Saul Abarbanel; Half a century of scientific work Bertil Gustafsson, - - PowerPoint PPT Presentation

Saul Abarbanel; Half a century of scientific work

Bertil Gustafsson, Uppsala University

Grew up in Tel Aviv Served in Israeli Army during the War of Independence 1948–1950

MIT 1952–1959

◮ Ph.D 1959, Theoretical Aerodynamics

Weizmann Insitute, 1960–1961

◮ Post Doc

Tel Aviv University, 1961–2017

◮ Professor ◮ Head of Appl. Math. Dept., 1964– (As Associate Professor) ◮ Dean of Science ◮ Vice Rector, ◮ Rector ◮ Chairman National Research Council ◮ Director Sackler Institute of Advanced Studies

ICASE (NASA Langley)

◮ Visitor

Brown University

◮ Visitor ◮ IBM Distinguished Visiting Research Professor

1959–1969 Heat transfer, gas dynamics Most part mathematical analysis, little numerics. Abarbanel: J. Math. and Physics (1960) Time Dependent Temperature Distribution in Radiating Solids. Abarbanel: Israel Journal of Technology (1966) The deflection of confining walls by explosive loads. Abarbanel–Zwas: J. Math. Anal. & Appl. (1969) The Motion of Shock Waves and Products of Detonation Confined between a Wall and a Rigid Piston. "...a detailed analytical solution of the piston motion and flow field is carried out..."

1969–

Construction and analysis of difference methods for PDE Stability of PDE and difference methods

◮ Lax–Wendroff type methods ◮ Compact high-order finite-difference schemes. ◮ Method of lines, Runge–Kutta methods ◮ PML methods

Law–Wendroff type methods and shocks

∂u ∂t = ∂f(u) ∂x

von Neumann–Richtmyer (1950): Add viscosity for numerical computation

∂u ∂t = ∂f(u) ∂x + ε∂2u ∂x2

Difference approximation "may be used for the entire calculation, just as though there were no shocks at all". 1954: Lax defines shocks as viscous limits ε → 0 Dissipative difference methods for computation 1960: Lax–Wendroff scheme, damping all frequencies 1969: MacCormack scheme, two stage, easier to apply Godunov methods (Riemann solvers), upwind methods, shock fitting

Lax-W methods: Possible oscillations near shock

97 il3 129 145 t6t t77 r95

Abarbanel–Zwas: Math. Comp. (1969): An iterative finite-difference method for hyperbolic systems. Lax–Wendroff type methods How to avoid oscillations near shocks? Wt + F(W)x = 0

⇐ ⇒

Wt + A(W)Wx = 0 Lax-W W n+1

j

= W n

j − λ 2 (F n j+1 − F n j−1)

+ λ2

2 [An j+1/2(F n j+1 − F n j ) − An j−1/2(F n j − F n j−1)]

W n+1 = W n + Q · W n Modify to W n+1 = W n + Q · [θW n+1 + (1 − θ)W n] with iteration W n+1,s+1 = W n+Q·[θW n+1,s+(1−θ)W n], s = 0, 1, . . . , k−1, W n+1,0 = Analysis for different θ and different k: Courant number λ = ∆t/∆x No oscillations for 1 and 2 iterations

97 il3 129 145 t6t t77 r95

Abarbanel-Goldberg: J. Comp. Phys. (1972) Numerical Solution of Quasi-Conservative Hyperbolic Systems; The Cylindrical Shock Problem. Wt + [F(W)]x = Ψ(x; W) General difference scheme W n+1 = W n + CW n (1) Implicit scheme External: W n+1,s+1 = W n + CW n + θ[CW n+1,s − CW n] Internal: W n+1,s+1 = W n + C(1 − θ)W n + θCW n+1,s Iterative solver as in Abarbanel–Zwas (1969), fixed number of iterations Larger timestep compared to explicit solver.

Standard scheme

i nt ,i iexocl) t1 (opprox.) 10 0.0

00

2 39 0.1976 02 3 82 0.3957 0 4 4 136 0.5996 0.6 5182 0,7988 0.8 6 ?17 0.9951 l.o 7 ?49 1.1959 1.2

Internal scheme

Use of time-dependent methods for computation of steady state. Abarbanel-Dwoyer-Gottlieb: J. Comp. Phys. (1986) Improving the Convergence Rate to Steady State of Parabolic ADI Methods. ut = uxx + uyy ADI-methods: Peaceman–Rachford (1955) ..... Beam–Warming (1976)

(1 − λδ2

x)(1 − λδ2 y)(v n+1 − v n) = αλ(δ2 x + δ2 y)v n,

λ = ∆t/h2

Improve convergence rate as n → ∞ by adding extra term

(1−λδ2

x)(1−λδ2 y)(v n+1−v n) = αλ(δ2 x+δ2 y)v n+γ

4 λ2δ2

xδ2 y(δ2 x + δ2 y)v n

Fourier analysis. Choose γ to minimize amplification factor. Model equation ⇒ γ = 0.8 independent of mesh-size.

