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.