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A Streamline Upwind Petrov-Galerkin Numerical Angular Stabilization Scheme for the Linear Boltzmann Transport Equation with Magnetic Field

A Swan1,2*, R Yang1,2 , J St-Aubin3 , (1) University of Alberta, Edmonton, AB, (2) Cross Cancer Institute, Edmonton, AB, (3) University of Iowa, Iowa City, IA


(Sunday, 7/29/2018) 3:00 PM - 6:00 PM

Room: Exhibit Hall

Purpose: To investigate a technique for angularly stabilizing a finite element method based deterministic solution to the Linear Boltzmann Transport Equation (LBTE) with magnetic fields. A novel streamline upwind Petrov-Galerkin method, or streamline diffusion, is derived and applied in angle.

Methods: An existing Discontinuous Galerkin Finite Element Method (DGFEM) was modified to include artificial diffusion in the advection direction as a stabilization scheme for angular advection. An upwind stabilization framework was applied in space in order to stabilize the spatial advection term. The multigroup method was used to handle energy discretization, with cartesian voxels used for spatial discretization, and triangular elements conformal to the unit sphere used for angular discretization. Linear basis functions were used in space, with cubic basis functions used in angle. Streamline diffusion in angle was included via a modification of the weighting function, and a new hybrid DGFEM spatial Petrov-Galerkin angular formulation was derived. Additionally, an analysis was performed to determine the effect of the streamline diffusion stabilization on convergence of the source iteration.

Results: The DGFEM system with the inclusion of the streamline diffusion terms was derived and implemented numerically. Simple preliminary simulations in lung indicated that the method did in fact offer additional stability in angle. However, some unexpected numerical challenges were encountered as a result of the angular diffusion term, which were observed at high values of the diffusion coefficient.

Conclusion: Incorporating a streamline diffusion modification into the weighting functions used in a hybrid DGFEM formulation offers numerical stability for low magnetic field strengths in lung phantoms. There is an advantage over existing angular sweeping methods, as the streamline diffusion method allows for simultaneous solution, and thus parallelization.

Funding Support, Disclosures, and Conflict of Interest: This work is partially supported by the Alberta Cancer Foundation and Alberta Innovates Health Solutions


Finite Element Analysis, Numerical Analysis


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