On the stability of IMEX schemes for singular hyperbolic PDEs - - PowerPoint PPT Presentation

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On the stability of IMEX schemes for singular hyperbolic PDEs - - PowerPoint PPT Presentation

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On the stability of IMEX schemes for singular hyperbolic PDEs Sebastian Noelle, RWTH Aachen joint with Klaus Kaiser, Ruth Sch obel, Jochen Sch utz, Hamed Zakerzadeh Paris, Nov. 2015 Sebastian Noelle AP Stability Paris, Nov. 2015 1

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On the stability of IMEX schemes for singular hyperbolic PDE’s

Sebastian Noelle, RWTH Aachen

joint with Klaus Kaiser, Ruth Sch¨

  • bel, Jochen Sch¨

utz, Hamed Zakerzadeh

Paris, Nov. 2015

Sebastian Noelle AP Stability Paris, Nov. 2015 1 / 25

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Key example

Isentropic gas dynamics ∂tρ + div(ρu) = 0, ∂t(ρu) + div(ρu ⊗ u) + 1 ε2 ∇p(ρ) = 0.

Sebastian Noelle AP Stability Paris, Nov. 2015 2 / 25

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Key example

Isentropic gas dynamics ∂tρ + div(ρu) = 0, ∂t(ρu) + div(ρu ⊗ u) + 1 ε2 ∇p(ρ) = 0. Mach number: ε = uref cref

Sebastian Noelle AP Stability Paris, Nov. 2015 2 / 25

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Challenges

Sebastian Noelle AP Stability Paris, Nov. 2015 3 / 25

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Challenges

ε ≪ 1: stiffness u ⋅ n, u ⋅ n ± c ε

Sebastian Noelle AP Stability Paris, Nov. 2015 3 / 25

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Challenges

ε ≪ 1: stiffness u ⋅ n, u ⋅ n ± c ε ε → 0: change of type compressible to incompressible flow Klainerman-Majda

Sebastian Noelle AP Stability Paris, Nov. 2015 3 / 25

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Challenges

Sebastian Noelle AP Stability Paris, Nov. 2015 4 / 25

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Challenges

Preserve the Asymptotics Asymptotic Consistency Asymptotic Stability ⇒ AP property (Shi Jin)

Sebastian Noelle AP Stability Paris, Nov. 2015 4 / 25

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Challenges

Preserve the Asymptotics Asymptotic Consistency Asymptotic Stability ⇒ AP property (Shi Jin) Efficiency: implicit for stiff part explicit for non-stiff part ⇒ IMEX

Sebastian Noelle AP Stability Paris, Nov. 2015 4 / 25

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Todays Talk

Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

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Todays Talk

  • a key stability structure

Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

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Todays Talk

  • a key stability structure
  • a new class of IMEX schemes

Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

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Todays Talk

  • a key stability structure
  • a new class of IMEX schemes
  • examples, applications, stability

Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

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Linear Stability Theory

Prototype linear system

Ut + AUx = 0. with stiff eigenvalues λmax ∶= max∣λ∣ = O (1 ε) λmin ∶= min∣λ∣ = O(1)

Sebastian Noelle AP Stability Paris, Nov. 2015 6 / 25

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Linear Stability Theory

Admissible Splittings

Definition A splitting A = ̃ A + ̂ A. is admissible, if (i) both ̃ A and ̂ A induce a hyperbolic system (ii) ̃ λ ∶= ρ(̃ A) = O (1 ε) ̂ λ ∶= ρ(̂ A) = O(1)

Sebastian Noelle AP Stability Paris, Nov. 2015 7 / 25

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Linear Stability Theory

CFL Conditions

ν ∶= λmax ∆t ∆x full CFLnumber ̂ ν ∶= ̂ λ ∆t ∆x nonstiff CFLnumber ν = O(1) ⇒ ̂ ν = O(ε) stable inefficient ν = O (1

ε)

⇐ ̂ ν = O(1) unstable efficient

Sebastian Noelle AP Stability Paris, Nov. 2015 8 / 25

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Linear Stability Theory

Flux-Splitting & IMEX Time-Discretization

Implicit-explicit discretization Klein 1996 Degond, Tang 2011 Haack, Jin, Liu 2011 Un+1 = Un + ̃ AUn+1

x

+ ̂ AUn

x

Sebastian Noelle AP Stability Paris, Nov. 2015 9 / 25

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Linear Stability Theory

Examples of stability/instability

Numerical experiments: IMEX schemes which are

  • based on admissible splittings
  • asymptotic consistent

can be stable and unstable Noelle, Bispen, Arun, Lukacova, Munz SISC 2014 Euler, Low Mach, IMEX, weakly AP Bispen, Arun, Lukacova, Noelle CiCP 2014 Shallow water, Low Froude, IMEX, AP