Compact Pade’ type difference methods Orzag 1971, Kreiss-Oliger 1972: pseudospectral methods high order accuracy. Number of points per wavelength? High order difference methods? Pade’ (1890): Approximation of functions by rational functions Lele 1992: "Compact Finite Difference Schemes with Spectral-like Resolution" v = ∂u/∂x vj+1 + 4vj + vj−1 = 1 h(3uj+1 − 3uj−1)

(✹t❤ ♦r❞❡r)

Approximation ˆ Q(ξ) of ξ in Fourier space 0 ≤ ξ ≤ π Standard 4th order, standard 6th order, compact 4th order

Boundary conditions? Stability? Lele: Numerical computation of eigenvalues of difference operators, fixed ∆x.

Carpenter-Gottlieb-Abarbanel, J. Comp. Phys. (1993) The stability of numerical boundary treatments for compact high-order finite-difference schemes. Normal mode stability analysis (GKS). "Weak point: complexity in its application to higher order numerical schemes." Extra consideration: Fixed ∆t: Growing solutions ||V(t)|| ≤ Ceαt||V(0)|| ? Time-stable if α = 0. Analysis and construction of boundary conditions leading to time stability. Extensive thorough analysis, but for scalar case.

SBP-operators (Summation By Parts) . Kreiss–Scherer (1977) ut = ux, 0 ≤ x ≤ 1, u(1, t) = g(t), u(x, 0) = f(x)

(v, ∂

∂x v) = 1

2(|v(1)|2 − |v(0)|2)

for all v

⇒

d dt u 2= |u(1, t)|2 − |u(0, t)|2

SBP: Construct scalar product (u, v)h and a difference operator D such that

(v, Dv)h =

1 2(|vN|2 − |v0|2)

Simultaneous Approximation Terms (SAT) Funaro 1988, Funaro–Gottlieb 1988: SAT for pseudospectral methods Add penalty term dv dt = Dv − τ

• vN − g(t)
• w

(2) Carpenter-Gottlieb-Abarbanel, J. Comp.Phys. (1994) Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. Previous article (1993) with stable and time-stable methods are constructed for the scalar case. Use SAT method based on SBP-operators for systems This article: A systematic way of constructing time-stable SAT.

Abarbanel–Ditkowski, J. Comp. Phys. (1997) Asymptotically Stable Fourth-Order Accurate Schemes for the Diffusion Equation on Complex Shapes 4-th order, nonsymmetric difference operators near boundaries, "SAT-type". Solution bounded by constant independent of t.

Method of lines Carpenter-Gottlieb-Abarbanel-Don: SIAM J. Sci. Comput. (1995) The theoretical accuracy of Runge–Kutta time discretizations for the initial boundary value problem: A study of the boundary error.

∂u ∂t + ∂u ∂t = 0,

0 ≤ x ≤ 1, u(0, t) = g(t) Physical boundary condition at each stage of the R-K method (4th order) v 1

0 = g(t + δt 2 )

. . . Theoretical analysis showing deterioration of accuracy. Use instead derivative boundary conditions derived from original b.c. v 1

0 = g(t) + δt 2 g′(t)

. . . Full accuracy for the linear case, only 3rd order in nonlinear case

Abarbanel–Gottlieb, J. Comp. Phys. (1981): Optimal Time Splitting for Two- and Three-Dimensional Navier-Stokes Equations with Mixed Derivatives (33 pages) Interview by Philip Davis 2003: "Perhaps the most important article" U = [ρ, ρu, ρv, ρw, e]T Ut + Fx + Gy + Hz = 0 V = [ρ, u, v, w, p]T Vt+AVx+BVy +JVz = CVxx+DVyy +KVzz+ExyVxy +EyzVyz+EzxVxz Similarity transformation such that S−1MS are symmetric for all matrixes M = A, B, . . . , Ezx

Ut + (FH + FP + FM)x + (GH + GP + GM)y + (HH + HP + HM)z = 0 Un+2 =

[Lx(∆tx)Ly(∆ty)Lz(∆tz)Lxyz(∆txyz)Lxx(∆txx)Lyy(∆tyy)Lzz(∆tzz)]· [Lzz(∆tzz)Lyy(∆tyy)Lxx(∆txx)Lxyz(∆txyz)Lz(∆tz)Ly(∆ty)Lx(∆tx)]Un

Lx . . . , Lxx . . . MacCormack solvers Lxyz “MacCormack-like” solver

Scalar equation: ut = aux + buy + juz + cuxx + duyy + kuzz + exyuxy + eyzuyz + ezxuzx Stability under the standard one-dimensional conditions

a∆tx

∆x

≤ 1, . . .

c∆txx

(∆x)2

≤ 1

2, . . .

and ∆txyz ≤ ∆tx. The same stability result for the Navier-Stokes equations due to symmetric coefficient matrices.

