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 ) + ε∂ 2 u ∂ x ∂ 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
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? W t + F ( W ) x = 0 ⇐ ⇒ W t + A ( W ) W x = 0 Lax-W W n + 1 j − λ = W n 2 ( F n j + 1 − F n j − 1 ) j + λ 2 2 [ A n j + 1 / 2 ( F n j + 1 − F n j ) − A n 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 · [ θ W n + 1 + ( 1 − θ ) W n ] with iteration W n + 1 , s + 1 = W n + Q · [ θ W n + 1 , s +( 1 − θ ) W n ] , W n + 1 , 0 = s = 0 , 1 , . . . , k − 1 , 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. W t + [ 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 + θ CW 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. u t = u xx + u yy ADI-methods: Peaceman–Rachford (1955) ..... Beam–Warming (1976) y )( v n + 1 − v n ) = αλ ( δ 2 ( 1 − λδ 2 x )( 1 − λδ 2 x + δ 2 y ) v n , λ = ∆ t / h 2 Improve convergence rate as n → ∞ by adding extra term y ) v n + γ y )( v n + 1 − v n ) = αλ ( δ 2 ( 1 − λδ 2 x )( 1 − λδ 2 x + δ 2 4 λ 2 δ 2 x δ 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.
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 1 v j + 1 + 4 v j + v j − 1 = h ( 3 u j + 1 − 3 u j − 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 ) || ≤ Ce α t || 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) u t = u x , 0 ≤ x ≤ 1 , u ( 1 , t ) = g ( t ) , u ( x , 0 ) = f ( x ) 2 ( | v ( 1 ) | 2 − | v ( 0 ) | 2 ) ( v , ∂ ∂ x v ) = 1 ⇒ for all v dt � u � 2 = | u ( 1 , t ) | 2 − | u ( 0 , t ) | 2 d SBP: Construct scalar product ( u , v ) h and a difference operator D such that 1 2 ( | v N | 2 − | v 0 | 2 ) ( v , Dv ) h =
Simultaneous Approximation Terms (SAT) Funaro 1988, Funaro–Gottlieb 1988: SAT for pseudospectral methods Add penalty term d v � � dt = D v − 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.
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) 0 = g ( t + δ t v 1 2 ) . . . Theoretical analysis showing deterioration of accuracy. Use instead derivative boundary conditions derived from original b.c. 0 = g ( t ) + δ t v 1 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, ρ u , ρ v , ρ w , e ] T U t + F x + G y + H z = 0 V = [ ρ, u , v , w , p ] T V t + A V x + B V y + J V z = C V xx + D V 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
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
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 a ∆ t x ≤ 1 , . . . ∆ x c ∆ t xx ≤ 1 2 , . . . (∆ x ) 2 and ∆ t xyz ≤ ∆ t x . 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.
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