DG-KFVS schemes for convection-diffusion equations Praveen. C Center for Applicable Mathematics Tata Institute of Fundamental Research Bangalore 560065 http://math.tifrbng.res.in/~praveen Dept. of Mathematics, IISER Pune 25 March, 2014 1 / 73
Compressible Navier-Stokes model • Mass conservation equation ∂ρ ∂t + ∇ · ( ρ u ) = 0 • Momentum equation ∂ ∂t ( ρ u ) + ∇ · ( pI + ρ uu ) = ∇ · τ • Energy conservation ∂E ∂t + ∇ · ( E + p ) u = ∇ · ( u · τ ) + ∇ · q 2 / 73
Compressible Navier-Stokes model • Equation of state: ideal gas γ − 1 + 1 p 2 ρ | u | 2 , E = γ > 1 • Constitutive laws τ ij = µ ( ∂ i u j + ∂ j u i ) − 2 3 µ ( ∂ k u k ) δ ij , q i = − κ∂ i T • System of conservation laws ∂U ∂t + ∂ F i ( U ) + ∂ G i ( U, ∇ U ) = 0 ∂x i ∂x i 3 / 73
Flow over NACA0012 airfoil: M = 0 . 85 , α = 1 Pressure 5 4 / 73
Flow through scramjet intake Density 5 / 73
Model problems • Linear convection equation ∂ρ ∂t + u∂ρ ∂x = 0 • Inviscid Burger’s equation ∂u ∂t + u∂u ∂x = 0 • Heat equation ∂t = µ∂ 2 u ∂u ∂x 2 6 / 73
Heat equation: Finite difference method ∂t = µ∂ 2 u ∂u ∂x 2 , x ∈ (0 , 1) u ( x, 0) = f ( x ) , u (0 , t ) = u (1 , t ) = 0 Partition domain (0 , 1) with grid points x j = jh, j = 0 , 1 , . . . , N + 1 , u j ( t ) = u ( x j , t ) Approximate second derivative by finite difference = ∂ 2 u u j − 1 − 2 u j + u j +1 ∂x 2 ( x j ) + O ( h 2 ) h 2 Semi-discrete scheme (system of ODE) d u j d t = µu j − 1 − 2 u j + u j +1 , j = 1 , 2 , . . . , N h 2 Central difference scheme, stencil ( j − 1 , j, j + 1) is symmetric 7 / 73
Heat equation: Finite element method Find u ( t ) ∈ V = H 1 0 (0 , 1) such that for all φ ∈ V � 1 � 1 ∂u ∂u ∂φ ∂t φ d x + µ ∂x d x = 0 ∂x 0 0 Main idea: Approximate V by a finite dimensional space V h . Finite element method: V h is made of piecewise polynomial functions. Partition (0 , 1) into disjoint elements I j = ( x j − 1 2 , x j + 1 2 ) , (0 , 1) = ∪ j I j V h = V k h = { φ ∈ C (0 , 1) : φ (0) = φ (1) = 0 , φ | I j ∈ P k ( I j ) ∀ j } ⊂ H 1 0 Galerkin method: Find u h ( t ) ∈ V h such that for all φ h ∈ V h � 1 � 1 ∂u h ∂u h ∂φ h ∂t φ h d x + µ ∂x d x = 0 ∂x 0 0 8 / 73
Linear convection equation Linear, scalar convection/advection equation (Initial value problem) u t + au x =0 x ∈ R , t > 0 (1) u ( x, 0) = f ( x ) x ∈ R Exact solution u ( x, t ) = f ( x − at ) Initial condition is convected with speed a without change of form. f ( x ) u ( x , t ) x at Hence the extrema of the solution do not change with time. Also the L 2 -norm of the solution does not change with time. If E ( t ) is the solution operator u ( x, t ) = E ( t ) f ( x ) = ⇒ � E ( t ) u � = � u � in both sup-norm and L 2 -norm. 9 / 73
