A Finite Volume Evolution Galerkin Scheme for Wave Propagation in - PowerPoint PPT Presentation

A Finite Volume Evolution Galerkin Scheme for Wave Propagation in Heterogeneous Media K. R. Arun Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, Germany 14th International Conference on Hyperbolic Problems: Theory, Numerics and Applications 25 June, 2012

with many thanks to... Prof. Maria Luk´ aˇ cov´ a-Medvid ’ov´ a, Institut f¨ ur Mathematik, Johannes Gutenberg-Universit¨ at Mainz, Prof. Sebastian Noelle, Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, to the people paying for me: the Alexander von Humboldt Foundation Arun (RWTH Aachen) FVEG Scheme HYP2012 2 / 27

Outline Introduction 1 Aim of the present work Governing Equations Arun (RWTH Aachen) FVEG Scheme HYP2012 3 / 27

Outline Introduction 1 Aim of the present work Governing Equations Bicharacteristics of Multi-dimensional Hyperbolic Systems 2 Characteristic Surfaces in Multi-dimensions Bicharacteristic Curves Arun (RWTH Aachen) FVEG Scheme HYP2012 3 / 27

Outline Introduction 1 Aim of the present work Governing Equations Bicharacteristics of Multi-dimensional Hyperbolic Systems 2 Characteristic Surfaces in Multi-dimensions Bicharacteristic Curves Integral Representation 3 Exact Evolution Operator Approximate Evolution Operator Arun (RWTH Aachen) FVEG Scheme HYP2012 3 / 27

Outline Introduction 1 Aim of the present work Governing Equations Bicharacteristics of Multi-dimensional Hyperbolic Systems 2 Characteristic Surfaces in Multi-dimensions Bicharacteristic Curves Integral Representation 3 Exact Evolution Operator Approximate Evolution Operator Finite Volume Evolution Galerkin Schemes 4 The Numerical Method Numerical Experiments Arun (RWTH Aachen) FVEG Scheme HYP2012 3 / 27

Aim To model the propagation of acoustic waves in heterogeneous media. To extend the Finite Volume Evolution Galerkin (FVEG) scheme for linear hyperbolic systems with spatially varying flux functions. To derive a genuinely multi-dimensional finite volume scheme for the acoustic wave equation system. Arun (RWTH Aachen) FVEG Scheme HYP2012 4 / 27

Governing Equations Propagation of acoustic waves in an ideal gas at rest initially is given by,  p   γp 0 u   γp 0 v  ρ 0 u + p + p = 0 , (1)       ρ 0 v 0 0 t x y ρ 0 = ρ 0 ( x, y ) , p 0 = const are the ambient density and pressure. In non-conservation form v t + A 1 v x + A 2 v y = 0 , (2)  p   0 γp 0 0  1 0 0 where v = u  , A 1 =  ,   ρ 0 v 0 0 0  0 0 γp 0  0 0 0 A 2 =  . Note that (2) is a linear system with spatially  1 0 0 ρ 0 varying coefficients. Arun (RWTH Aachen) FVEG Scheme HYP2012 5 / 27

Definition of a characteristic surface Definition A characteristic surface Ω: ϕ ( x, y, t ) = 0 of (1) is a surface of discontinuity of the first derivatives. The one parameter family of characteristic surfaces ϕ ( x, y, t ) = const is goverened by the characteristic partial differential equation Q ( x , ϕ t , ∇ ϕ ) ≡ det ( ϕ t I + ϕ x A 1 + ϕ y A 2 ) = 0 , (3) A 1 and A 2 are the flux Jacobian matrices. Note that (3) is a nonlinear first order PDE for ϕ , of the Hamilton-Jacobi type. This is a generalization of the eikonal equation in optics. Arun (RWTH Aachen) FVEG Scheme HYP2012 6 / 27

Definition of a bicharacteristic curve Definition The characteristic curves of (3) are called bicharacteristic curves These are curves in ( x, y, t ) -space. The generators of characteristic surfaces. Advection curves, stream lines for Euler equations. A hyperbolic system of m equations has m families of bicharacteristic curves. [Courant-Hilbert 1962, Prasad 2001] Arun (RWTH Aachen) FVEG Scheme HYP2012 7 / 27

Examples For the wave equation u tt − a 2 0 ( u xx + u yy ) = 0 , (4) the eikonal is � 1 / 2 = 0 . ϕ 2 x + ϕ 2 � ϕ t − a 0 (5) y An important solution is the characterestic conoid t ( x − x 0 ) 2 + ( y − y 0 ) 2 � 1 / 2 = 0 t − t 0 ± 1 � a 0 P ( x 0 , y 0 , t 0 ) y x Arun (RWTH Aachen) FVEG Scheme HYP2012 8 / 27 Figure: Characteretic conoid

Examples contd (x P , y P , t n+1 ) 0 −0.2 −0.4 −0.6 t −0.8 −1 −1 3 0 2 1 1 2 0 3 −1 y x Figure: Characteristic conoid for the acoustic wave equation system Arun (RWTH Aachen) FVEG Scheme HYP2012 9 / 27

P ( x, y, t + ∆ t ) Q ( θ ) θ P ′ t y x Figure: Bicharacteristics along the Mach cone through P and Q ( θ ) Can we get the solution at P using the values at Q ( θ ) ? As Q ( θ ) moves along the circle we get contributions from infinitely many directions! Arun (RWTH Aachen) FVEG Scheme HYP2012 10 / 27

