Perfectly Matched Layers for Linear Transport

Perfectly Matched Layers for Linear Transport

Herbert Egger (TU Darmstadt), Matthias Schlottbom

Department of Applied Mathematics

Introduction

Linear Boltzmann equations are hyperbolic integro-partial di↵erential equations de-scribing the dynamics of a single-particle probability distribution in phase space. The dynamics is governed by streaming, damping and scattering. The two main challenges in the numerical approximation of solutions to linear Boltzmann equations are (i) the high dimensionality of the phase space, and (ii) the anisotropic structure of the so-lution. Item (i) refers to the seven space-velocity-time dimensions, making standard simulation methods extremely costly. Item (ii) stems from the interplay of transport and multiple scattering, such that it is impossible to find solutions analytically.

The linear Boltzmann equation is equivalent to a mixed variational problem that in-corporates boundary conditions naturally [1, 2]. However, the natural inclusion of boundary conditions introduces a non-smooth coupling of spatial and velocity vari-ables, which is inconvenient for practical implementations. To overcome this difficulty, we introduce an absorbing layer and consider a perturbed problem such that the re-sulting discretizations can be implemented easily and stored efficiently.

Figure 1: quoting: B. Seibold, M. Frank, StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer, ACM Trans. Math. Software, Vol. 41, No. 1, 2014.

Linear Transport Equation

I (r , s) probability of finding a photon at point r with direction s

I _{R ⇢ R}3 _{bounded convex domain}

I supp(q) _{[ supp( s}) _{⇢ R} I (K )(r , s) = R_S k(r , s0 _{· s) (r, s}0) ds0 I C = ( a + s) sK n s R @_R s _{· r + C = q in R ⇥ S} = 0 on = _{{(r, s) 2 @R ⇥ S : s · n(r) < 0}}

Theorem: Let q 2 Lp(_{R ⇥ S). The transport problems has a unique solution} 2 Lp(_{R ⇥ S) with s · r 2 L}p(_{R ⇥ S), and | + 2 L}p( ₊; _{|s · n|).}

Perfectly Matched Layers

s ⌘ 0 q ⌘ 0 a = 1/✏ X L _R n(r ) _s X R r s _{· r ✏} + ( _a + _s) _✏ = _sK _✏ + q in X _{⇥ S} ✏(r , s) = ⇢(r , s) ✏(r , s) on L

Theorem: The modified problem has a unique solution _{✏ 2 L}p(X _{⇥ S) with} s _{· r ✏ 2 L}p(X _{⇥ S) and ✏|} L

+ 2 L

p₍ L

+; |s · n|).

Lemma: For ⇢ = 0 we have _✏|R⇥S = _|R⇥S and Z L + | ✏|2|s · n| d  e 2L/✏ Z + | |2|s · n| d .

9↵ = ↵(X , R) > 0 such that (r, s) = 0 for (r, s) 2 L+ with |s · n|  ↵.

References

[1] Herbert Egger and Matthias Schlottbom.

A mixed variational framework for the radiative transfer equation.

Math. Mod. Meth. Appl. Sci., 22:1150014, 2012.

[2] Herbert Egger and Matthias Schlottbom.

A class of Galerkin Schemes for Time-Dependent Radiative Transfer.

SIAM J. Numer. Anal., 54(6):3577–3599, 2016.

[3] Herbert Egger and Matthias Schlottbom.

An Lp theory for stationary radiative transfer.

Appl. Anal., 93(6):1283–1296, 2014.

Mixed variational framework

Even-odd parities

±_{(s) =} 1

2( (s) ± ( s))

Transport problem with reflection b.c. is equivalent to the system s _{· r ✏} + _C✏ +_✏ = q+ in X _{⇥ S}

s _{· r} +_✏ + _{C✏ ✏} = q in X _{⇥ S} (1 ⇢) +_✏ + (1 + ⇢) _✏ = 0 on L

Find _✏ such that for all sufficiently smooth 2_h1 ⇢ 1 + ⇢ + ✏ , +|s · n|i L + (s · r +✏ , ) ( ✏ , s · r +) + (C✏ ✏, ) = (q, ) Observations I odd part _{✏ 2 L}2(X _{⇥ S) =: V}

I even part +_{✏ 2 W has more regularity: s · r} +_{✏ 2 L}2, regular trace

I boundary conditions are incorporated naturally ⇢(r , s) = |s · n| 1 |s · n| + 1 =) 2h 1 ⇢ 1 + ⇢ +_, + |s · n|i L = h +, +i_@X⇥S

Theorem: Let k k2_C✏ = (_C✏ , ). The mixed variational problem has a unique solution _{✏ 2 W}+ V , and the error e_✏ = _✏ satisfies the estimate

ks · re✏+kC_✏ 1 + ke✏kC✏ + ke✏| L₊kL2( L₊)  C k | ₊L kL2( L₊).

Galerkin approximations

I _W+

h ⇢ W+ and Vh ⇢ V finite dimensional spaces

I s _{· rW}+_{h ⇢ Vh}

Find _{✏,h 2 W}+_h V_h such that for all _{h 2 W}+_h V_h

h +✏,h, h+i L + (s · r +✏,h, h ) ( ✏,h, s · r h+) + (C✏ ✏,h, h) = (q, h)

I Galerkin problem is well-posed

I Quasi-best approximation error estimate in energy norm for _{✏ ✏,h}

I Error estimate for _✏,h via triangle inequality

Tensor products and P_N approximation

I _S+

N = span{Y2lm : 0  l  N, l  m  l} spherical harmonics

I _X+

h = span{'j : j 2 J+} = P1(Th) \ H1(X )

I Approximation space W+_h = S+_{N ⌦ X}+_h , i.e.,

+ ✏,h(r , s) = N X 2l =0 l X m= l X j_2J+ a_{2l ,m}j '_j(r )Y_2lm(s) _{2 S}+_{N ⌦ X}+_h I dimX+_{h ⇠ h} d, dimS+_{N ⇠ N}d 1 h +✏,h, h+i L = X l ,m,i X k,n,j ✓ a_{2l ,m}i b_2k,nj Z @X '_i'_j Z S Y_2lmY n_2k ◆ = aTMb

I Storage complexity for M = O(h d).

Checkerboard example

Figure 2: # vertices: 47,089. Angular dof: 2,704. total dof: 191,023,950.