Sebastian Noelle AP Stability Paris, Nov. 2015 10 / 25

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Linear Stability Theory

Modified equation, cf. Warming/Hyett 1974

Theorem (Sch¨ utz, Noelle JSC 2014) The modified equation of the IMEX scheme is wt + Awx = ∆t 2 C wxx with diffusion matrix C ∶= (̂ α + ̃ α) ∆x ∆t I −(̂ A − ̃ A)(̂ A + ̃ A) and numerical upwind viscosities ̂ α, ̃ α.

Sebastian Noelle AP Stability Paris, Nov. 2015 11 / 25

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Linear Stability Theory

the crucial commutator

Is C positive definite? C = ((̂ α + ̃ α) ∆x

∆t I − ̂

A2) + (̃ Â A − ̂ Ã A) + ̃ A2 = O(1) + O (1

ε)

+ O ( 1

ε2 )

Certainly yes, if commutator [̃ A, ̂ A] = 0

Sebastian Noelle AP Stability Paris, Nov. 2015 12 / 25

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Linear Stability Theory

Example (Sch¨ utz, Noelle 2014)

Fourier stability analysis for prototype system A = ⎛ ⎜ ⎝ a 1

1 ε2

a

1 ε2

1 a ⎞ ⎟ ⎠ a > 0, eigenvalues λ = a,a ± √ 2 ε

Sebastian Noelle AP Stability Paris, Nov. 2015 13 / 25

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Linear Stability Theory

Euler: classical versus characteristic splitting

10−6 10−5 10−4 10−3 10−2 10−1 10−7 10−4 10−1 102 105 ε ν Allowable timestep sizes - A comparison

Splitting by Arun, Noelle... Characteristic splitting

Comparison of classical versus characteristic splitting

Sebastian Noelle AP Stability Paris, Nov. 2015 14 / 25

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Linear Stability Theory

How to recover stability

Need e.g. ̂ A and ̃ A symmetric

  • r

̃ Â A − ̂ Ã A = O(1)

  • r

̂ A = O(ε)

Sebastian Noelle AP Stability Paris, Nov. 2015 15 / 25

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Linear Stability Theory

Theorem (Stability for Haack-Jin-Liu (Zakerzadeh 2015)) For the isentropic Euler equations, the Haack-Jin-Liu scheme with Mach-uniform CFL condition, has (strictly) stable modified equation in the sense of Majda-Pego, i.e. it is AP stable.

Sebastian Noelle AP Stability Paris, Nov. 2015 16 / 25

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RS-IMEX

Reference-Solution IMEX

Nonlinear hyperbolic system of balance laws ∂tU(x,t;ε) + ∇ ⋅ F(U,x,t;ε) = S(U,x,t;ε) with U ∶ Rd × R+ × (0,1] → Rm, (x,t;ε) ↦ U(x,t;ε)

  • Challenge: Stiffness as ε → 0
  • Goal:

Asymptotic stability

Sebastian Noelle AP Stability Paris, Nov. 2015 17 / 25

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RS-IMEX

Reference solution and scaled perturbation: U = U + D V U ∶ Rd × R+ → Rm, (x,t) ↦ U(x,t) V ∶ Rd × R+ × (0,1] → Rm, (x,t;ε) ↦ U(x,t;ε) and D = diag(εk1,...,εkm) Taylor expansion with remainder of F and S around U: F = F(U) + A(U) DV + ̂ F(U,V ) = D(G + ̃ G + ̂ G) S = S(U) + S

′V DV + ̂

S(U,V ) = D(Z + ̃ Z + ̂ Z) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

RS+IM+EX

Sebastian Noelle AP Stability Paris, Nov. 2015 18 / 25

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RS-IMEX

Theorem (Modified equation for RS-IMEX (Noelle 2014)) B0Wt = −∇ ⋅ B1 + B2 + ∇ ⋅ (B3 ⋅ ∇W ) with B0 ∶= I − ∆t 2 (̃ Z ′ − ̂ Z ′), B1 ∶= ̃ G + ̂ G + ∆t 2 ((̃ G ′ − ̂ G ′)(̃ Z ′ + ̂ Z ′ − ̃ Gx − ̂ Gx)), B2 ∶= ̃ Z + ̂ Z + ∆t 2 (̃ Zt − ̂ Zt), B3 ∶= (̂ α + ¯ α)∆x 2 I + ∆t 2 (̃ G ′ − ̂ G ′)(̃ G ′ + ̂ G ′). Study this for each application!

