Relativistic Burgers equations on a curved spacetime Baver - PowerPoint PPT Presentation

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Relativistic Burgers equations on a curved spacetime Baver Okutmustur Middle East Technical University (METU) HYP2012, June 25–29, 2012 Joint work with Philippe LeFloch and Hasan Makhlof, Universit´ e Paris 6 1 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation 1 Introduction-preliminaries Euler equations ⇒ Burgers equation Hyperbolic Balance law Lorentz invariance and derivation of the new model Derivation from relativistic Euler equations 2 Burgers equations with geometric effects Relativistic Euler Equations on Schwarzschild spacetime Burgers equation on Schwarzshild spacetime Relativistic Burgers equation on Schwarzshild spacetime 3 Well-balanced finite volume approximation Geometric formulation of finite volume schemes Finite volume methods in coordinates Numerical experiments 2 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Euler equations ⇒ Burgers equation Euler equations of compressible fluids ∂ t ( ρ u ) + ∂ x ( ρ u 2 + p ( ρ )) = 0 ∂ t ρ + ∂ x ( ρ u ) = 0 , ρ : density, u : velocity of the fluid, p ( ρ ) : pressure Rewrite the second equation combining with the first one 0 = u ∂ t ( ρ ) + ρ ∂ t ( u ) + u 2 ∂ x ( ρ ) + 2 u ρ ∂ x ( u ) = ρ ( ∂ t u + 2 u ∂ x u ) + u ( ∂ t ρ + u ∂ x ρ ) = ρ ( ∂ t u + 2 u ∂ x u ) − u ρ∂ x u = ρ ( ∂ t u + u ∂ x u ) The (inviscid) Burgers equation ∂ t u + ∂ x ( u 2 / 2) = 0 , u = u ( t , x ) , t > 0 , x ∈ R 3 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Hyperbolic Balance law Hyperbolic balance laws Balance laws div ω ( T ( v )) = S ( v ) M = ( M , ω ) : ( n + 1)-dimensional curved spacetime (with boundary) div ω : the divergence operator associated with the volume form ω v : M → R unknown function (scalar field) T = T ( v ) flux vector field on M , S = S ( v ) a scalar field on M . The manifold M is assumed to be foliated by hypersurfaces : � M = H t , t ≥ 0 such that each slice H t is an n -dimensional manifold. H t : spacelike, H 0 : initial slice 4 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model Lorentz invariant conservation law Supposing that n = 1, S ( v ) ≡ 0 M = [0 , + ∞ ) × R covered by a single coordinate chart ( x 0 , x 1 ) = ( t , x ) with ω = dx 0 dx 1 , it follows that Hyperbolic conservation law ∂ 0 T 0 ( v ) + ∂ 1 T 1 ( v ) = 0 , where ∂ 0 = ∂/∂ x 0 , ∂ 1 = ∂/∂ x 1 , x 0 ∈ [0 , ∞ ) and x 1 ∈ R . We search for the flux vector fields T = T ( v ) for which solutions to the above equation satisfy Lorentz invariant property. 5 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model Derivation of a Lorentz invariant model Lorentz transformations ( x 0 , x 1 ) �→ (¯ x 0 , ¯ x 1 ) x 0 := γ ǫ ( V ) ( x 0 − ǫ 2 Vx 1 ) , ¯ � 1 − ǫ 2 V 2 � − 1 / 2 , x 1 := γ ǫ ( V ) ( − V x 0 + x 1 ) , ¯ γ ǫ ( V ) = ǫ ∈ ( − 1 , 1) denotes the inverse of the (normalized) speed of light, γ ǫ ( V ) is the so-called Lorentz factor associated with a given speed V ∈ ( − 1 /ǫ, 1 /ǫ ) v : fluid velocity component in the coordinate system ( x 0 , x 1 ) x 0 , ¯ x 1 ) related to the component v in the coordinates (¯ v − V v = 1 − ǫ 2 V v . 6 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model Relativistic Burgers equations on Minkowski spacetime Theorem The conservation law ∂ 0 T 0 ( v ) + ∂ 1 T 1 ( v ) = 0 , (1) is invariant under Lorentz transformations if and only if after suitable normalization one has � � v T 1 ( v ) = 1 1 T 0 ( v ) = √ 1 − ǫ 2 v 2 , √ 1 − ǫ 2 v 2 − 1 , ǫ 2 where the scalar field v takes its value in ( − 1 /ǫ, 1 /ǫ ) . 