Conservation Laws & Finite Volume Methods Achim Schroll, - PDF document

Conservation Laws & Finite Volume Methods Achim Schroll, Conservation Laws & FVM

conservation laws � � d u ( t, x ) d x + f ( u ( t, x )) · n d S = 0 d t Ω ∂ Ω f(u) n u t + ∇ · f ( u ) = 0 | u − k | t + ( sign ( u − k )( f ( u ) − f ( k ))) x ≤ 0 , ∀ k ∈ R Z ∞ Z | u − k | φt + sign ( u − k )( f ( u ) − f ( k )) φx d x d t ≥ 0 , ∀ k, φ . . . 0 R Achim Schroll, Conservation Laws & FVM 1

conservation laws shallow water ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ h hu hv hu 2 + h 2 / 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ hu huv + + = 0 hv 2 + h 2 / 2 hv huv t x y Achim Schroll, Conservation Laws & FVM 2

conservation laws Euler equations ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ρ m n ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ m um + p vm ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + + = 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ n un vn + p E u ( E + p ) v ( E + p ) t x y u = m/ρ v = n/ρ � � E − m 2 + n 2 p = ( γ − 1) 2 ρ Achim Schroll, Conservation Laws & FVM 3

conservation laws applications ⊲ acoustic waves in the atmosphere, the ocean, or solids ⊲ shock waves and rarefaction waves in gas dynamics ⊲ electromagnetic waves, visible light, radar ⊲ shallow water waves ⊲ ultrasound waves ⊲ traffic dynamics ⊲ porous media flow, (oil) reservoirs, blood flow ⊲ waves arising from chemical reactions ⊲ combustion of gases, detonation, deflagration ⊲ waves in plasmas and ionized gases (MHD) ⊲ gravitational waves, colliding black holes . . . Achim Schroll, Conservation Laws & FVM 4

conservation laws tentative outline (linear systems) ⊲ examples: – gas dynamics (2.6) – linear accoustics (2.7) – shallow water (13.1) ⊲ linear hyperbolicity (3) ⊲ finite–volume methods: – first–order (4) – high–resolution (6) ⊲ stability, convergence, accuracy (8) Achim Schroll, Conservation Laws & FVM 5

conservation laws tentative outline (nonlinear problems) ⊲ scalar nonlinear conservation laws: – traffic flow (11.1) – Burgers’ equation (11.3) ⊲ weak solutions: shocks, rarefaction waves, entropy (11) ⊲ finite–volume methods for nonlinear problems (12) ⊲ nonlinear systems: – shallow water (13) – gas dynamics (14) ⊲ finite–volume methods for nonlinear systems (15) ⊲ stability and convergence (16) ⊲ relaxation and kinetic methods Achim Schroll, Conservation Laws & FVM 6

conservation laws isentropic gas � � � � ρ ρu + = 0 ρu 2 + p ρu t x p = κρ γ , ( air: γ = 1 . 4) polytropic gas ⎛ ⎞ ⎛ ⎞ ρ m ⎝ ⎠ ⎝ ⎠ m + um + p = 0 E u ( E + p ) t x � � E − m 2 m = ρu , p = ( γ − 1) 2 ρ Achim Schroll, Conservation Laws & FVM 7

conservation laws linear acoustics � p � � p � � � u 0 K 0 + = 0 u 1 /ρ 0 u 0 u t x K 0 = ρ 0 p ′ ( ρ 0 ) Achim Schroll, Conservation Laws & FVM 8

conservation laws shallow water y h(t,x) v(t,x) x h t + ( vh ) x = 0 ( v 2 h + gh 2 / 2) x ( vh ) t + = 0 Achim Schroll, Conservation Laws & FVM 9

Godunov’s method (1959) Riemann problem (1860) u x x x i+1 i−1 i t x x x i+1 i−1 i � 1 � CFL condition (1928): ∆ t ∆ x | λ | m ax ≤ 2 ≤ 1 Achim Schroll, Conservation Laws & FVM 10

Godunov’s method . . . advection equation u (0 , x ) = u 0 ( x ) u t + au x = 0 , travelling wave solution u ( t, x ) = u 0 ( x − at ) u x x x i+1 i−1 i Achim Schroll, Conservation Laws & FVM 11

Godunov’s method . . . REA algorithm (1959): ⊲ Reconstruct a piecewise constant function q ( x, t n ) = Q n i , x ∈ C i � ⊲ Evolve the conservation law with this data q ( x, t n ) � � q ( x, t n +1 ) � ⊲ Average the solution at t n +1 � = 1 Q n +1 q ( x, t n +1 ) d x � i ∆ x C i Achim Schroll, Conservation Laws & FVM 12

High resolution method ⊲ Reconstruct a piecewise linear function q ( x, t n ) = Q n i + σ n i ( x − x i ) , x ∈ C i � such that TV ( q ( · , t n )) ≤ TV ( Q n ) ⊲ Evolve the conservation law with this data q ( x, t n ) � � � q ( x, t n +1 ) scalar case: TV ( � q ( · , t n +1 )) ≤ TV ( � q ( · , t n )) ⊲ Average the solution at t n +1 � = 1 Q n +1 q ( x, t n +1 ) d x � i ∆ x C i TV ( Q n +1 ) ≤ TV ( � q ( · , t n +1 )) Achim Schroll, Conservation Laws & FVM 13

High resolution method limited slopes: u x x x x x x Achim Schroll, Conservation Laws & FVM 14