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High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

  • Zhi Wang , Yongbin Ge EMAIL logo and Hai-Wei Sun
Published/Copyright: November 8, 2022
Computational Methods in Applied Mathematics
From the journal Computational Methods in Applied Mathematics Volume 23 Issue 2

Abstract

In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

Keywords: Helmholtz Equations; Compact Finite Difference Method; Sixth-Order Accuracy; Truncation Error
MSC 2010: 65N06; 65N12; 65N40

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12161067

Award Identifier / Grant number: 11772165

Award Identifier / Grant number: 11961054

Award Identifier / Grant number: 11902170

Funding source: Natural Science Foundation of Ningxia Province

Award Identifier / Grant number: 2022AAC02023

Award Identifier / Grant number: 2020AAC03059

Funding source: Key Research and Development Program of Ningxia

Award Identifier / Grant number: 2018BEE03007

Funding source: Universidade de Macau

Award Identifier / Grant number: MYRG2020-00224-FST

Funding statement: The second author Yongbin Ge is supported by NSFC (Grant No. 12161067, 11772165, 11961054, 11902170), National Natural Science Foundation of Ningxia (2022AAC02023, 2020AAC03059), the Key Research and Development Program of Ningxia (2018BEE03007), National Youth Top-notch Talent Support Program of Ningxia, and the First Class Discipline Construction Project in Ningxia Universities: Mathematics, and the third author Hai-Wei Sun is supported by Science and Technology Development Fund of Macao SAR (Grant No. 0122/2020/A3) and MYRG2020-00224-FST from University of Macau.

A The Sixth-Order Compact Difference Operators of the Second-Order Derivatives

By the Taylor expansion, u 1 ( u i + 1 , j , k ) and u 3 ( u i - 1 , j , k ) at point ( i , j , k ) have the following approximation:

u 1 = u 0 + h ⁢ ( ∂ ⁡ u ∂ ⁡ x ) 0 + h 2 2 ⁢ ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 + h 3 3 ! ⁢ ( ∂ 3 ⁡ u ∂ ⁡ x 3 ) 0 + h 4 4 ! ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 + h 5 5 ! ⁢ ( ∂ 5 ⁡ u ∂ ⁡ x 5 ) 0 + h 6 6 ! ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + h 7 7 ! ⁢ ( ∂ 7 ⁡ u ∂ ⁡ x 7 ) 0 + O ⁢ ( h 8 ) , u 3 = u 0 - h ⁢ ( ∂ ⁡ u ∂ ⁡ x ) 0 + h 2 2 ⁢ ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 - h 3 3 ! ⁢ ( ∂ 3 ⁡ u ∂ ⁡ x 3 ) 0 + h 4 4 ! ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 - h 5 5 ! ⁢ ( ∂ 5 ⁡ u ∂ ⁡ x 5 ) 0 + h 6 6 ! ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 - h 7 7 ! ⁢ ( ∂ 7 ⁡ u ∂ ⁡ x 7 ) 0 + O ⁢ ( h 8 ) .

Adding the above two formulas, the following equation holds:

(A.1) u 1 + u 3 - 2 ⁢ u 0 h 2 = ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 + h 2 12 ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 + h 4 360 ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + O ⁢ ( h 6 ) .

Substituting the central difference operator, (A.1) can be written as

(A.2) ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 = D x ⁢ x ⁢ u 0 - h 2 12 ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 - h 4 360 ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + O ⁢ ( h 6 ) .

Replacing u 0 with ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 and dropping the sixth-order derivative term and O ⁢ ( h 6 ) , we have

(A.3) ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 = D x ⁢ x ⁢ ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 - h 2 12 ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + O ⁢ ( h 4 ) ,

where

(A.4) ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 = [ ∂ 2 ∂ ⁡ x 2 ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) ] 0 = D x ⁢ x ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 + O ⁢ ( h 2 ) .

