Saul Abarbanel; Half a century of scientific work

Bertil Gustafsson, Uppsala University

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

MIT 1952–1959

I Ph.D 1959, Theoretical Aerodynamics

Weizmann Insitute, 1960–1961

I Post Doc

Tel Aviv University, 1961–2017

I Professor

I Head of Appl. Math. Dept., 1964– (As Associate Professor)

I Dean of Science

I Vice Rector,

I Rector

I Chairman National Research Council

I Director Sackler Institute of Advanced Studies

ICASE (NASA Langley)

I Visitor

Brown University

I Visitor

I IBM Distinguished Visiting Research Professor

1959–1969Heat 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 Confinedbetween a Wall and a Rigid Piston."...a detailed analytical solution of the piston motion and flow field iscarried out..."

1969–

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

I Lax–Wendroff type methods

I Compact high-order finite-difference schemes.

I Method of lines, Runge–Kutta methods

I PML methods

Law–Wendroff type methods and shocks

∂u

∂t=∂f(u)

∂xvon Neumann–Richtmyer (1950): Add viscosity for numericalcomputation

∂u

∂t=∂f(u)

∂x+ ε

∂2u

∂x2

Difference approximation "may be used for the entire calculation, just asthough there were no shocks at all".

1954: Lax defines shocks as viscous limits ε→ 0Dissipative difference methods for computation

1960: Lax–Wendroff scheme, damping all frequencies1969: 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 methodsHow to avoid oscillations near shocks?

Wt + F(W)x = 0 ⇐⇒ Wt + A(W)Wx = 0

Lax-WW n+1

j = W nj − λ

2(F n

j+1 − F nj−1)

+λ2

2[An

j+1/2(F nj+1 − F n

j )− Anj−1/2(F n

j − F nj−1)]

W n+1 = W n + Q ·W n

Modify toW 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 = W n

Analysis for different θ and different k :Courant number λ = ∆t/∆xNo 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; TheCylindrical Shock Problem.

Wt + [F(W)]x = Ψ(x; W)

General difference scheme

W n+1 = W n + CW n (1)

Implicit schemeExternal:

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 iterationsLarger timestep compared to explicit solver.

Standard scheme

i nt ,i iexocl) t1 (opprox.)

10 0.0 002 39 0.1976 023 82 0.3957 0 44 136 0.5996 0.6

5182 0,7988 0.8

6 ?17 0.9951 l.o7 ?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 ADIMethods.

ut = uxx + uyy

ADI-methods: Peaceman–Rachford (1955) .....Beam–Warming (1976)

(1− λδ2x )(1− λδ2

y )(vn+1 − vn) = αλ(δ2x + δ2

y )vn, λ = ∆t/h2

Improve convergence rate as n→∞ by adding extra term

(1−λδ2x )(1−λδ2

y )(vn+1−vn) = αλ(δ2x +δ2

y )vn+γ

4λ2δ2

xδ2y (δ2

x + δ2y )vn

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 orderaccuracy.Number of points per wavelength?High order difference methods?

Pade’ (1890): Approximation of functions by rational functionsLele 1992: "Compact Finite Difference Schemes with Spectral-likeResolution"

v = ∂u/∂x

vj+1 + 4vj + vj−1 =1

h(3uj+1 − 3uj−1) (4th order)

Approximation Q(ξ) of ξ in Fourier space 0 ≤ ξ ≤ πStandard 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-orderfinite-difference schemes.Normal mode stability analysis (GKS)."Weak point: complexity in its application to higher order numericalschemes."

Extra consideration:Fixed ∆t : Growing solutions ||V(t)|| ≤ Ceαt ||V(0)|| ?Time-stable if α = 0.Analysis and construction of boundary conditions leading to timestability.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) = 12(|v(1)|2 − |v(0)|2) for all v ⇒

ddt‖ u ‖2= |u(1, t)|2 − |u(0, t)|2

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

(v ,Dv)h =1

2(|vN|2 − |v0|2)

Simultaneous Approximation Terms (SAT)

Funaro 1988, Funaro–Gottlieb 1988: SAT for pseudospectral methodsAdd 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 solvinghyperbolic systems: Methodology and application to high-order compactschemes.Previous article (1993) with stable and time-stable methods areconstructed for the scalar case.Use SAT method based on SBP-operators for systemsThis article: A systematic way of constructing time-stable SAT.

Abarbanel–Ditkowski, J. Comp. Phys. (1997)Asymptotically Stable Fourth-Order Accurate Schemes for the DiffusionEquation 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 theinitial 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)

v10 = g(t + δt

2)

...

Theoretical analysis showing deterioration of accuracy.

Use instead derivative boundary conditions derived from original b.c.

v10 = g(t) + δt

2g′(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-StokesEquations 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 +K Vzz +Exy Vxy +EyzVyz +EzxVxz

Similarity transformation such that S−1MS are symmetric for all matrixesM = 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 solversLxyz “MacCormack-like” solver

Scalar equation:

ut = aux + buy + juz + cuxx + duyy + kuzz + exy uxy + eyzuyz + ezxuzx

Stability under the standard one-dimensional conditions

a∆tx∆x

≤ 1, . . .

c∆txx

(∆x)2 ≤ 12, . . .

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

Abarbanel-Duth-Gottlieb: Computers & Fluids (1989) Splitting methodsfor low Mach number Euler and Navier-Stokes equationsStiff systemSplittingSymmetrizingStiffness isolated to linear system ("may be solved implicitly with ease")

Abarbanel-Chertock: J. Comp. Phys. (2000)Strict Stability of High-Order Compact Implicit Finite-DifferenceSchemes: 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√

for leftgoing wave

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

∂y2= 0

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

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Absorbing layer

x

y

Outer boundaries of computational domain

Maxwell equations 2DW = [Ex , Ey , Hz]T

∂W

∂t= A

∂W

∂x+ B

∂W

∂y+ CW

Can be symmetrized.PML formulationWb = [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.

|Hx(t)| ∼ (αω1 + βω2)t

Requires bounded derivatives, but still growth in time.

Even worse:Perturbation

0 0 −δ δ0 0 −δ δ0 0 0 00 0 0 0

Compute eigenvalues λ

λ1 ∼√ωδ

⇓W(t) ∼ eωδt

Ill posed!Similar results for semi-discrete and fully discrete approximations.

Abarbanel-Gottlieb, Appl. Numer. Math., 1998On 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 intoaccount).Still another formulation constructed, strongly well posed.

Abarbanel-Gottlieb-Hesthaven, J. Comp. Phys., 1999Well-posed Perfectly Matched Layers for Advective Acoustics

Development based on Abarbanel-Gottlieb (1998)."...somewhat lengthy algebraic manipulations..."Strongly well posedNumerical method: 4th order in space, Runge–Kutta in time

Abarbanel-Gottlieb-Hesthaven, J. Sci. Comp. 2002Long Time Behavior of the Perfectly Matched Layer Equations inComputational Electromagnetics

PML-method of Abarbanel–Gottlieb (1998) shows long time growth(after the initial pulse has left the original domain).

0 ≤ t ≤ 70

aal0

0

.10

-20

"):

X

0 ≤ t ≤ 5000

Analysis of source of the problemDouble eigenvalue, one eigenvectorCure: Split the eigenvalues by introducing small perturbation εUncertainty about damping properties in the PML-layer

Abarbanel-Quasimov-Tsynkov: J. Sci. Comp. (2009)Long-Time Performance of Unsplit PMLs with Explicit Second OrderSchemes.

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 mediumunder the effect of gravity.

Saul 84 years old.