On Energy Stable dG Approximation of the PML for the Wave Equation - PowerPoint PPT Presentation

On Energy Stable dG Approximation of the PML for the Wave Equation Monash Workshop on Numerical Differential Equations and Applications February 2020 Kenneth Duru Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications1 /

Waves are everywhere Simulations of seismic waves to quantify and assess earthquake risks and hazards. Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications2 /

Wave propagation: Forward Modeling Efficient Time Domain Wave Propagation Tool Accurate and stable Efficient and reliable volume discretizations absorbing boundaries Accurate source generation Efficient and scalable implementation on modern HPC platforms. My research has penetrated all components. Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications3 /

Truncated Domain Which boundary conditions ensure that numerical simulations converge to the infinite domain problem? (old but relevant: Engquist and Majda (1977)). Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications4 /

Reflections from boundaries A solution of the acoustic pressure with a point source. 0.4 ∆ x = 5/9 ∆ x = 5/27 0.2 Analytical p[MPa] 0 -0.2 -0.4 0 2 4 6 8 10 t[s] Classical absorbing boundary condition Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications5 /

Absorbing Layer . B´ Equations must be perfectly matched: J. P erenger (1994). Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications6 /

Anechoic chamber ”non-reflective”, ”non-echoing”, ”echo-free”. A room designed to completely absorb reflections of either sound or electromagnetic waves. Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications7 /

Complex coordinate stretching: Chew and Weedon (1994) S x := d ˜ ∂/∂ x → 1 / S x ∂/∂ x , x / dx = 1 + d x ( x ) / s , d x ≥ 0. Simplifies PML construction for hyperbolic systems 1 0.5 U PML 0 −0.5 PML −1 −2 −1 0 1 2 3 x Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications8 /

Acoustic wave equation in first order form 1 ∂ p ρ ∂ v ∂ t + ∇ · v = 0 , ∂ t + ∇ p = 0 . κ Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications9 /

Acoustic wave equation in first order form 1 ∂ p ρ ∂ v ∂ t + ∇ · v = 0 , ∂ t + ∇ p = 0 . κ ( x , y , z ) ∈ Ω = [ − 1 , 1 ] 3 , 1 − r η Zv η ∓ 1 + r η BCs: p = 0 , | r η | ≤ 1 , at η = ± 1 . 2 2 Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications9 /

Acoustic wave equation in first order form 1 ∂ p ρ ∂ v ∂ t + ∇ · v = 0 , ∂ t + ∇ p = 0 . κ ( x , y , z ) ∈ Ω = [ − 1 , 1 ] 3 , 1 − r η Zv η ∓ 1 + r η BCs: p = 0 , | r η | ≤ 1 , at η = ± 1 . 2 2   � � dxdydz = 1 dE  1 κ | p | 2 + ρ | v η | 2  > 0 , E ( t ) = dE > 0 . 2 Ω η = x , y , z � � 1 � 1 � d BT ( η ) dydzdx dt E ( t ) = − p ( n · v ) dS = − ≤ 0 , d η ∂ Ω − 1 − 1 η = x , y , z | χ ( − η ) | 2 � | χ (+ η ) | 2 � BT ( η ) = 1 − | r η | 2 η = − 1 + 1 − | r η | 2 � � � � η = 1 ≥ 0 . Z Z Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications9 /

Derive PML in the Laplace domain � ∞ e − st u ( x , y , z , t ) dt , � u ( x , y , z , s ) = s = a + ib , Re { s } = a > 0 , 0 Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10

Derive PML in the Laplace domain � ∞ e − st u ( x , y , z , t ) dt , � u ( x , y , z , s ) = s = a + ib , Re { s } = a > 0 , 0 1 κ s � p + ∇ · � ρ s � v + ∇ � v = 0 , p = 0 . Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10

