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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Imagine a bucket of water. It contains many molecules, of the order of 1026 molecules. Thus, although it contains discrete elements (molecules) at the microscopic level, macroscopically, it can be considered continuous. Small volume elements of water, infinitesimal compared to the bulk of the bucket's volume, still contain many molecules. Under this framework, quantized matter is approximated as continuous for practical purposes.
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Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann

Paul Houston1, Matthew E Hubbard1, Thomas J Radley1,2

  • 1School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK.

Journal of Scientific Computing
|July 5, 2024
PubMed
Summary
This summary is machine-generated.

We developed a new hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. This accurate, parallelizable method works with existing software and handles complex geometries.

Keywords:
Discontinuous Galerkin methodsDiscrete ordinates methodsLinear Boltzmann transport problemPolytopic mesheshp-finite element methods

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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.