Forward time, backward space (FTBS) u n +1 − u n + au n j − u n j j j − 1 u n +1 = (1 − aλ ) u n j + aλu n = 0 ⇒ j j − 1 ∆ t h Forward time, central space (FTCS) u n +1 − u n + au n j +1 − u n j + aλ j j j − 1 u n +1 = u n 2 ( u n j − 1 − u n = 0 ⇒ j +1 ) j ∆ t 2 h Forward time, forward space (FTFS) u n +1 − u n + au n j +1 − u n j j j u n +1 = (1 + aλ ) u n j − aλu n = 0 ⇒ j j +1 ∆ t h λ = ∆ t h 10 / 73
u u u x x x Backward Forward Central Upwind scheme: switch between backward and forward difference u n +1 − u n + a + u n j − u n + a − u n j +1 − u n j j j − 1 j = 0 ∆ t h h 11 / 73
Stable in maximum norm for any a provided | a | λ ≤ 1 is satisfied. Courant-Friedrichs-Levy (CFL) number, CFL condition CFL = | a | λ = | a | ∆ t , CFL ≤ 1 , ∆ t = O ( h ) h 12 / 73
Hyperbolic conservation law ∂u ∂t + ∂ ∂xf ( u ) = 0 e.g., Burger’s equation u 2 ∂u ∂t + ∂ 2 = 0 ∂x d d d tu ( x ( t ) , t ) = 0 along d tx ( t ) = u ( x ( t ) , t ) Solution at times t � 0, t � tc and t � tc 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.5 1.0 1.5 2.0 2.5 1.5 2.0 2.5 13 / 73
Hyperbolic conservation law Definition: Weak solution A function u : R × R + → R is a weak solution of the IVP ( x, t ) ∈ R × R + , u t + f ( u ) x = 0 , u ( x, 0) = u 0 ( x ) together with locally integrable initial data u 0 if u is locally integrable and satisfies ∞ ∞ ∞ � � � ∀ φ ∈ C 1 0 ( R × R + ) ( uφ t + f ( u ) φ x )d x d t + u 0 ( x ) φ ( x, 0)d x = 0 , 0 −∞ −∞ Rankine-Hugoniot condition: f ( u + ) − f ( u − ) = s ( u + − u − ) , s = speed of discontinuity 14 / 73
Entropy function Consider a convex scalar conservation law u t + f ( u ) x = 0 Assume that there exists a convex function η ( u ) and another function θ ( u ) such that η ′ ( u ) f ′ ( u ) = θ ′ ( u ) Such a pair ( η, θ ) is called an entropy-entropy flux pair. For Burgers equation, we can choose θ ( u ) = 2 η ( u ) = u 2 , 3 u 3 For smooth solutions u t + f ′ ( u ) u x = 0 , η ′ ( u ) u t + η ′ ( u ) f ′ ( u ) u x = 0 , � �� � θ ′ ( u ) leads to another conservation law η t + θ x = 0 15 / 73
Entropy function In reality, the conservation law includes some dissipation η ′ ( u ) u t + η ′ ( u ) f ′ ( u ) u x = ǫη ′ ( u ) u xx u t + f x = ǫu xx = ⇒ leads to the entropy equation η t + θ x = ǫ ( η ( u ) u x ) x − ǫη ′′ ( u ) u 2 η ′′ ( u ) > 0 x ≤ ǫ ( η ( u ) u x ) x since In the limit of ǫ → 0 , we get η t + θ x ≤ 0 This condition must be satisfied in weak sense for all φ ∈ C 1 0 ( R × R + ) , φ ≥ 0 � ∞ � � ( η ( u ) φ t + θ ( u ) φ x )d x d t + η ( u 0 ( x )) φ ( x, 0)d t ≥ 0 0 R R 16 / 73
Kruzkov’s result The scalar Cauchy problem f ∈ C 1 ( R ) u t + f ( u ) x = 0 , with initial condition u 0 ∈ L ∞ ( R ) u (0 , x ) = u 0 ( x ) , has a unique entropy solution u ∈ L ∞ ( R + × R ) which fulfills (important for numerics) 1 Stability: || u ( t, · ) || L ∞ ≤ || u 0 || L ∞ , a.e. in t ∈ R + 2 Monotone: if u 0 ≥ v 0 a.e. in R , then u ( t, · ) ≥ v ( t, · ) a.e. in R , a.e. in t ∈ R + 17 / 73
