Saul Abarbanel; Half a century of scientific work. Bertil Gustafsson, Uppsala University

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1 Saul Abarbanel; Half a century of scientific work Bertil Gustafsson, Uppsala University

2 Grew up in Tel Aviv Served in Israeli Army during the War of Independence

3 MIT Ph.D 1959, Theoretical Aerodynamics

4 Post Doc Weizmann Insitute,

5 Tel Aviv University, 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

6 Visitor ICASE (NASA Langley)

7 Brown University Visitor IBM Distinguished Visiting Research Professor

8 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..."

9 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

10 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 2 u x 2 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

11 Lax-W methods: Possible oscillations near shock 97 il t6t t77 r95

12 Abarbanel Zwas: Math. Comp. (1969): An iterative finite-difference method for hyperbolic systems. Lax Wendroff type methods How to avoid oscillations near shocks? W t + F(W) x = 0 W t + A(W)W x = 0 Lax-W = W n j λ(f n 2 j+1 Fj 1) n + λ2 2 [An (F n j+1/2 j+1 F n j ) A n (F n j 1/2 j W n+1 j F n j 1)]

13 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

14 97 il t6t t77 r95

15 Abarbanel-Goldberg: J. Comp. Phys. (1972) Numerical Solution of Quasi-Conservative Hyperbolic Systems; The Cylindrical Shock Problem. General difference scheme Implicit scheme External: Internal: W t + [F(W)] x = Ψ(x; W) W n+1 = W n + CW n (1) W n+1,s+1 = W n + CW n + θ[cw n+1,s CW n ] 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.

16 Standard scheme i nt,i iexocl) t1 (opprox.) , ? l.o 7?

17 Internal scheme

18 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. u t = u xx + u yy ADI-methods: Peaceman Rachford (1955)... Beam Warming (1976) (1 λδ 2 x)(1 λδ 2 y)(v n+1 v n ) = αλ(δ 2 x + δ 2 y)v n, λ = t/h 2 Improve convergence rate as n by adding extra term (1 λδ 2 x)(1 λδ 2 y)(v n+1 v n ) = αλ(δ 2 x+δ 2 y)v n + γ 4 λ2 δ 2 xδ 2 y(δ 2 x + δ 2 y)v n Fourier analysis. Choose γ to minimize amplification factor. Model equation γ = 0.8 independent of mesh-size.

19 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 v j+1 + 4v j + v j 1 = 1 h (3u j+1 3u j 1 ) (4th order)

20 Approximation ˆQ(ξ) of ξ in Fourier space 0 ξ π Standard 4th order, standard 6th order, compact 4th order

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

22 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) Ce αt V(0)? Time-stable if α = 0. Analysis and construction of boundary conditions leading to time stability. Extensive thorough analysis, but for scalar case.

23 . SBP-operators (Summation By Parts) Kreiss Scherer (1977) u t = u x, 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 ( v N 2 v 0 2 )

24 Simultaneous Approximation Terms (SAT) Funaro 1988, Funaro Gottlieb 1988: SAT for pseudospectral methods Add penalty term dv dt = Dv τ ( v N 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.

25 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.

26 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 Theoretical analysis showing deterioration of accuracy. Use instead derivative boundary conditions derived from original b.c. v ) = g(t) + δt 2 g (t) Full accuracy for the linear case, only 3rd order in nonlinear case.

27 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 U t + F x + G y + H z = 0 V = [ρ, u, v, w, p] T V t +AV x +BV y +JV z = CV xx +DV yy +K V zz +E xy V xy +E yz V yz +E zx V xz Similarity transformation such that S 1 MS are symmetric for all matrixes M = A, B,..., E zx

28 U t + (F H + F P + F M ) x + (G H + G P + G M ) y + (H H + H P + H M ) z = 0 U n+2 = [L x ( t x )L y ( t y )L z ( t z )L xyz ( t xyz )L xx ( t xx )L yy ( t yy )L zz ( t zz )] [L zz ( t zz )L yy ( t yy )L xx ( t xx )L xyz ( t xyz )L z ( t z )L y ( t y )L x ( t x )]U n L x..., L xx... MacCormack solvers L xyz MacCormack-like solver

29 Scalar equation: u t = au x + bu y + ju z + cu xx + du yy + ku zz + e xy u xy + e yz u yz + e zx u zx Stability under the standard one-dimensional conditions and t xyz t x. a t x x 1,... c t xx 1,... ( x) 2 2 The same stability result for the Navier-Stokes equations due to symmetric coefficient matrices.

30 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")

31 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.

32 Absorbing boundary conditions Enquist Majda (1977): Wave equation u tt = u xx + u yy, < x, y < Boundary conditions for finite domain x x 0? Fourier transform ω 2 = ξ 2 + η 2 ξ = ±ω 1 η 2 /ω 2, +ω for leftgoing wave Pseudo-differential equation. η/ω small 1 η 2 /ω 2 1 η2 2ω 2 ξω ω η2 = 0 boundary condition at x = x 0 2 u x t = t 2 2 y 0 2

33 Berenger (1994): (Centre d Analyse de Dèfense, France) Perfectly Matched Layers (PML). Outer boundaries of computational domain Absorbing layer y x

34 Maxwell equations 2D W = [E x, E y, H z ] T W t = A W x + B W y + CW Can be symmetrized. PML formulation W b = [E x, E y, H zx, H zy ] T W b t = A b W b x + B W b b y + C bw b

35 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 iω 1 / y iω 2 Explicit form of transformed system is derived. Ĥ x (t) (αω 1 + βω 2 )t Requires bounded derivatives, but still growth in time.

36 Even worse: Perturbation 0 0 δ δ 0 0 δ δ Compute eigenvalues λ Ill posed! λ 1 ωδ Ŵ(t) e ωδt Similar results for semi-discrete and fully discrete approximations.

37 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) E x t J t = Hz y = σ Hz y J P = J + σe x P t = σp + σ 2 E x Strongly well posed (even when the outer boundary is taken into account). Still another formulation constructed, strongly well posed.

38 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

39 Abarbanel-Gottlieb-Hesthaven, J. Sci. Comp 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).

40 0 t 70

41 a l a X "): 0 t 5000

42 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

43 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).

44 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.

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