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A unified discontinuous Galerkin framework for time integration.

Shan Zhao1, G W Wei2

  • 1Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA.

Mathematical Methods in the Applied Sciences
|November 11, 2014
PubMed
Summary

A new discontinuous Galerkin approach unifies time integration methods for ordinary differential equations. This framework optimizes schemes for accuracy, sparseness, and stability, enhancing numerical simulations.

Keywords:
discontinuous Galerkinfinite element time discretizationoptimized Runge–Kutta methodstime integration schemes

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Time integration of ordinary differential equations (ODEs) is crucial in various scientific fields.
  • Existing numerical methods often have limitations in accuracy, stability, or applicability to nonlinear problems.
  • A unified framework for deriving and optimizing time-stepping schemes is needed.

Purpose of the Study:

  • To introduce a novel discontinuous Galerkin (DG) approach for time integration of ODEs.
  • To establish a unified framework for deriving various time-stepping schemes, including Runge-Kutta and symplectic methods.
  • To enable optimization of these schemes based on accuracy, sparseness, and stability criteria.

Main Methods:

  • The method of weighted residuals and numerical quadratures are employed for finite element time discretization.
  • Variational analysis enforces explicit, implicit, and symplectic conditions on test functions.
  • Optimization strategies are developed for accuracy, sparseness (related to compressive sensing), and stability (Courant-Friedrichs-Lewy conditions).

Main Results:

  • A unified framework is presented for deriving diverse time-stepping schemes.
  • Explicit and symplectic Runge-Kutta methods of various orders are constructed.
  • Optimized schemes demonstrate improved accuracy, sparseness, and stability, validated by numerical experiments.

Conclusions:

  • The proposed discontinuous Galerkin approach offers a versatile and powerful tool for time integration.
  • The framework facilitates the development of tailored numerical schemes for specific problems.
  • Optimized schemes enhance the efficiency and reliability of solving differential equations in scientific computing.