Darwin and higher order approximations to Maxwells equations in R 3 - PowerPoint PPT Presentation

Darwin and higher order approximations to Maxwell’s equations in R 3 Sebastian Bauer Universit¨ at Duisburg-Essen in close collaboration with the Maxwell group around Dirk Pauly Universit¨ at Duisburg-Essen Special Semester on Computational Methods in Science and Engineering RICAM, October 20, 2016 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equations Electro-and magnetostatics div E = ρ rot B = µ 0 j ε 0 rot E = 0 div B = 0 Faraday’s law of induction, no charge conservation, Eddy current model div E = ρ rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Maxwell’s displacement current, charge conservation, Lorentz invariance div E = ρ − 1 c 2 ∂ t E + rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equations Electro-and magnetostatics div E = ρ rot B = µ 0 j ε 0 rot E = 0 div B = 0 Faraday’s law of induction, no charge conservation, Eddy current model div E = ρ rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Maxwell’s displacement current, charge conservation, Lorentz invariance div E = ρ − 1 c 2 ∂ t E + rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equations Electro-and magnetostatics div E = ρ rot B = µ 0 j ε 0 rot E = 0 div B = 0 Faraday’s law of induction, no charge conservation, Eddy current model div E = ρ rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Maxwell’s displacement current, charge conservation, Lorentz invariance div E = ρ − 1 c 2 ∂ t E + rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Another system with charge conservation but elliptic equations Maxwell’s equations div E = ρ − 1 c 2 ∂ t E + rot B = µ 0 j ε 0 ∂ t B + rot E = 0 div B = 0 Darwin equations E = E L + E T with rot E L = 0 and div E T = 0 div E L = ρ − 1 c 2 ∂ t E L + rot B = µ 0 j ∂ t B + rot E T = 0 ε 0 rot E L = 0 div E T = 0 div B = 0 charge conservation, three elliptic equations which can be solved successively Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Problems/Questions and Outline of the talk Questions Dimensional analysis: In which situations is the Darwin system a reasonable approximation? What are lower order and what are higher order approximations? solution theory for all occuring problems rigorous estimates for the error between solutions of approximate equations and solutions of Maxwell’s equations Outline of the talk dimensional analysis and asymptotic expansion bounded domains exterior domains Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

In which situations is the approximation reasonable? – dimensionless equations x characteristic length-scale of the charge and current distributions ¯ ¯ t characteristic time-scale, in which a charge moves over a distant ¯ x , slow time-scale ρ characteristic charge density ¯ v = ¯ x ¯ t characteristic velocity of the problem ¯ jj ′ , E ′ ( t ′ ) = E (¯ tt ′ , E = ¯ EE ′ , B = ¯ tt ′ ) ρρ ′ , j = ¯ xx ′ , t = ¯ BB ′ , ρ = ¯ x = ¯ ... ¯ E Maxwell’s dimensionless equations ε 0 ¯ v ¯ ¯ E ¯ E j ¯ x ̺ div ′ E ′ = ̺ ′ B ∂ t ′ E ′ − rot ′ B ′ = − µ 0 B j ′ c 2 ¯ ¯ x ¯ ¯ v ¯ ¯ B E ∂ t ′ B ′ + rot ′ E ′ = 0 div ′ B ′ = 0 ¯ charge conservation ̺ ¯ ¯ v j ∂ t ′ ̺ ′ + div ′ j ′ = 0 ¯ Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

