September  2010, 3(3): 373-394. doi: 10.3934/krm.2010.3.373

Galerkin methods for primary ion transport in inhomogeneous media

1. 

Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, SE-412 96, Göteborg, Sweden

2. 

Medical Radiation Physics, Department of Oncology-Pathology, Karolinska Institute, Stockholm, Sweden

3. 

Department of Mathematics, University of Karlsruhe (TH), 76128 Karlsruhe, Germany

Received  May 2009 Revised  February 2010 Published  July 2010

This paper concerns the energy deposition of high-energy (e.g., $\approx 50-500$ MeV) proton and carbon ions and high-energy electrons (of $\approx 50$ MeV), in inhomogeneous media. Our goal is to develop a flexible model incorporated with the analytic theory for ions based on bipartition and Fokker-Planck developments. Both procedures are leading to convection dominated convection diffusion equations. We study convergence for semi-discrete and fully discrete approximations of a such obtained equation, for a broad beam model, using the standard Galerkin and streamline diffusion finite element methods. The analytic broad beam model of the light ion absorbed dose were compared with the results of the modified Monte Carlo (MC) code SHIELD-HIT+ and those of Galerkin streamline diffusion approach.
Citation: Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic and Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373
[1]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447

[2]

Mathieu Lutz. Application of Lie transform techniques for simulation of a charged particle beam. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 185-221. doi: 10.3934/dcdss.2015.8.185

[3]

Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu. Study of adaptive symplectic methods for simulating charged particle dynamics. Journal of Computational Dynamics, 2019, 6 (2) : 429-448. doi: 10.3934/jcd.2019022

[4]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic and Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251

[5]

Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339

[6]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

[7]

Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315

[8]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[9]

Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159

[10]

Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021

[11]

Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114

[12]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic and Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037

[13]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic and Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395

[14]

Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Problems and Imaging, 2007, 1 (2) : 289-298. doi: 10.3934/ipi.2007.1.289

[15]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034

[16]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126

[17]

Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034

[18]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843

[19]

Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065

[20]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]