Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann...

Area of Science:

• Computational Physics
• Numerical Analysis
• Transport Theory

Background:

• The linear Boltzmann transport equation describes particle transport in various physical systems.
• Existing numerical methods, like discrete ordinates, have limitations in handling complex geometries and achieving arbitrary-order accuracy.
• Efficient and accurate solutions are crucial for simulating complex physical phenomena.

Purpose of the Study:

• Introduce a novel hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem.
• Demonstrate the method's capability for arbitrary-order convergence and efficient parallel implementation.
• Enable accurate solutions for problems with complex spatial geometries.

Main Methods:

• Developed a unified hp-DGFEM discretizing space, angle, and energy domains.
• Incorporated local mesh refinement and local polynomial degree variation.
• Utilized general polytopic elements for complex geometries.
• Performed stability and hp-version a priori error analysis, including a novel inf-sup bound.

Main Results:

• The hp-DGFEM offers arbitrary-order convergence rates.
• The method is compatible with standard multigroup discrete ordinates implementations for efficient, parallel computation.
• Numerical experiments confirm the method's performance for polyenergetic and monoenergetic problems.
• The approach effectively handles complex spatial geometries.

Conclusions:

• The proposed hp-DGFEM provides an efficient, accurate, and flexible approach for solving the linear Boltzmann transport equation.
• This method allows for high-accuracy solutions in parallel using existing software infrastructures.
• It offers significant advantages for problems involving complex geometries and demanding accuracy requirements.