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

Conservation Laws & Finite Volume Methods

Achim Schroll, Conservation Laws & FVM

conservation laws

d dt

• Ω

u(t, x) dx +

• ∂Ω

f(u(t, x)) · n dS = 0

n f(u)

ut + ∇ · f(u) = 0

|u − k|t + (sign(u − k)(f(u) − f(k)))x ≤ 0 , ∀k ∈ R

Z ∞ Z R |u − k|φt + sign(u − k)(f(u) − f(k))φxdxdt ≥ 0 , ∀k, φ . . . Achim Schroll, Conservation Laws & FVM 1

conservation laws

shallow water ⎛ ⎝ h hu hv ⎞ ⎠

t

+ ⎛ ⎝ hu hu2 + h2/2 huv ⎞ ⎠

x

+ ⎛ ⎝ hv huv hv2 + h2/2 ⎞ ⎠

y

= 0

Achim Schroll, Conservation Laws & FVM 2

conservation laws

Euler equations ⎛ ⎜ ⎜ ⎝ ρ m n E ⎞ ⎟ ⎟ ⎠

t

+ ⎛ ⎜ ⎜ ⎝ m um + p un u(E + p) ⎞ ⎟ ⎟ ⎠

x

+ ⎛ ⎜ ⎜ ⎝ n vm vn + p v(E + p) ⎞ ⎟ ⎟ ⎠

y

= 0 u = m/ρ v = n/ρ p = (γ − 1)

• E − m2+n2

2ρ

• 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

• t

+

• ρu

ρu2 + p

• x

= 0 p = κργ , (air: γ = 1.4) polytropic gas ⎛ ⎝ ρ m E ⎞ ⎠

t

+ ⎛ ⎝ m um + p u(E + p) ⎞ ⎠

x

= 0 m = ρu , p = (γ − 1)

• E − m2

2ρ

• Achim Schroll, Conservation Laws & FVM

7

conservation laws

linear acoustics p u

• t

+

• u0

K0 1/ρ0 u0 p u

• x

= 0 K0 = ρ0p′(ρ0)

Achim Schroll, Conservation Laws & FVM 8

conservation laws

shallow water

x y v(t,x) h(t,x)

ht + (vh)x = (vh)t + (v2h + gh2/2)x =

Achim Schroll, Conservation Laws & FVM 9

Godunov’s method (1959)

Riemann problem (1860)

x x

i i−1

x i+1 u x x

i i−1

x i+1 t

CFL condition (1928): ∆t ∆x |λ|max ≤ 1 2 ≤

• 1

Achim Schroll, Conservation Laws & FVM 10

Godunov’s method . . .

advection equation ut + aux = 0 , u(0, x) = u0(x) travelling wave solution u(t, x) = u0(x − at)

x x

i i−1

x i+1 u

Achim Schroll, Conservation Laws & FVM 11

Godunov’s method . . .

REA algorithm (1959): ⊲ Reconstruct a piecewise constant function

• q(x, tn) = Qn

i ,

x ∈ Ci ⊲ Evolve the conservation law with this data

• q(x, tn)

q(x, tn+1) ⊲ Average the solution at tn+1 Qn+1

i

= 1 ∆x

• Ci
• q(x, tn+1) dx

Achim Schroll, Conservation Laws & FVM 12

High resolution method

⊲ Reconstruct a piecewise linear function

• q(x, tn) = Qn

i + σn i (x − xi) ,

x ∈ Ci such that TV (q(·, tn)) ≤ TV (Qn) ⊲ Evolve the conservation law with this data

• q(x, tn)

q(x, tn+1) scalar case: TV ( q(·, tn+1)) ≤ TV ( q(·, tn)) ⊲ Average the solution at tn+1 Qn+1

i

= 1 ∆x

• Ci
• q(x, tn+1) dx

TV (Qn+1) ≤ TV ( q(·, tn+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