Abarbanel-Duth-Gottlieb: Computers & Fluids (1989) Splitting methods for low Mach number Euler and Navier-Stokes equations Stiff system Splitting Symmetrizing Stiffness isolated to linear system ("may be solved implicitly with ease")

Abarbanel-Chertock: J. Comp. Phys. (2000) Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs, I,II Derivation of general compact implicit methods.

Absorbing boundary conditions Enquist–Majda (1977): Wave equation utt = uxx + uyy,

−∞ < x, y < ∞

Boundary conditions for finite domain x ≥ x0 ? Fourier transform

ω2 = ξ2 + η2 ξ = ±ω

• 1 − η2/ω2,

+ω√ ❢♦r ❧❡❢t❣♦✐♥❣ ✇❛✈❡

Pseudo-differential equation.

η/ω small ⇒

• 1 − η2/ω2 ≈ 1 − η2

2ω2 ⇒ ξω − ω2 + 1 2η2 = 0 ⇒ boundary condition at x = x0

∂2u ∂x∂t − ∂2 ∂t2 +

1 2

∂2 ∂y 2 = 0

Berenger (1994): (Centre d’Analyse de Dèfense, France) Perfectly Matched Layers (PML).

• Absorbing layer

x

y

Outer boundaries of computational domain

Maxwell equations 2D W = [Ex, Ey, Hz]T

∂W ∂t = A∂W ∂x + B∂W ∂y + CW

Can be symmetrized. PML formulation Wb = [Ex, Ey, Hzx, Hzy]T

∂Wb ∂t = Ab ∂Wb ∂x + Bb ∂Wb ∂y + CbWb

Abarbanel-Gottlieb, J. Comp. Phys. (1997) A mathematical analysis of the PML method New system cannot be symmetrized. Shown in the article: Initial value problem weakly well posed: Fourier transform

∂/∂x → iω1 ∂/∂y → iω2

Explicit form of transformed system is derived.

|ˆ

Hx(t)| ∼ (αω1 + βω2)t Requires bounded derivatives, but still growth in time.

Even worse: Perturbation

    −δ δ −δ δ    

Compute eigenvalues λ

λ1 ∼ √ ωδ ⇓ ˆ

W(t) ∼ eωδt Ill posed! Similar results for semi-discrete and fully discrete approximations.

Abarbanel-Gottlieb, Appl. Numer. Math., 1998 On the construction and analysis of absorbing layers in CEM. New PML type formulation. Introduce new variable polarization current J (Zilkowski 1997)

∂Ex ∂t = ∂Hz ∂y − J

· ·

∂J ∂t = −σ ∂Hz ∂y

P = J + σEx

∂P ∂t = −σP + σ2Ex

Strongly well posed (even when the outer boundary is taken into account). Still another formulation constructed, strongly well posed.

Abarbanel-Gottlieb-Hesthaven, J. Comp. Phys., 1999 Well-posed Perfectly Matched Layers for Advective Acoustics Development based on Abarbanel-Gottlieb (1998). "...somewhat lengthy algebraic manipulations..." Strongly well posed Numerical method: 4th order in space, Runge–Kutta in time

Abarbanel-Gottlieb-Hesthaven, J. Sci. Comp. 2002 Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics PML-method of Abarbanel–Gottlieb (1998) shows long time growth (after the initial pulse has left the original domain).

0 ≤ t ≤ 70

a a

l0 .10

• 20

"):

X

0 ≤ t ≤ 5000

Analysis of source of the problem Double eigenvalue, one eigenvector Cure: Split the eigenvalues by introducing small perturbation ε Uncertainty about damping properties in the PML-layer

Abarbanel-Quasimov-Tsynkov: J. Sci. Comp. (2009) Long-Time Performance of Unsplit PMLs with Explicit Second Order Schemes. Long-time growth with PML analyzed. Sensitive to choice of numerical method. Perturbation may or may not enter the original domain from PML-layer. "Lacunae based stabilization" by Qasimov-Tsynkov (2008).

Last publication: Abarbanel-Ditkowski: Appl. Numer.Math. (2015) Wave propagation in advected acoustics within a non-uniform medium under the effect of gravity. Saul 84 years old.