Hybrid discontinuous Galerkin method for the hyperbolic linear Boltzmann transport equation for multiscale problems

Area of Science:

• Computational physics
• Numerical analysis
• Transport phenomena

Background:

• The linear Boltzmann transport equation is crucial for modeling various physical phenomena.
• Existing numerical methods face challenges in efficiently handling multiscale and thick diffusive regimes.

Purpose of the Study:

• To develop a novel numerical method for the first-order hyperbolic linear Boltzmann transport equation.
• To enhance the efficiency and accuracy of simulations in complex physical scenarios.

Main Methods:

• A hybrid discontinuous Galerkin (HDG) method is proposed, utilizing primal variables and numerical traces.
• The method employs projection matrices and constructs the global matrix system from numerical traces to reduce degrees of freedom.
• An upwind sweep sequence is used for efficient solution of the resulting block-lower-triangular matrix system.

Main Results:

• The upwind-HDG method demonstrates flexibility in spatial order selection.
• Asymptotic analysis in the thick diffusion limit shows convergence is linked to the response matrix L.
• Numerical experiments confirm the accuracy and stability of the upwind-HDG method, outperforming even-parity HDG in challenging regimes.

Conclusions:

• The proposed upwind-HDG method is accurate and stable for thick diffusive and multiscale heterogeneous problems.
• The method offers significant computational advantages by reducing degrees of freedom and enabling efficient solution strategies.
• This work provides a robust numerical tool for advanced transport phenomena simulations.