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Complete centered finite difference method for Helmholtz equation.

Gustavo B Alvarez1, Helder F Nunes2, Welton A Menezes1

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Summary
This summary is machine-generated.

A novel finite difference method minimizes dispersion errors by constructing a local approximation subspace basis. This approach eliminates pollution errors in 1D and matches finite element methods in 2D.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Finite difference methods are widely used for solving differential equations.
  • Dispersion errors can limit the accuracy of numerical solutions.
  • Existing methods often struggle to balance accuracy and computational cost.

Purpose of the Study:

  • To introduce a new finite difference framework that minimizes dispersion errors.
  • To develop schemes for constructing local approximation subspace bases.
  • To demonstrate the method's consistency and applicability across dimensions and stencil sizes.

Main Methods:

  • Developing a three-step approach: subspace dimension selection, basis construction, and coefficient determination.
  • Deriving new schemes for local approximation subspace basis construction by approximating the k^2u term of the Helmholtz equation.
  • Analyzing the dispersion relation for various dimensionalities and stencil configurations.

Main Results:

  • The new method is consistent and minimizes dispersion for all stencils and dimensions.
  • Pollution error is eliminated in the one-dimensional, 3-point stencil case.
  • In 2D, the method achieves dispersion relations comparable to established finite element techniques (Galerkin/Least-Squares and Quasi Stabilized).
  • A link between linear system coefficients and stencil symmetry was identified.

Conclusions:

  • The proposed finite difference framework offers enhanced accuracy by minimizing dispersion.
  • The method provides a unified approach capable of representing various finite difference schemes.
  • It achieves competitive accuracy with advanced finite element methods, particularly in 2D.
  • The findings suggest potential for improved numerical simulations in various scientific domains.