Result Lemma (Extended lemma on bicharacteristics, Prasad, 1993) For a hyperbolic system d � u t + ( f j ( u )) x j = 0 , (6) j =1 the evolution of the p -th bicharacteristic family is given by d x j d t = l ( p ) A j r ( p ) , (7) � d d � �� d n j ∂ A s d t = l ( p ) � λ ( p ) � r ( p ) , n k n s (8) ∂η j k =1 s =1 k where l ( p ) and r ( p ) are left and right eigenvectors corresponding to the eigenvalue λ ( p ) of the matrix pencil A := � d j =1 n j A j . Arun (RWTH Aachen) FVEG Scheme HYP2012 11 / 27

Compatibility conditions Result (Prasad and Ravindran, 1984) For a hyperbolic system of quasilinear equations d � ∂ t u + A j ( u ) ∂x j u = 0 , (9) j =1 The transport equation along the p th family of bicharacteristics is given by d l ( p ) d u l ( p ) � � A j − χ ( p ) � dt + ∂x j u = 0 , (10) j I m j =1 where χ ( p ) = l ( p ) A j r ( p ) is the ray velocity vector and j dt ≡ ∂ t + � d j =1 χ ( p ) j ∂x j is the derivative along the p th bicharacteristic. d Arun (RWTH Aachen) FVEG Scheme HYP2012 12 / 27

Exact integral representation Result (Ostkamp, 1995) For a linear hyperbolic system an exact integral representation of the solution is given by m 1 � � r ( k ) ( P ) l ( k ) ( Q k ) u ( Q k )d S u ( P ) = (11) | S d − 1 | S d − 1 k =1 � t n +1 m r ( k ) ( P )d l ( k ) 1 � � dt ( ˜ Q k ) u ( ˜ + Q k )d τ d S | S d − 1 | S d − 1 t n k =1 � t n +1 m d � ∂ u 1 � � A j − χ ( k ) � � r ( k ) ( P ) l ( k ) ( ˜ − Q k ) j I d τ d | S d − 1 | ∂x j S d − 1 t n k =1 j =1 Arun (RWTH Aachen) FVEG Scheme HYP2012 13 / 27

Examples For the acoustic wave equation system,       p γp 0 u γp 0 v ρ 0 u + p + p = 0 , (12)       ρ 0 v 0 0 t x y � 2 π p ( P ) = 1 ( p − z 0 u cos θ − z 0 v sin θ ) ( Q 1 )d ω 2 π 0 � 2 π � t n +1 − 1 ( z 0 ( a 0 x u + a 0 y v )) ( ˜ Q 1 )d τ d ω (13) 2 π 0 t n � 2 π � t n +1 − 1 ( z 0 S )( ˜ Q 1 )d τ d ω. 2 π 0 t n Arun (RWTH Aachen) FVEG Scheme HYP2012 14 / 27

� 2 π 1 u ( P ) = ( − p + z 0 u cos θ + z 0 v sin θ ) ( Q 1 ) cos ω d ω 2 πz 0 ( P ) 0 � 2 π � t n +1 1 z 0 ( a 0 x u + a 0 y v ) ( ˜ + Q 1 ) cos ω d τ d ω 2 πz 0 ( P ) 0 t n (14) � t n +1 + 1 1 p x ( ˜ 2 u ( Q 2 ) − Q 2 )d τ 2 ρ 0 ( P ) t n � 2 π � t n +1 1 ( z 0 S )( ˜ + Q 1 ) cos ω d τ d ω. 2 πz 0 ( P ) 0 t n The expression for v ( P ) is analogous. � Q ) sin 2 θ − ( u y ( ˜ Q ) cos 2 θ � S ( ˜ u x ( ˜ Q ) + v x ( ˜ Q )) sin θ cos θ + v y ( ˜ Q ) := a 0 , (15) is a geometric source term. Arun (RWTH Aachen) FVEG Scheme HYP2012 15 / 27

Approximation of the exact evolution operators The exact evolution operator is an implicit relation. It involves the time integrals of the unknown and its derivatives. The integrals along the Mach cone are to be simplified. We freeze the time integrals at t = t n to get an explicit relation. The geometric source term S contains only tangential derivatives, thanks to this special structure. Arun (RWTH Aachen) FVEG Scheme HYP2012 16 / 27

Approximate evolution operators � � 2 π p ( P ) = 1 ( p − z 0 ( u cos ω + v sin ω ))( Q )d ω 2 π 0 � 2 π − ∆ t ( z 0 [ u sin ω − v cos ω ][ − a 0 x sin ω + a 0 y cos ω ])( Q )d ω 0 � 2 π − ∆ t ( z 0 ( a 0 x u + a 0 y v ))( Q )d ω 0 �� φ j +1 3 1 � − γp 0 ( u cos ω + v sin ω )( Q )d ω a j φ j 0 j =0 φ j = jπ/ 2 + ( u sin φ j − v cos φ j )( Q ( φ + j )) �� + O (∆ t 2 − ( u sin φ j +1 − v cos φ j +1 )( Q ( φ − j +1 )) Arun (RWTH Aachen) FVEG Scheme HYP2012 17 / 27

� � 2 π 1 u ( P ) = ( − p + z 0 ( u cos ω + v sin ω ))( Q ) cos ω d ω πz 0 ( P ) 0 � 2 π + ∆ t ( z 0 [ u sin ω − v cos ω ][ − a 0 x sin ω + a 0 y cos ω ])( Q ) 0 � 2 π + ∆ t ( z 0 ( a 0 x u + a 0 y v ))( Q ) cos ω d ω 0 �� φ j +1 3 1 ( u (2 cos 2 ω − 1) + 2 v cos ω sin ω )( Q � + γp 0 a j φ j 0 j =0 φ j = jπ/ 2 + ( u cos φ j sin φ j − v cos 2 φ j )( Q ( φ + j )) − ( u (cos φ j +1 sin φ j +1 ) − v cos 2 φ j +1 )( Q ( φ − j +1 Arun (RWTH Aachen) FVEG Scheme HYP2012 18 / 27