Convergence of Filtered Spherical Harmonic Equations for Radiation - PowerPoint PPT Presentation

Convergence of Filtered Spherical Harmonic Equations for Radiation Transport Martin Frank (RWTH) Cory Hauck (ORNL) Kerstin K¨ upper (RWTH) MMKTII, Fields Institute, October 2014 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 1/34

Outline & References • Filtered P N equations 1 • Convergence analysis • Modified equation 2 • Galerkin estimate 3 • Convergence estimates • Numerical experiments using StaRMAP 4 1 McClarren, Hauck, JCP 2010 2 Radice et al., JCP 2013 3 Schmeiser, Zwirchmayr, SINUM 1999 4 Seibold, Frank, TOMS 2014 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 2/34

Checkerboard: P 5 versus FP 5 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 3/34

Line Source: P 9 versus FP 9 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 4/34

Challenges Challenges in radiation transport: • Highly heterogeneous media • Media/initial conditions/sources lead to non-smooth solutions • Preserve realizability, rotational invariance • Capture beams Challenges for spectral methods: • Spectral methods achieve fast convergence for smooth solutions • But suffer from the Gibbs phenomenon • Idea of filtering: dampen the coefficients in the expansion • Con: Some adjustments of the filter strength may be required for different problems • Pro: Speed, overall accuracy, and simplicity Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 5/34

FILTERED P N Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 6/34

Radiation Transport ∂ t ψ ( t , x , Ω)+Ω ·∇ x ψ ( t , x , Ω)+ σ a ( x ) ψ ( t , x , Ω) − ( Q ψ )( t , x , Ω) = S ( t , x , Ω) • ψ ( t , x , Ω): density of particles, with respect to the measure d Ω dx , which at time t ∈ R are located at position x ∈ R 3 and move in the direction Ω ∈ S 2 . • Scattering operator �� � S 2 g ( x , Ω · Ω ′ ) ψ ( t , x , Ω ′ ) d Ω ′ − ψ ( t , x , Ω) ( Q ψ )( t , x , Ω) = σ s ( x ) T ψ = S Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 7/34

Sphercial Harmononic P N equations Notation: • Real-valued spherical harmonic m k ℓ , ℓ = 0 , 1 , . . . , k = − ℓ, . . . , ℓ � • Angular integration �·� = S 2 ( · ) d Ω Spectral Galerkin method: • Expand unknown ψ ≈ ψ PN ≡ m T u PN • Plug into equation and project residual � m T ( m T u PN ) � = � m S � =: s . • Other combinations of ansatz and projection can be used! P N equations ∂ t u PN + A · ∇ x u PN + σ a u PN − σ s Gu PN = s , where A := � mm T Ω � and G is diagonal Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 8/34

Filtering • Filtering well-known in spectral methods • A filter of order α is a function f ∈ C α ( R + ), which fulfills f (0) = 1 , f ( k ) (0) = 0 , for k = 1 , . . . , α − 1 , and f ( α ) (0) � = 0 • Additional condition f ( η ) ≥ C (1 − η ) k , η ∈ [ η 0 , 1] • Filtering the expansion after every time step ℓ N � � �� β ∆ t � � ℓ u k ℓ m k f ℓ . N +1 ℓ =0 k = − ℓ Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 9/34

NUMERICAL ANALYSIS Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 10/34

Main Result Galerkin estimate � ψ ( t , · , · ) − ψ FPN ( t , · , · ) � L 2 ( R 3 ; L 2 ( S 2 )) ≤ � ψ ( t , · , · ) − P N ψ ( t , · , · ) � L 2 ( R 3 ; L 2 ( S 2 )) � + t � a N +1 · ∇ x � m N +1 ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) � + β � G f � m ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) Rates � ψ ( t , · , · ) − P ψ ( t , · , · ) � L 2 ( R 3 ; L 2 ( S 2 )) ≤ CN − q � ψ � C ([0 , T ]; L 2 ( R 3 ; H q ( S 2 ))) � a N +1 · ∇ x � m N +1 ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) ≤ CN − r �∇ x ψ � C ([0 , T ]; L 2 ( R 3 ; H r ( S 2 ))) � α > q − 1 CN − q +1 / 2 , 2 � G f � m ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) ≤ α ≤ q − 1 CN − α + ε , 2 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 11/34

Sobolev Spaces • H q ( S 2 ) Sobolev space on the unit sphere with norm   1 / 2 �  � S 2 | D α Φ(Ω) | 2 d Ω � Φ � H q ( S 2 ) :=  | α |≤ q • Spherical harmonics are eigenfunctions of Laplace-Beltrami operator L m k ℓ = − λ ℓ m k ℓ , λ ℓ = ℓ ( ℓ + 1) • Expansion coefficients Φ k ℓ := � m k ℓ Φ � of any function Φ ∈ H 2 q ( S 2 ) satisfy 1 1 Φ k ℓ = � m k ( − λ ℓ ) q � ( L q m k ( − λ ℓ ) q � m k ℓ L q Φ � ℓ Φ � = ℓ )Φ � = Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 12/34

Spectral Convergence • L 2 -orthogonal projection of a generic function Φ ∈ L 2 ( S 2 ) onto P N P N Φ = m T � mm T � − 1 � m Φ � = m T � m Φ � • Projection onto polynomials of exact degree ℓ ( P ℓ − P ℓ − 1 )Φ = m T ℓ � m ℓ m T ℓ � − 1 � m ℓ Φ � = m T ℓ � m ℓ Φ � • Spectral convergence �� m ℓ Φ �� 2 R n ℓ = � ( P ℓ − P ℓ − 1 )Φ � 2 L 2 ( S 2 ) ≤ � ( I − P ℓ )Φ � 2 L 2 ( S 2 ) ∞ ∞ 1 � � | φ ℓ | 2 = ( − λ ℓ ) 2 q |� m ℓ L q Φ �| 2 = k = ℓ +1 k = ℓ +1 1 ( ℓ ( ℓ + 1)) 2 q � φ � 2 ≤ H 2 q ( S 2 ) Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 13/34

Step 1: Modified Equation • Time step u n +1 , ∗ FPN = u n FPN − ∆ t ( A · ∇ x u n FPN + σ a u n FPN − σ s Gu n FPN − s n ) • Filtering FPN + ∆ t exp( β log( f )∆ t ) − 1 FPN = f β ∆ t u n +1 , ∗ FPN = u n +1 , ∗ u n +1 , ∗ u n +1 FPN ∆ t • Operator split discretization of Modified equation ∂ t u FPN + A · ∇ x u FPN + σ a u FPN − σ s Gu FPN − β G f u FPN = s , � �� � ℓ where G f is diagonal with entries log , ℓ = 0 , . . . , N . f N +1 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 14/34

Step 2: Galerkin Estimate • Residual ψ − ψ FPN = ( ψ − P N ψ ) + P N ψ − ψ FPN = ( ψ − P N ψ ) + m T r • Multiply by m T r and integrate in angle and space � � 1 R 3 | r | 2 dx = − R 3 r T 2 ∂ t N a N +1 · ∇ x � m N +1 ψ � dx � � R 3 r T G f � m ψ � dx − R 3 r T Mr dx . − σ f • M := σ a I − σ s G − σ f G f is positive definite • This yields ∂ t � r � L 2 ( R 3 ; R n ) ≤� a N +1 · ∇ x � m N +1 ψ �� L 2 ( R 3 ; R 2 N +1 ) + σ f � G f � m ψ �� L 2 ( R 3 ; R n ) • Control error by projection error + residual r Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 15/34

Step 3: Convergence Estimate • Estimate filter term � G f � m ψ ( t , · , · ) �� 2 L 2 ( R 3 ; R n ) N log 2 � �� � � ℓ �� m ℓ ψ ( t , · , · ) �� 2 = f L 2 ( R 3 ; R n ℓ ) N +1 ℓ =0 N log 2 � �� � � � ( P ℓ − P ℓ − 1 ) ψ ( t , · , · ) � 2 ℓ = f L 2 ( R 3 ; L 2 ( S 2 )) N +1 ℓ =1 N log 2 � �� � � ℓ � ( I − P ℓ − 1 ) ψ ( t , · , · ) � 2 = C f L 2 ( R 3 ; L 2 ( S 2 )) N +1 ℓ =1 �� 1 N log 2 � � � ℓ ℓ 2 q � ψ ( t , · , · ) � 2 ≤ C f L 2 ( R 3 ; H q ( S 2 )) N +1 ℓ =1 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 16/34

Step 3: Convergence Estimate • For θ ≤ 2 q �� 1 N log 2 � � � ℓ f N +1 ℓ 2 q ℓ =1 N 1 1 log 2 � � �� � N +1 � � θ ℓ ≤ f ( N + 1) θ − 1 N +1 ℓ N + 1 ℓ =1 � �� � =:Σ • Interpret as Riemann sum � 1 log 2 ( f ( η )) η − θ d η Σ ∼ 0 • Around η = 0, log f ( η ) ≤ C η α • Σ Integrable for θ < 2 α + 1 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 17/34

Step 3: Convergence Estimate Two cases: Case 1: α > q − 1 2 . Choose θ = 2 q , convergence limited by the regularity of ψ � G f � m ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) ≤ CN − q +1 / 2 Case 2: α ≤ q − 1 2 . Choose θ = 2 α + 1 − δ , where δ > 0 is arbitrary, convergence limited by the filter order � G f � m ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) ≤ CN − α + ε , where ε = δ/ 2 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 18/34

Main Result Galerkin estimate � ψ ( t , · , · ) − ψ FPN ( t , · , · ) � L 2 ( R 3 ; L 2 ( S 2 )) ≤ � ψ ( t , · , · ) − P ψ ( t , · , · ) � L 2 ( R 3 ; L 2 ( S 2 )) � + t � a N +1 · ∇ x � m N +1 ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) � + β � G f � m ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) , Rates � ψ ( t , · , · ) − P ψ ( t , · , · ) � L 2 ( R 3 ; L 2 ( S 2 )) ≤ CN − q � ψ � C ([0 , T ]; L 2 ( R 3 ; H q ( S 2 ))) � a N +1 · ∇ x � m N +1 ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) ≤ CN − r �∇ x ψ � C ([0 , T ]; L 2 ( R 3 ; H r ( S 2 ))) � α > q − 1 CN − q +1 / 2 , 2 � G f � m ψ �� C ([0 , T ]; L 2 ( R 3 ; R n )) ≤ α ≤ q − 1 CN − α + ε , 2 Convergence of Filtered Spherical Harmonic Equations for Radiation Transport 19/34