units and dimensionless equations Degond, Raviart (’92): E = ¯ ¯ x ¯ ρ ¯ ¯ x ¯ ¯ ρ c ε 0 and η = ¯ v ε 0 , j = c ¯ ρ, B = c leads to div E = ρ − η ∂ t E + rot B = j η ∂ t B + rot E = 0 div B = 0 together with charge conservation η ∂ t ρ + div j = 0. Schaeffer (’86), plasma physics with Vlasov matter E = ¯ ¯ x ¯ ρ ¯ ¯ x ¯ ¯ ρ c ε 0 and η = ¯ v ε 0 , j = ¯ v ¯ ρ, B = c leads to div E = ρ − η ∂ t E + rot B = η j η ∂ t B + rot E = 0 div B = 0 together with charge conservation 1 ∂ t ρ + div j = 0. Assumption: η ≪ 1 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Formal expansion in powers of η and equations in the orders of η div E η = ρ η − η ∂ t E η + rot B η = η j η η ∂ t B η + rot E η = 0 div B η = 0 E η = E 0 + η E 1 + η 2 E 2 + . . . , B η = B 0 + η B 1 + η 2 B 2 + . . . Ansatz: ρ η = ρ 0 , j η = j 0 ∂ t ρ 0 + div j 0 = 0 For simplicity: with resulting equations (for the plasma scaling) � η 0 � div E 0 = ρ 0 , rot B 0 O = 0 rot E 0 div B 0 = 0 , = 0 � η 1 � = j 0 + ∂ t E 0 div E 1 rot B 1 O = 0 , rot E 1 = − ∂ t B 0 , div B 1 = 0 , � η 2 � div E 2 rot B 2 = ∂ t E 1 , O = 0 , rot E 2 = − ∂ t B 1 , div B 2 = 0 , � η k � = ∂ t E k − 1 , div E k rot B k O = 0 , rot E k = − ∂ t B k − 1 , div B k = 0 , Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Comparsion with eddy current and Darwin, plasma case We can consistently set : E 1 = E 2 k − 1 = 0 and B 0 = B 2 k = 0 first order : Set E = E 0 + η E 1 = E 0 and B = B 0 + η B 1 = η B 1 div E = ρ 0 rot B = j 0 η∂ t B + rot E = 0 div B = 0 second order: Set E L = E 0 , E T = η 2 E 2 , and B = η B 1 , then div E L = ρ 0 rot B = j 0 + η ∂ t E L rot E T = − η ∂ t B rot E L = 0 div E T = 0 div B = 0 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Formal expansion in powers of η and equations in the orders of η , Degond Raviart scaling div E η = ρ η − η ∂ t E η + rot B η = j η η ∂ t B η + rot E η = 0 div B η = 0 E η = E 0 + η E 1 + η 2 E 2 + . . . , B η = B 0 + η B 1 + η 2 B 2 + . . . Ansatz: ρ η = ρ 0 , j η = j 0 + η j 1 . For simplicity: resulting equations � η 0 � div E 0 = ρ 0 , rot B 0 = j 0 O rot E 0 div B 0 = 0 , = 0 � η 1 � = j 1 + ∂ t E 0 div E 1 rot B 1 O = 0 , rot E 1 = − ∂ t B 0 , div B 1 = 0 , � η 2 � div E 2 rot B 2 = ∂ t E 1 , O = 0 , rot E 2 = − ∂ t B 1 , div B 2 = 0 , � η k � = ∂ t E k − 1 , div E k rot B k O = 0 , = − ∂ t B k − 1 , rot E k div B k = 0 , Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Comparsion with eddy current and Darwin, Degond Raviart scaling zeroth order: quasielectrostatic and quasimagnetostatic, div j 0 = 0 div E 0 = ρ 0 rot B 0 = j 0 rot E 0 = 0 div B 0 = 0 second order: E = E 0 + η E 1 + η 2 E 2 , E L = E 0 , E T = η E 1 + η 2 E 2 B = B 0 + η B 1 and j = j 0 + η j 1 div E L = ρ 0 rot B = j 0 + η ∂ t E L rot E T = − η ∂ t B rot E L = 0 div E T = 0 div B = 0 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Solution theory Maxwell’s time-dependent equations: L 2 setting, selfadjoint operator, spectral calculus or halfgroup theory or Picard’s theorem, independently of the domain, very flexible. Iterated rot-div systems. Solution of the previous step enters as source term. Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

the general L 2 -setting for rot, div and grad ◦ C ∞ (Ω) ⊂ L 2 → L 2 , R (Ω) = D (rot ∗ ) = H (curl , Ω) rot : ◦ rot = rot ∗ : R (Ω) ⊂ L 2 → L 2 , R (Ω) = D (rot ∗ ) = { E ∈ R (Ω) | E ∧ ν = 0 } ◦ ◦ ◦ ◦ rot= rot ∗ : R (Ω) ⊂ L 2 → L 2 , rot = rot ∗∗ = rot rot ∗ = rot and L 2 -decomposition ◦ ◦ L 2 = rot R ⊕ R 0 = rot R⊕ R 0 In the same manner ◦ D = H (div , Ω) = D (grad ∗ ) D = D (grad ∗ ) = { E ∈ D | E · ν = 0 } ◦ ◦ H 1 = D (div ∗ ) H 1 = D ( div ∗ ) L 2 decompositions ◦ ◦ L 2 = grad H 1 ⊕ D 0 = grad H 1 ⊕ D 0 ◦ L 2 = div D = div D ⊕ Lin { 1 } Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

L 2 -decompositions in bounded domains Let Ω ⊂ R 3 be a bounded domain. The following embeddings are compact, if the boundary is suffenciently regular (weakly Lipschitz is enough). ◦ ◦ → L 2 , → L 2 R ∩D ֒ R∩ D ֒ If these embeddings are compact we can skip the bars: ◦ ◦ R 0 D 0 � �� � � �� � ◦ ◦ L 2 = R ⊕H N ⊕ grad H 1 = rot R ⊕ H 1 rot H D ⊕ grad ◦ L 2 = div D = div D ⊕ H 1 0 ◦ ◦ Dirichlet fields H D = R 0 ∩D 0 and Neumann fields H N = R 0 ∩ D 0 refinement of the decomposition � ◦ � � � ◦ ◦ L 2 = rot ⊕ H N ⊕ grad H 1 = rot H 1 R ∩D 0 R∩ ⊕ H D ⊕ grad D 0 � ◦ � � � ◦ L 2 = div D∩ = div D ∩R 0 ⊕ Lin { 1 } R 0 Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

rot-div-problems in bounded domains � ◦ � � � ◦ ◦ ⊕ H N ⊕ grad H 1 = rot L 2 (Ω) 3 H 1 = rot R ∩D 0 R∩ ⊕ H D ⊕ grad D 0 � ◦ � � � ◦ L 2 (Ω) = div D∩ = div D ∩R 0 ⊕ Lin { 1 } R 0 The problems   rot E = rot B = F G       div E = div B = f g and E ∧ ν = 0 B · ν = 0       E ⊥ H D B ⊥ H N ◦ � are uniquely solvable iff F ∈ D 0 , F ⊥ H N , G ∈ D 0 , G ⊥ H D and g dx = 0. Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen