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Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
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From h to p efficiently: optimal implementation strategies for explicit time-dependent problems using the spectral/hp

A Bolis1, C D Cantwell1, R M Kirby2

  • 1Department of Aeronautics, Imperial College London South Kensington Campus, London, UK.

International Journal for Numerical Methods in Fluids
|April 21, 2015
PubMed
Summary

This study compares numerical methods for solving 2D advection problems. Higher-order spectral/hp element methods with appropriate time-stepping schemes significantly reduce computation time while maintaining accuracy.

Keywords:
discontinuous Galerkinexplicit time-integration methodshyperbolic problemsspectral/hp element method

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Area of Science:

  • Computational Fluid Dynamics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Accurate and efficient numerical methods are crucial for solving complex fluid dynamics problems.
  • Time-stepping schemes and spatial discretization significantly impact computational performance and solution accuracy.

Purpose of the Study:

  • To compare the performance of Adams-Bashforth and Runge-Kutta time-stepping schemes with spectral/hp element discretization.
  • To investigate the influence of mesh size, polynomial order, and time integration length on numerical solution accuracy and computational cost.

Main Methods:

  • Implemented second-order Adams-Bashforth and second/fourth-order Runge-Kutta schemes.
  • Utilized spectral/hp element technique for discretizing a 2D linear advection problem.
  • Conducted numerical experiments on uniform and non-uniform meshes with varying polynomial orders and time integration durations.

Main Results:

  • The Courant-Friedrichs-Lewy (CFL) condition dictates the maximum stable time step, influenced by both discretization and time-stepping scheme.
  • Higher polynomial orders in spectral/hp elements substantially reduce CPU time without compromising accuracy, particularly on uniform meshes.
  • Systematic analysis highlights the interplay between spatial resolution, time integration choice, and computational performance.

Conclusions:

  • Higher-order spectral/hp elements offer significant advantages in reducing computational time for achieving desired accuracy.
  • Guidelines are provided for selecting optimal combinations of spatial discretization and time-stepping schemes for efficient simulations.
  • The choice of numerical methods critically affects the balance between accuracy and computational efficiency in fluid dynamics simulations.