Kruzkov’s result 3 TV-diminishing: if u 0 ∈ BV ( R ) then u ( t, · ) ∈ BV ( R ) and TV ( u ( t, · )) ≤ TV ( u 0 ) 4 Conservation: if u 0 ∈ L 1 ( R ) then � � u ( t, x )d x = u 0 ( x )d x, a.e. in t ∈ R + R R 5 Finite domain of dependence: if u , v are two entropy solutions corresponding to u 0 , v 0 ∈ L ∞ and φ {| f ′ ( φ ) | : | φ | ≤ max( || u 0 || L ∞ , || v 0 || L ∞ ) } M = max then � � | u ( t, x ) − v ( t, x ) | d x ≤ | u 0 ( x ) − v 0 ( x ) | d x | x |≤ R | x |≤ R + Mt 18 / 73
Finite volume method Divide space domain into finite volumes x j = 1 � Ω = ( x j − 1 2 , x j + 1 2 ) , h j = x j + 1 2 − x j − 1 2 , 2( x j − 1 2 + x j + 1 2 ) j 2 , x j + 1 2 ) and time slab Integrate conservation law over finite volume ( x j − 1 ( t n , t n +1 ) � t n +1 � x j + 1 � ∂u � ∂t + ∂f 2 d x d t = 0 ∂x t n x j − 1 2 Cell average value � x j + 1 u j ( t ) = 1 2 u ( x, t )d x h j x j − 1 2 gives conservation law (exact) � t n +1 ( u n +1 − u n j ) h j + [ f ( x j + 1 2 , t ) − f ( x j − 1 2 , t )]d t = 0 j t n 19 / 73
Finite volume method Approximate time integral of flux using solution at t n (explicit scheme) � t n +1 2 , t n )∆ t f ( x j + 1 2 , t )d t ≈ f ( x j + 1 t n leads to finite volume method f n 2 − f n v n +1 − v n j + 1 j − 1 j j + 2 = 0 ∆ t h j Cell average values are the unknowns in the finite volume method. � x j + 1 j ≈ u j ( t n ) = 1 v n u ( x, t n )d x 2 h j x j − 1 2 20 / 73
Godunov’s idea: Riemann problem Riemann problem defined at each cell face x j + 1 2 � u n x < x j + 1 j w ( x, t n ) = 2 u n x > x j + 1 j +1 2 Find exact solution 2 ) / ( t − t n ); u n j , u n w ( x, t ) = w R (( x − x j + 1 j +1 ) Compute flux f n 2 = f ( w R (0; u n j , u n j +1 )) j + 1 21 / 73
Linear convection eqn: upwind scheme u t + au x = 0 Upwind scheme u n +1 − u n + a + u n j − u n + a − u n j +1 − u n j j j − 1 j = 0 ∆ t h h can be written as finite volume scheme f n 2 − f n u n +1 − u n j + 1 j − 1 j j + 2 = 0 ∆ t h where � au j a > 0 1 2( au j + au j +1 ) − 1 f j + 1 2 = = 2 | a | ( u j +1 − u j ) au j +1 a < 0 22 / 73
Kinetic description of gases • Gas composed of many molecules • Describe state of gas by a velocity distribution function x ∈ R 3 , v ∈ R 3 f ( x, v, t ) , • Number density of gas molecules � n ( x, t ) = R 3 f ( x, v, t )d v =: � f � • Mass density ρ ( x, t ) = mn ( x, t ) , m = mass of one molecule We will assume that f is rescaled to give mass density, � f � = ρ . 23 / 73
Kinetic description of gases • Basic conserved quantities ρ = � f � 1 ρu = � vf � U = � ψf � , ψ = v 2 | v | 2 1 � � ( v 2 / 2) f ρe = • Evolution of f governed by Boltzmann equation ∂f ∂t + v · ∇ f = J ( f, f ) • Collisions do not change mass, momentum, energy � ψJ ( f, f ) � = 0 ψ are the only collisional invariants. 24 / 73
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