Click here to


Are you sure ?

Yes, do it No, cancel

A Spectral Analysis of a Linear Streamline Upwind Petrov Galerkin Scheme for the Linear Boltzmann Transport Equation with Magnetic Field

A Swan1*, R Yang1 , O Zelyak2 , J St-Aubin3 , (1) Cross Cancer Institute, Edmonton, AB, (2) The Ottawa Hospital Cancer Centre, Ottawa, ON, (3) University of Iowa, Iowa City, IA


(Sunday, 7/14/2019) 4:00 PM - 4:30 PM

Room: Exhibit Hall | Forum 7

Purpose: To derive and analyze a novel linear Streamline Upwind Petrov Galerkin scheme to stabilize the angular advection in the Linear Boltzmann Transport Equation with magnetic field. Analysis will include a spectral radius characterization and measured convergence rates as compared to an angular upwinding scheme.

Methods: A linear SUPG angular discretization was implemented within an existing spatial discretization by modifying the angular weighting functions to incorporate an artificial diffusion component in the advection direction. A spectral analysis on the continuous form of the equation was initially performed to validate the use of an angular linear SUPG method. A further spectral analysis on the discretized space-angle equation using Cartesian spatial voxels and triangular elements conformal to the unit sphere, respectively, was then performed. Finally, the stability of the linear SUPG method was determined via analysis of the spectral radius results, and the convergence rate was observed empirically.

Results: The spectral radius analysis from the continuous form of the equation indicated that the SUPG method eliminated the impact of the magnetic field on the spectral radius. This suggested that the convergence rate should be independent of magnetic field strength, as well as the SUPG weighting parameter. The spectral radius calculations on the discretized equation confirmed these findings. As the spectral radius results were always less than unity, the linear SUPG method was shown to be unconditionally stable. It was also shown that the linear SUPG method required a lower number of iterations to converge compared to the angular upwind method.

Conclusion: A linear SUPG method is a promising angular stabilization scheme for the LBTE with magnetic fields with benefits including the potential for parallelization, and an increased convergence rate over existing angular sweeping methods.

Funding Support, Disclosures, and Conflict of Interest: Alberta Cancer Foundation Graduate Studentship - Amanda Swan


Image-guided Therapy, Finite Element Analysis, Dose


IM/TH- MRI in Radiation Therapy: MRI/Linear accelerator combined dose computation

Contact Email