Derive PML in the Laplace domain � ∞ e − st u ( x , y , z , t ) dt , u ( x , y , z , s ) = � s = a + ib , Re { s } = a > 0 , 0 1 κ s � p + ∇ · � ρ s � v + ∇ � v = 0 , p = 0 . S η = 1 + d η ( η ) PML : ∂/∂η → 1 / S η ∂/∂η, , d η ( η ) ≥ 0 , η = x , y , z . s Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10

Derive PML in the Laplace domain � ∞ e − st u ( x , y , z , t ) dt , u ( x , y , z , s ) = � s = a + ib , Re { s } = a > 0 , 0 1 κ s � p + ∇ · � ρ s � v + ∇ � v = 0 , p = 0 . S η = 1 + d η ( η ) PML : ∂/∂η → 1 / S η ∂/∂η, , d η ( η ) ≥ 0 , η = x , y , z . s 1 κ s � p + ∇ d · � v = 0 , ρ s � v + ∇ d � p = 0 , ∇ d = ( 1 / S x ∂/∂ x , 1 / S y ∂/∂ y , 1 / S z ∂/∂ z ) T . Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10

Time-domain PML � � ∂ � � � ∂ � S x v S x w s � Auxiliary variables: s � σ = d x − d y ∂ y , ψ = d x − d z ∂ z . S y S z Modified PDE:   � ∂ p � � ∂ v � d x 0 0 1   ∂ t + d x p + ∇ · v − σ − ψ = 0 , ρ ∂ t + dv + ∇ p = 0 , d = 0 d y 0  .  κ 0 0 d z Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications11

Time-domain PML � � ∂ � � � ∂ � S x v S x w s � Auxiliary variables: s � σ = d x − d y ∂ y , ψ = d x − d z ∂ z . S y S z Modified PDE:   � ∂ p � � ∂ v � d x 0 0 1   ∂ t + d x p + ∇ · v − σ − ψ = 0 , ρ ∂ t + dv + ∇ p = 0 , d = 0 d y 0  .  κ 0 0 d z Auxiliary differential equation: ODE � ∂σ � � ∂ψ � + ( d y − d x ) ∂ v + ( d z − d x ) ∂ w ∂ t + d y σ ∂ y = 0 , ∂ t + d z ψ ∂ z = 0 , 1 − r η Zv η ∓ 1 + r η p = 0 , η = ± 1 . BCs: at 2 2 Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications11

2D: SBP finite difference approximation 0 0 10 10 2nd−order 2nd−order 4th−order 4th−order 6th−order 6th−order −1 −1 10 10 � E z � h � E z � h −2 −2 10 10 −3 −3 10 10 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 time time (b) Discrete PML (c) No PML ( d x = 0 ) . Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications12

A nightmare for finite element and dG practitioners! t=150 s 50 0.1 40 0.08 30 0.06 y[km] 20 0.04 10 0.02 0 0 -60 -40 -20 0 20 40 60 x[km] GLL. t=35 s 50 0.1 40 0.08 30 0.06 y[km] 20 0.04 10 0.02 0 0 -60 -40 -20 0 20 40 60 x[km] GL. t=10 s 50 0.1 40 0.08 30 0.06 y[km] 20 0.04 10 0.02 0 0 -60 -40 -20 0 20 40 60 x[km] GLR. Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications13

Some pioneering work: Cauchy problem Abarbanel and Gottlieb (1997 & 98), Hesthaven et al. (1999), Collino and Tsogka (2001), B´ ecache et al. (2003): Geometric stability condition, Appel¨ o, Hagstrom and Kreiss (2006), Diaz and Joly (2006), Halpern et al. (2011), Duru and Kreiss (2012), ..., Duru (2016) Skelton, et al. (2007): Destabilizing effects of boundary conditions Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications14

Summary of literature: IBVP Assume constant coefficients and consider: 1. IVP PML : ( x , y ) ∈ ( −∞ , ∞ ) × ( −∞ , ∞ ) . Geometric stability condition B´ ecache et al. 2003. Classical Fourier analysis Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications15