24th International Conference on Transport Theory, Taormina, ITALY, 7-11 September 2015

An Accurate Flux-limited Diffusive PN Closure: the PN Quasi-Transient (PN QT) Model for Radiation Transport Weixiong Zheng and Ryan G. McClarren [email protected], [email protected] Texas A&M University, College Station, TX

1. Introduction In simulating time dependent particle transport problems the truncated spherical harmonics method (PN ), with a zero closure, has the issue of oscillations in the solution known as wave effects. These effects arise from the particles moving at finite number of discrete speeds[1,2]. As a result, in some problems, e.g. the plane source problem, delta functions of uncollided particles create non-physical waves[3]. There have been efforts on modifying the PN closure were annuounced to be effective in dampening and smoothening the delta functions based on new closures [3–6]. Yet, in short-time transients none of these closures capture the flux profile accurately. In this work, we derive a new closure based on indication from minimizing a functional measuring residual of transport equation. The new closure basically takes a similar form of flux limiter in Larsen-type flux limited diffusion (n = 1) plus a temporal limiter on flux. Numerical tests with the plane source problem indicate generally the delta functions are effectively smoothened with the main-mode magnitudes preserved well. Due to the accurate estimation of flux profile in short time transient, we name the model PN QT model. 2. Closure Principle and Results Define a cost function J measuring the residual of the transport equation with isotropic scattering and source as a functional of moments in PN approximation: ′

Z1

dµ R2 ,

(1a)

X N 1 ∂ 2l′ + 1 1 ∂t + µ + σt φl′ Pl′ (µ) − σs φ0 − q(µ), v ∂z 2 2 ′

(1b)

J ({φl′ : l = 0, · · · }) =

−1

R ≡ Lφ¯N (µ) − q(µ) =

l =0

where the functional is defined in moment space. Before continuing the analysis, we generalize the PN equations as N + 1 unclosed equations with a closure equation: l ∂φl−1 l + 1 ∂φl+1 1 ∂φl + (1 − δ0,l ) + (σt − σs δ0,l )φl + = qext δ0,l , l = 0 ≤ N v ∂t 2l + 1 ∂z 2l + 1 ∂z φM = F({φl′ : l′ = 0, · · · }),

M ≥ N + 1,

(2a) (2b)

where the δ is the Kronecker delta. In the conventional PN approximation, a zero closure, or truncation, is used, i.e. φM = 0, M ≥ N + 1. To minimize the residual, one would look for moments such that ∂J /∂φl = 0. On the other hand, the form of ∂J /∂φl could also be used to predict if, or where, an approximation could minimize the residual in L-2 norm sense. To examine the behavior of the zero closure we introduce Eq. (2) into ∂J /∂φl to get ∂ ∂J ∂ N + 1 ∂φN ∂J φN , (3) = 0, l < N and = (N + 1) ∂φl ∂φN ∂φN ∂z 2N + 3 ∂z

W. Zheng R. McClarren

which indicates that if the residual of PN approximation could be minimized depends on the smoothness of the solution itself. Furthermore there is no boundedness enforced on ∂J /∂φl for l = N because of the presence of the derivative term.

We can do the same analysis on Levermore’s diffusive closure [6], which sets φM = −

N + 1 ∂φN δM,N +1 . (2N + 3)σt ∂z

The resulting derivative is ∂J ∂ = (N + 1) ∂φl ∂φN

∂ φN ∂z

1 ∂ N +2 1 ∂ φN − δl,N . 2N + 3 v ∂t σt ∂z

(4)

The form of this minimizer could explain previous observations that this closure results in high accuracy after short time transients decay away. Yet, in short time transient, since ∂t (·) 6= 0, there is also no boundedness imposed on the functional derivative ∂J /∂φl . From another point of view, if we leave the PN system unclosed, i.e., one does not impose anything on F in Eq. (2b), ∂J /∂φl we could search for the closure that makes the functional derivative zero: ∂ ∂ N + 1 ∂φN N + 2 ∂φN +2 1 ∂J =(N + 1) φN ∂t φN +1 + + σt φN +1 + , (5) ∂φN ∂φN ∂z v 2N + 3 ∂z 2N + 3 ∂z which indicates a “closure” with the following form would lead to zero value of ∂J /∂φl : φN +1 = −

∂t φN +1 σt + vφN +1

N +1 ∂ 1 φ . (N + 2)∂z φN +2 2N + 3 ∂z N + (2N + 3)φN +1

(6)

The formulation shares a similar form with diffusive closure except solution based terms are added to correct dissipation in the closure. Of course this is only a formal closure at this point because it depends on the values of φN +1 and φN +2 . Nevertheless, the form implies adding spatial and temporal flux limiters to diffusive closure could help minimize ∂J /∂φl . Therefore, we propose the following Larsen-flux-limiter like closure: N +1 ∂ 1 (7) φN +1 = − ∂t φ0 α∂z φ0 2N + 3 ∂z φN + σt + vφ0 φ0 A desirable property is that it could be realizable that at any time t, ∃β(t) > 0, such that |φN +1 (t)|

An Accurate Flux-limited Diffusive PN Closure: the PN Quasi-Transient (PN QT) Model for Radiation Transport Weixiong Zheng and Ryan G. McClarren [email protected], [email protected] Texas A&M University, College Station, TX

1. Introduction In simulating time dependent particle transport problems the truncated spherical harmonics method (PN ), with a zero closure, has the issue of oscillations in the solution known as wave effects. These effects arise from the particles moving at finite number of discrete speeds[1,2]. As a result, in some problems, e.g. the plane source problem, delta functions of uncollided particles create non-physical waves[3]. There have been efforts on modifying the PN closure were annuounced to be effective in dampening and smoothening the delta functions based on new closures [3–6]. Yet, in short-time transients none of these closures capture the flux profile accurately. In this work, we derive a new closure based on indication from minimizing a functional measuring residual of transport equation. The new closure basically takes a similar form of flux limiter in Larsen-type flux limited diffusion (n = 1) plus a temporal limiter on flux. Numerical tests with the plane source problem indicate generally the delta functions are effectively smoothened with the main-mode magnitudes preserved well. Due to the accurate estimation of flux profile in short time transient, we name the model PN QT model. 2. Closure Principle and Results Define a cost function J measuring the residual of the transport equation with isotropic scattering and source as a functional of moments in PN approximation: ′

Z1

dµ R2 ,

(1a)

X N 1 ∂ 2l′ + 1 1 ∂t + µ + σt φl′ Pl′ (µ) − σs φ0 − q(µ), v ∂z 2 2 ′

(1b)

J ({φl′ : l = 0, · · · }) =

−1

R ≡ Lφ¯N (µ) − q(µ) =

l =0

where the functional is defined in moment space. Before continuing the analysis, we generalize the PN equations as N + 1 unclosed equations with a closure equation: l ∂φl−1 l + 1 ∂φl+1 1 ∂φl + (1 − δ0,l ) + (σt − σs δ0,l )φl + = qext δ0,l , l = 0 ≤ N v ∂t 2l + 1 ∂z 2l + 1 ∂z φM = F({φl′ : l′ = 0, · · · }),

M ≥ N + 1,

(2a) (2b)

where the δ is the Kronecker delta. In the conventional PN approximation, a zero closure, or truncation, is used, i.e. φM = 0, M ≥ N + 1. To minimize the residual, one would look for moments such that ∂J /∂φl = 0. On the other hand, the form of ∂J /∂φl could also be used to predict if, or where, an approximation could minimize the residual in L-2 norm sense. To examine the behavior of the zero closure we introduce Eq. (2) into ∂J /∂φl to get ∂ ∂J ∂ N + 1 ∂φN ∂J φN , (3) = 0, l < N and = (N + 1) ∂φl ∂φN ∂φN ∂z 2N + 3 ∂z

W. Zheng R. McClarren

which indicates that if the residual of PN approximation could be minimized depends on the smoothness of the solution itself. Furthermore there is no boundedness enforced on ∂J /∂φl for l = N because of the presence of the derivative term.

We can do the same analysis on Levermore’s diffusive closure [6], which sets φM = −

N + 1 ∂φN δM,N +1 . (2N + 3)σt ∂z

The resulting derivative is ∂J ∂ = (N + 1) ∂φl ∂φN

∂ φN ∂z

1 ∂ N +2 1 ∂ φN − δl,N . 2N + 3 v ∂t σt ∂z

(4)

The form of this minimizer could explain previous observations that this closure results in high accuracy after short time transients decay away. Yet, in short time transient, since ∂t (·) 6= 0, there is also no boundedness imposed on the functional derivative ∂J /∂φl . From another point of view, if we leave the PN system unclosed, i.e., one does not impose anything on F in Eq. (2b), ∂J /∂φl we could search for the closure that makes the functional derivative zero: ∂ ∂ N + 1 ∂φN N + 2 ∂φN +2 1 ∂J =(N + 1) φN ∂t φN +1 + + σt φN +1 + , (5) ∂φN ∂φN ∂z v 2N + 3 ∂z 2N + 3 ∂z which indicates a “closure” with the following form would lead to zero value of ∂J /∂φl : φN +1 = −

∂t φN +1 σt + vφN +1

N +1 ∂ 1 φ . (N + 2)∂z φN +2 2N + 3 ∂z N + (2N + 3)φN +1

(6)

The formulation shares a similar form with diffusive closure except solution based terms are added to correct dissipation in the closure. Of course this is only a formal closure at this point because it depends on the values of φN +1 and φN +2 . Nevertheless, the form implies adding spatial and temporal flux limiters to diffusive closure could help minimize ∂J /∂φl . Therefore, we propose the following Larsen-flux-limiter like closure: N +1 ∂ 1 (7) φN +1 = − ∂t φ0 α∂z φ0 2N + 3 ∂z φN + σt + vφ0 φ0 A desirable property is that it could be realizable that at any time t, ∃β(t) > 0, such that |φN +1 (t)|