plasmas Lise-Marie Imbert-Gérard, Bruno Després. July 27th 2011 Laboratoire J.-L. LIONS, Université Pierre et Marie Curie, Paris.

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1 Lise-Marie Imbert-Gérard, Bruno Després. Laboratoire J.-L. LIONS, Université Pierre et Marie Curie, Paris. July 27th 2011

2 1 Physical and mathematical motivations 2 approximation of the solutions 3 4

3 Plan 1 Physical and mathematical motivations 2 approximation of the solutions 3 4

4 Waves in - Nuclear fusion : ITER project Plasma Physics and Fusion Energy, Jeffrey Friedberg B. Despre s. Reflectometry : a radar technique for plasma density measurements using the reflection of electromagnetic waves by a plasma cutoff

5 Choice of a model to describe the plasma 1 ω2 p ω cωp i 2 ω 2 ωc 2 ω ε(x) = P(x) cωp i 2 ω(ω 2 ωc 2) 1 ω2 p ω(ω 2 ω 2 c ) 0 ω 2 ω 2 c 0 P(x)T ω2 p ω 2 ωp 2 = e2 n e(x) ε 0 m e, ωc 2 = eb 0 m e P(x) unitary, slow variations, n e (x) : plasma turbulences plasma cutoff where eigenvalues vanish Maxwell s harmonic equations : t = iω curl E i ω c H = 0, curl H + i ω c εe = 0, E ν + b (H ν) ν = Q (E ν + b (H ν) ν) + g.

6 Case P(x) = I i.e. B 0 //e z, orthogonal propagation modes Maxwell s equations can be split into a system of two equations and one equation, standing for different propagation modes in 2 dimensions. X mode : E B 2 y E x + 2 xye y ω2 c 2 (ε 11E x + ε 12 E y ) = 0, 2 x E y + 2 xye x ω2 c 2 (ε 21E x + ε 22 E y ) = 0. O mode : E//B E z ω 2 ε z E z = 0.

7 Focus on a typical O mode equation E z ω 2 ε z (x, y)e z = 0 Model equation in two dimensions ( ) β t 2 E z E z = 0. u + βu = f This is a wave equation with variable coefficient. sign(β) = ±1 B.C. ( ν + iγ) u = Q ( ν + iγ) u + g, γ > 0. Example : β(x, y) = x, f = 0, then the solutions to be approximated are the Airy functions. Plane waves numerical this equation

8 Plan 1 Physical and mathematical motivations 2 approximation of the solutions 3 4

9 Notations Special type of high order DG method Σ jj Ω j Σ kj Ω k e iω(cosθx+sinθy) - T. Huttunen, M. Malinen ja P. Monk, Solving Maxwell s equations using the UWVF. JCP (2007) - A. Moiola, R. Hiptmair and I. Perugia, by plane waves, (2009) under interest : ( ν + iγ) u on all interfaces

10 of the discrete problem, β(x) = x Test functions are solutions of the homogeneous adjoint equation : ( + β ) ϕ = 0. solutions are Airy functions, which are transcendental (3a) 1/3 πai((3a) 1/3 x) = 0 cos(at 3 + xt)dt, obtained by series calculus : if y = 0 a kx k, then k 2, (k + 2)(k + 1)a k+2 = a k 1, and a 0 = 0. construction of new basis functions with a non linear process, much more easy to implement We propose to consider generalized plane waves : exponentials of polynomials ( e P) = ( P + P 2) e P

11 Basis functions design in 1D Locally on Ω k we look for two basis functions ui k ( ) + βi k ui k = 0, satisfying for some β i close to the coefficient β in the following sense β k i β L (Ω k ) h q. If x k denotes the center of Ω k then we propose to define the basis function as u k,i = e P k,i such that β k i = P k,i + (P k,i )2, = β + O((x x k ) q ). Chose P k,i s coefficients to fit with β s Taylor expansion in the vicinity of x k.

12 Plan 1 Physical and mathematical motivations 2 approximation of the solutions 3 4

13 in 1D Proposition If max i {1,2},k [[1,N h ]] βk i β L (Ω k ) = O(h q ) is small enough, then the corresponding discrete problem has a unique solution. Then the problem is close to a non conforming FE problem. Solution Adapt second Strang lemma. Convergence result x x h q = O(h q 3/2 ), where q is a norm adapted to the discrete problem.

14 Plan 1 Physical and mathematical motivations 2 approximation of the solutions 3 4

15 Two examples with β(x) = x { u + xu = 0 ( ν + iγ) u = g β i = β + O(h) β i h constant on each element of size h PW basis functions β i = β + O(h 3 ) on an element centered on x k, denote y = x x k P 1 = ( 1/2x k y 2 + 1/6y 3) corresponds to β 1 = x k + y + x 2 k y 2 + x k y 3 + 1/4y 4, and P 2 = y + (x k 1)/2 y 2 + (3 2x k )/6 y 3 to β 2 = x k + y + (x k 1) 2 y 2 + (x k 1)(3 2x k )y 3 + (3 2x k ) 2 /4y 4

16 Fig.: Example for y + xy = 0, with ( ν + i) y = g, q = 2. Convergence order : 1.86.

17 Examples of numerical order of h-convergence q denotes the order of approximation of the coefficient : β i β C q h q q ( ν + i)u = g ( ν + i x )u = g Fig.: order of convergence of the error l 2 depending on the number of subdivision of the domain, on the interval [ 1, 1]. The error considered here is i u th(i) u num (i) 2 i u th(i) 2

18 h-convergence of the method Convergence of the y + xy = 0, with different boundary conditions, on the interval [ 1, 1]. ( ν + i) y = g ( ν + i x )u = g

19 Some ideas concerning q-convergence Even for small N h, q-convergence is observed. Plausible explanation : even for large h values the basis functions are uniformly adapted to the exact solution on a subdomain [x 0 α, x 0 + α]. of Airy function Ai : Projection on the space spanned by the two basis functions centered on x 0 = 0 and on x 0 = 4, for several values of q. We expect to use large values of h as has be done in the case of piecewise constant coefficients.

20 First test in dimension 2 { ( + x)u(x, y) = 0 ( ν + i) u(x, y) = g structured triangular and square mesh Simpson formula for integral computations Basis functions : generalized plane waves e P(x,y) we perform a 1-d reduction P(x, y) = iy sin θ + P(x) for different values of θ, following the case of plane waves ( 2 x + (x + sin 2 θ) ) P 0 reconstruction of the approximated solution u h on an element Ω k with 2p basis functions per element (u num ) Ωk = 2p l=1 (X num ) l k ϕl k

21 First test in dimension 2 with triangular mesh. error considered k u th(x k, y k ) u num (x k, y k ) 2 k u th(x k, y k ) 2. β i β C q h q 2p basis functions per element high order quadrature formulas q p = 1, simpson p = 3, boole Fig.: Order of the error

22 Summary & Perspectives A new generalized plane wave method adapted to problems with variable coefficients dimension 1 and 2 high order achieved numericallyin 1D and 2D Theoretical study of the method done in 1D Perspectives more test problems in dimension 1 and 2 develop accurate quadrature formulas for truncated exponential series adapt to the X-mode equation for applications