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Sixth order compact multi-phase block-AGE iteration methods for computing 2D Helmholtz equation.

R K Mohanty1, Niranjan1

  • 1Department of Mathematics, South Asian University, Rajpur Road, Maidan Garhi, New Delhi 110068, India.

Methodsx
|April 25, 2024
PubMed
Summary
This summary is machine-generated.

New block alternating group explicit (block-AGE) methods offer efficient, sixth-order accurate solutions for the 2D Helmholtz equation. These methods demonstrate faster convergence and improved precision compared to traditional block successive over relaxation techniques.

Keywords:
9-point compact meshBlock-AGE Multi-stage Iteration MethodBlock-SOR iteration methodError analysisHelmholtz equationMulti-phase block-AGE iteration methodsSixth order approximationThree-diagonal solver

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • The 2D Helmholtz equation is crucial in various scientific fields, including acoustics and electromagnetics.
  • Efficient numerical methods are needed to solve this equation accurately and quickly.
  • Existing methods like block successive over relaxation (block-SOR) have limitations in speed and accuracy.

Purpose of the Study:

  • To introduce novel sixth-order accurate 9-point compact 2- and 3-phase block alternating group explicit (block-AGE) iteration methods.
  • To analyze the error and convergence properties of these new methods.
  • To compare the performance of block-AGE methods against block-SOR for solving the 2D Helmholtz equation.

Main Methods:

  • Development of sixth-order accurate 9-point compact 2- and 3-phase block-AGE iteration schemes.
  • Implementation using Dirichlet boundary conditions without fictitious points.
  • Error analysis and convergence study of the proposed methods.
  • Comparative experiments with block-SOR, focusing on iteration count and CPU time.

Main Results:

  • The proposed 2- and 3-phase block-AGE methods achieve sixth-order accuracy.
  • These methods require fewer sweeps (two and three, respectively) for computation.
  • Block-AGE methods show superior performance over block-SOR in terms of convergence speed and computational time.
  • Optimal relaxation parameters significantly enhance the convergence and precision of block-AGE methods.

Conclusions:

  • The developed block-AGE iteration methods provide a highly accurate and efficient alternative for solving the 2D Helmholtz equation.
  • The study highlights the importance of optimal relaxation parameters in achieving superior numerical results.
  • These methods offer a promising advancement in computational solutions for Helmholtz-type problems.