7 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model Sketch of the proof Use Lorentz transformation with related change of coordinates Apply the chain rule to write ∂ 0 T 0 , ∂ 1 T 1 Substitute them in the conservation law equation (1) Checking the Lorentz invariance property Determine the general expression of T 0 and T 1 T 0 ( v ) = T 0 ( φ ǫ )( u ) = e ǫ u − e − ǫ u = 1 ǫ sinh( ǫ u ) = u + O ( ǫ 2 u 3 ), 2 ǫ � � = u 2 T 1 ( v ) = T 1 ( φ ǫ ( u )) = e ǫ u + e − ǫ u − 2 = 1 2 + O ( ǫ 2 u 4 ) , cosh( ǫ u ) − 1 2 ǫ 2 ǫ 2 e 2 ǫ u − 1 where v = 1 e 2 ǫ u +1 = φ ǫ ( u ) . ǫ T 0 and T 1 are linear and quadratic, respectively. 2 ǫ ln 1+ ǫ v 1 Substitute u = 1 − ǫ v , we get the desired result. (Note that in the limiting case ǫ → 0, we recover the (inviscid) Burgers equation). 8 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model Properties of the relativistic Burgers equation � � � �� � 1 1 v ∂ 0 + ∂ 1 1 − ǫ 2 v 2 − 1 = 0 (1) √ √ ǫ 2 1 − ǫ 2 v 2 1 The map w = T 0 ( v ) = v 1 − ǫ 2 v 2 ∈ R is increasing and one-to-one √ from ( − 1 /ǫ, 1 /ǫ ) onto R . 2 In terms of the new unknown w ∈ R , (1) is equivalent to ∂ 0 w + ∂ 1 f ǫ ( w ) = 0 , � (2) � 1 + ǫ 2 w 2 � f ǫ ( w ) = 1 − 1 ± , ǫ 2 3 In the non-relativistic limit ǫ → 0, one recovers the Burgers equation ∂ 0 u + ∂ 1 ( u 2 / 2) = 0 , where u ∈ R . 9 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model The proposed equation retains several key features of the relativistic Euler equations : Like the conservation of mass-energy in the Euler system, it has a conservative form. Like the velocity component in the Euler system, our unknown v is constrained to lie in the interval ( − 1 /ǫ, 1 /ǫ ) limited by the light speed parameter. Like the Euler system, by sending the light speed to infinity one recover the classical (non-relativistic) model. 10 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Lorentz invariance and derivation of the new model The non-relativistic limit We recover the Galilean transformations by relativistic case with ǫ → 0 given by x 0 = x 0 , x 1 = x 1 − Vx 0 , ¯ ¯ v = v − V . We have the following : The conservation law ∂ 0 T 0 ( v ) + ∂ 1 T 1 ( v ) = 0 , is invariant under Galilean transformations iff the flux functions T 0 and T 1 are linear and quadratic, respectively. If T 0 ( v ) = v , then after a suitable normalization one gets T 1 ( v ) = v 2 / 2. 11 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Derivation from relativistic Euler equations Relativistic Euler Equations � p + ρ c 2 � � � v 2 v ( p + ρ c 2 ) ∂ 0 c 2 − v 2 + ρ + ∂ 1 = 0 , c 2 − v 2 c 2 � � � � v 2 v ( p + ρ c 2 ) ( p + ρ c 2 ) ∂ 0 + ∂ 1 c 2 − v 2 + p = 0 , c 2 − v 2 p , ρ, v and c denote the pressure, density, velocity and speed of light. Set ρ as a constant (and thus the pressure p ) in the second equation : � � � � v 2 v ∂ 0 + ∂ 1 = 0 . c 2 − v 2 c 2 − v 2 v Use change of variable z = 1 − ǫ 2 v 2 , with c = 1 /ǫ , we get � � 1 + 4 ǫ 2 z 2 � ∂ 0 z + 1 2 ǫ 2 ∂ 1 − 1 ± = 0 . (3) 12 / 30

Introduction-preliminaries Burgers equations with geometric effects Well-balanced finite volume approximation Relativistic Euler Equations on Schwarzschild spacetime Vanishing pressure on flat spacetime Assume that the pressure vanishes identically in the relativistic Euler equations, i.e. � � � � ρ ρ v ∂ 0 + ∂ 1 = 0 , c 2 − v 2 c 2 − v 2 � � � � ρ v 2 ρ v ∂ 0 + ∂ 1 = 0 . c 2 − v 2 c 2 − v 2 Rewriting these two equations : ( c 2 − v 2 )( ∂ 0 ρ + v ∂ 1 ρ ) + ρ (2 v ∂ 0 v + ( v 2 + c 2 ) ∂ 1 v ) = 0 , v ( c 2 − v 2 )( ∂ 0 ρ + v ∂ 1 ρ ) + ρ (( v 2 + c 2 ) ∂ 0 v + 2 vc 2 ∂ 1 v ) = 0 . Combining these equations we recover the classical Burgers equation (in flat spacetime) ∂ 0 v + ∂ 1 ( v 2 / 2) = 0 13 / 30 .