Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Development of a new sixth order accurate compact scheme for two and three dimensional Helmholtz equation

Neelesh Kumar, Ritesh Kumar Dubey

TL;DR

The main significance of the proposed scheme is that its sixth order leading truncation error term does not explicitly depend on the associated wave number, which makes the scheme robust to work for the Helmholtz equation even with large wave numbers.

Abstract

In this work, a new compact sixth order accurate finite difference scheme for the two and three-dimensional Helmholtz equation is presented. The main significance of the proposed scheme is that its sixth order leading truncation error term does not explicitly depend on the associated wave number. This makes the scheme robust to work for the Helmholtz equation even with large wave numbers. The convergence analysis of the new scheme is given. Numerical results for various benchmark test problems are given to support the theoretical estimates. These numerical results confirm the accuracy and robustness of the proposed scheme.

Development of a new sixth order accurate compact scheme for two and three dimensional Helmholtz equation

TL;DR

The main significance of the proposed scheme is that its sixth order leading truncation error term does not explicitly depend on the associated wave number, which makes the scheme robust to work for the Helmholtz equation even with large wave numbers.

Abstract

In this work, a new compact sixth order accurate finite difference scheme for the two and three-dimensional Helmholtz equation is presented. The main significance of the proposed scheme is that its sixth order leading truncation error term does not explicitly depend on the associated wave number. This makes the scheme robust to work for the Helmholtz equation even with large wave numbers. The convergence analysis of the new scheme is given. Numerical results for various benchmark test problems are given to support the theoretical estimates. These numerical results confirm the accuracy and robustness of the proposed scheme.

Paper Structure

This paper contains 14 sections, 8 theorems, 136 equations, 25 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Formulation of a new sixth-order accurate approximation
  3. Two-Dimensional case
  4. Three-Dimensional case
  5. High order accurate approximation of Neumann boundary
  6. Two-dimensional case
  7. Three-dimensional case
  8. Convergence analysis
  9. Solvability of the Difference Scheme
  10. Error estimate of the Scheme
  11. Results and Discussion
  12. Two-dimensional test cases
  13. Three-dimensional test cases
  14. Conclusions

Key Result

Lemma 4.1

Let $A=[a_{i,j}], i,j=1(1)n, n=(N-1)^2$ be a matrix with its elements $a_{i,j}$ given by (a4)-(a6). When $Kh$ is sufficiently small, where $K$ is the wave number and $h$ is the grid length, then the directed graph $\mathcal{G}(A)$ of the matrix $A$ is strongly connected.

Figures (25)

  • Figure 1: Directed graph for its adjacency matrix $A$ for $N=4$.
  • Figure 2:
  • Figure 3:
  • Figure 5:
  • Figure 6:
  • ...and 20 more figures

Theorems & Definitions (15)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 5 more