Sebastian Noelle AP Stability Paris, Nov. 2015 19 / 25

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RS-IMEX

RS-IMEX is AP for isentropic Euler

Theorem (Consistency for RS-IMEX (Zakerzadeh 2015)) For isentropic Euler equations, the RS-IMEX scheme is consistent with the asymptotic limit in the fully-discrete settings, i.e. it is AP consistent. (Sch¨

  • bel 2015) AP consistency for semi-discrete scheme

Theorem (Stability for RS-IMEX (Zakerzadeh 2015)) For isentropic Euler equations, the RS-IMEX scheme with Mach-uniform CFL condition, has (strictly) stable modified equation in the sense of Majda-Pego, i.e. it is AP stable.

Sebastian Noelle AP Stability Paris, Nov. 2015 20 / 25

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RS-IMEX

van der Pol and IMEX (Sch¨ utz, Kaiser 2015)

−2 −1 1 2 −5 5

ε = 1 ε = 0.5 ε = 0.3

Prototype example (y′ z′) = ( z

g(y,z) ε

). ’Traditional’ splitting: (

g(y,z) ε

) + (z 0)

Sebastian Noelle AP Stability Paris, Nov. 2015 21 / 25

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RS-IMEX

van der Pol and IMEX

’Reference solution’ (RS) ε → 0: (y′

(0)

0 ) = ( z(0) g(y(0),z(0))). RS-IMEX splitting based on w(0): f (w) = f (w(0)) + f ′(w(0))(w − w(0)) + Rest Motivation: w − w(0) = O(ε).

Sebastian Noelle AP Stability Paris, Nov. 2015 22 / 25

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RS-IMEX

RS-IMEX + Runge-Kutta

10−3 10−2 10−1 10−8 10−6 10−4 10−2 Size of ∆t Error

ε = 10−1 ε = 10−3 ε = 10−5 ε = 10−7

10−3 10−2 10−1 10−11 10−9 10−7 10−5 10−3 10−1 Size of ∆t Error

ε = 10−1 ε = 10−3 ε = 10−5 ε = 10−7

10−3 10−2 10−1 10−11 10−9 10−7 10−5 10−3 10−1 Size of ∆t Error

ε = 10−1 ε = 10−3 ε = 10−5 ε = 10−7

10−3 10−2 10−1 10−8 10−6 10−4 10−2 Size of ∆t Error

ε = 10−1 ε = 10−3 ε = 10−5 ε = 10−7

10−3 10−2 10−1 10−11 10−9 10−7 10−5 10−3 10−1 Size of ∆t Error

ε = 10−1 ε = 10−3 ε = 10−5 ε = 10−7

10−3 10−2 10−1 10−11 10−9 10−7 10−5 10−3 10−1 Size of ∆t Error

ε = 10−1 ε = 10−3 ε = 10−5 ε = 10−7

(Left to right) DPA-242, BHR-553, BPR-353. (Top to bottom) Standard / RS-IMEX

IMEX Runge-Kutta (Pareschi, Russo, Boscarino ...) standard splitting looses convergence order RS-IMEX gives full order of accuracy

Sebastian Noelle AP Stability Paris, Nov. 2015 23 / 25

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RS-IMEX

Numerical Comparison for Isentropic Euler Equations

Haack-Jin-Liu scheme RS-IMEX scheme initial data from Degond-Tang (CiCP 2011) / Haak-Jin-Liu (JSC 2012) ρ(x,y,0) = 1 + ε2 sin2(2π(x + y)) ρu(x,y,0) = sin(2π(x − y)) + ε2 sin(2π(x + y)) ρv(x,y,0) = sin(2π(x − y)) + ε2 cos(2π(x + y))

Sebastian Noelle AP Stability Paris, Nov. 2015 24 / 25

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RS-IMEX

Results

(a) Density (b) Momentum

Sebastian Noelle AP Stability Paris, Nov. 2015 25 / 25

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RS-IMEX

Results

10−2 10−1 10−3 10−2 10−1 100 L1-error ε = 1 HJL RS-IMEX 10−2 10−1 10−3 10−2 10−1 100 L1-error ε = 10−6 HJL RS-IMEX

Numerical comparison of HJL vs. RS-IMEX

Sebastian Noelle AP Stability Paris, Nov. 2015 25 / 25