( v ) − ∇ η ( v )( f ( u ) − f ( v )) . For K ⊂ U convex and compact, η strictly convex, there exists c > 0 such that η ( u | v ) ≥ c | u − v | 2 ∀ u , v ∈ K . 7 / 22

Relative entropy Theorem (Dafermos): Let a system of conservation laws be endowed with a strictly convex entropy/entropy flux pair ( η, q ) . Let u be an entropy solution and v a Lipschitz solution. Assume that u and v take values in some convex and compact K ⊂ U . Then there exist a , b > 0 such that for t > 0 � u ( t , · ) − v ( t , · ) � L 2 ( I ) ≤ ae bt � u 0 − v 0 � L 2 ( I ) . b depends on the Lipschitz constant of v . Remark 1: Uniqueness of Lipschitz solutions in the class of entropy solutions. Remark 2: In the scalar case the abundance of entropies leads to L 1 -contraction: � u ( t , · ) − v ( t , · ) � L 1 ( I ) ≤ � u ( s , · ) − v ( s , · ) � L 1 ( I ) for t > s for entropy solutions u , v . 8 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T t D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) η ( u | v ) t + q ( u | v ) x ≤ − v T 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T t D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) η ( u | v ) t + q ( u | v ) x ≤ − v T η ( u | v ) t + q ( u | v ) x ≤ ( D f ( v ) v x ) T D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T t D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) η ( u | v ) t + q ( u | v ) x ≤ − v T η ( u | v ) t + q ( u | v ) x ≤ ( D f ( v ) v x ) T D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) Existence of an entropy flux implies ( D f ) T D 2 η = D 2 η D f 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T t D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) η ( u | v ) t + q ( u | v ) x ≤ − v T η ( u | v ) t + q ( u | v ) x ≤ ( D f ( v ) v x ) T D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) Existence of an entropy flux implies ( D f ) T D 2 η = D 2 η D f η ( u | v ) t + q ( u | v ) x = − v T x D 2 η ( v ) � � f ( u ) − f ( v ) − D f ( v )( u − v ) 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T t D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) η ( u | v ) t + q ( u | v ) x ≤ − v T η ( u | v ) t + q ( u | v ) x ≤ ( D f ( v ) v x ) T D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) Existence of an entropy flux implies ( D f ) T D 2 η = D 2 η D f η ( u | v ) t + q ( u | v ) x = − v T x D 2 η ( v ) � � f ( u ) − f ( v ) − D f ( v )( u − v ) d I η ( u | v ) d x ≤ c � u − v � 2 � L 2 ≤ b � I η ( u | v ) d x d t 9 / 22

Idea of the proof (formal) For the rigorous proof, see Dafermos’ book. t D 2 η ( v )( u − v ) − ∇ η ( v ) u t η ( u | v ) t = η ( u ) t − v T x D 2 η ( v )( f ( u ) − f ( v )) − ∇ η ( v ) f ( u ) x q ( u | v ) x = q ( u ) x − v T t D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) η ( u | v ) t + q ( u | v ) x ≤ − v T η ( u | v ) t + q ( u | v ) x ≤ ( D f ( v ) v x ) T D 2 η ( v )( u − v ) − v T x D 2 η ( v )( f ( u ) − f ( v )) Existence of an entropy flux implies ( D f ) T D 2 η = D 2 η D f η ( u | v ) t + q ( u | v ) x = − v T x D 2 η ( v ) � � f ( u ) − f ( v ) − D f ( v )( u − v ) d I η ( u | v ) d x ≤ c � u − v � 2 � L 2 ≤ b � I η ( u | v ) d x d t c � u ( t , · ) − v ( t , · ) � 2 � � I η ( u 0 | v 0 ) d x e bt L 2 ≤ I η ( u ( t , · ) | v ( t , · )) d x ≤ 9 / 22

The case with residuals Let v solve a perturbed problem: u t + f ( u ) x = 0 , v t + f ( v ) x = R . Then, � � � T D 2 η ( v ) v x � η ( u | v ) t d x = − f ( u ) − f ( v ) − D f ( v )( u − v ) I I − R T D 2 η ( v )( u − v ) d x , such that � η ( u | v ) t d x ≤ b � u − v � 2 L 2 ( I ) + � R � 2 L 2 ( I ) . I 10 / 22

A first estimate Proposition Let u be an entropy solution of (CLAW) and v a Lipschitz solution of a perturbed problem with residual R . Let u , v take values in a compact, convex K ⊂ U . Then there exist a , b > 0 such that for t > 0 � � � u ( t , · ) − v ( t , · ) � 2 � u 0 − v 0 � 2 L 2 ( I ) + � R � 2 e bt . L 2 ( I ) ≤ a L 2 ([0 , t ) × I ) b is related to the Lipschitz constant of v . 11 / 22