Substituting (A.3) and (A.4) into (A.2), we obtain

( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 = D x ⁢ x ⁢ u 0 - h 2 12 ⁢ [ D x ⁢ x ⁢ ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 - h 2 12 ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + O ⁢ ( h 4 ) ] - h 4 360 ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + O ⁢ ( h 6 ) = D x ⁢ x ⁢ u 0 - h 2 12 ⁢ D x ⁢ x ⁢ ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 + h 4 240 ⁢ ( ∂ 6 ⁡ u ∂ ⁡ x 6 ) 0 + O ⁢ ( h 6 ) = D x ⁢ x ⁢ u 0 - h 2 12 ⁢ D x ⁢ x ⁢ ( ∂ 2 ⁡ u ∂ ⁡ x 2 ) 0 + h 4 240 ⁢ D x ⁢ x ⁢ ( ∂ 4 ⁡ u ∂ ⁡ x 4 ) 0 + O ⁢ ( h 6 ) .

Finally, we obtain the sixth-order difference operator for the second-order derivative at the 𝑥 direction

∂ 2 ⁡ u ∂ ⁡ x 2 = ( 1 + h 2 12 ⁢ D x ⁢ x - h 4 240 ⁢ ∂ 4 ∂ ⁡ x 4 ) - 1 ⁢ D x ⁢ x ⁢ u + O ⁢ ( h 6 ) .

By the same method, the sixth-order difference operators for the second-order derivatives in the 𝑦 and 𝑧 directions can be written as

∂ 2 ⁡ u ∂ ⁡ y 2 = ( 1 + h 2 12 ⁢ D y ⁢ y - h 4 240 ⁢ ∂ 4 ∂ ⁡ y 4 ) - 1 ⁢ D y ⁢ y ⁢ u + O ⁢ ( h 6 ) , ∂ 2 ⁡ u ∂ ⁡ z 2 = ( 1 + h 2 12 ⁢ D z ⁢ z - h 4 240 ⁢ ∂ 4 ∂ ⁡ z 4 ) - 1 ⁢ D z ⁢ z ⁢ u + O ⁢ ( h 6 ) .

B The Truncation Error of the Sixth-Order Scheme

Using the Taylor series expansions for function ϕ ∈ { u , f } , we carry out the derivations below:

(B.1) ∑ l = 1 6 ϕ l = 6 ⁢ ϕ 0 + h 2 ⁢ ( ∂ 2 ⁡ ϕ ∂ ⁡ x 2 + ∂ 2 ⁡ ϕ ∂ ⁡ y 2 + ∂ 2 ⁡ ϕ ∂ ⁡ z 2 ) 0 + h 4 12 ⁢ ( ∂ 4 ∂ ⁡ x 4 + ∂ 4 ⁡ ϕ ∂ ⁡ y 4 + ∂ 4 ⁡ ϕ ∂ ⁡ z 4 ) 0 + h 6 360 ⁢ ( ∂ 6 ⁡ ϕ ∂ ⁡ x 6 + ∂ 6 ⁡ ϕ ∂ ⁡ y 6 + ∂ 6 ⁡ ϕ ∂ ⁡ z 6 ) 0 + h 8 20160 ⁢ ( ∂ 8 ⁡ ϕ ∂ ⁡ x 8 + ∂ 8 ⁡ ϕ ∂ ⁡ y 8 + ∂ 8 ⁡ ϕ ∂ ⁡ z 8 ) 0 + O ⁢ ( h 10 ) ,
(B.2) ∑ l = 7 18 ϕ l = 12 ⁢ ϕ 0 + 4 ⁢ h 2 ⁢ ( ∂ 2 ⁡ ϕ ∂ ⁡ x 2 + ∂ 2 ⁡ ϕ ∂ ⁡ y 2 + ∂ 2 ⁡ ϕ ∂ ⁡ z 2 ) 0 + h 4 3 ⁢ [ ∂ 4 ⁡ ϕ ∂ ⁡ x 4 + ∂ 4 ⁡ ϕ ∂ ⁡ y 4 + ∂ 4 ⁡ ϕ ∂ ⁡ z 4 + 3 ⁢ ( ∂ 4 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 + ∂ 4 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 2 + ∂ 4 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 ) ] 0 + h 6 180 [ 2 ( ∂ 6 ⁡ ϕ ∂ ⁡ x 6 + ∂ 6 ⁡ ϕ ∂ ⁡ y 6 + ∂ 6 ⁡ ϕ ∂ ⁡ z 6 ) + 15 ( ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 + ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ y 4 ) ] 0 + h 8 5040 [ ∂ 8 ⁡ ϕ ∂ ⁡ x 8 + ∂ 8 ⁡ ϕ ∂ ⁡ y 8 + ∂ 8 ⁡ ϕ ∂ ⁡ z 8 + 14 ( ∂ 8 ⁡ ϕ ∂ ⁡ x 6 ⁢ ∂ ⁡ y 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 6 + ∂ 8 ⁡ ϕ ∂ ⁡ y 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 6 + ∂ 8 ⁡ ϕ ∂ ⁡ x 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 6 ) ] 0 + O ( h 10 ) ,
(B.3) ∑ l = 19 26 ϕ l = 8 ⁢ ϕ 0 + 4 ⁢ h 2 ⁢ ( ∂ 2 ⁡ ϕ ∂ ⁡ x 2 + ∂ 2 ⁡ ϕ ∂ ⁡ y 2 + ∂ 2 ⁡ ϕ ∂ ⁡ z 2 ) 0 + h 4 3 ⁢ [ ∂ 4 ⁡ ϕ ∂ ⁡ x 4 + ∂ 4 ⁡ ϕ ∂ ⁡ y 4 + ∂ 4 ⁡ ϕ ∂ ⁡ z 4 + 6 ⁢ ( ∂ 4 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 + ∂ 4 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 2 + ∂ 4 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 ) ] 0 + h 6 90 [ ∂ 6 ⁡ ϕ ∂ ⁡ x 6 + ∂ 6 ⁡ ϕ ∂ ⁡ y 6 + ∂ 6 ⁡ ϕ ∂ ⁡ z 6 + 15 ( ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 + ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ y 4 ) + 90 ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 ] 0 + h 8 5040 [ ∂ 8 ⁡ ϕ ∂ ⁡ x 8 + ∂ 8 ⁡ ϕ ∂ ⁡ y 8 + ∂ 8 ⁡ ϕ ∂ ⁡ z 8 + 28 ⁢ ( ∂ 8 ⁡ ϕ ∂ ⁡ x 6 ⁢ ∂ ⁡ y 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 6 + ∂ 8 ⁡ ϕ ∂ ⁡ y 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 6 + ∂ 8 ⁡ ϕ ∂ ⁡ x 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 6 ) + 420 ⁢ ( ∂ 8 ⁡ ϕ ∂ ⁡ x 4 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 ) + 70 ( ∂ 8 ⁡ ϕ ∂ ⁡ x 4 ⁢ ∂ ⁡ y 4 + ∂ 8 ⁡ ϕ ∂ ⁡ x 4 ⁢ ∂ ⁡ z 4 + ∂ 8 ⁡ ϕ ∂ ⁡ y 4 ⁢ ∂ ⁡ z 4 ) ] 0 + O ( h 10 ) ,
(B.4) ( ∂ 2 ⁡ ϕ ∂ ⁡ x 2 ) 1 + ( ∂ 2 ⁡ ϕ ∂ ⁡ x 2 ) 3 + ( ∂ 2 ⁡ ϕ ∂ ⁡ y 2 ) 2 + ( ∂ 2 ⁡ ϕ ∂ ⁡ y 2 ) 4 + ( ∂ 2 ⁡ ϕ ∂ ⁡ z 2 ) 5 + ( ∂ 2 ⁡ ϕ ∂ ⁡ z 2 ) 6 = 2 ⁢ ( ∂ 2 ⁡ ϕ ∂ ⁡ x 2 + ∂ 2 ⁡ ϕ ∂ ⁡ y 2 + ∂ 2 ⁡ ϕ ∂ ⁡ z 2 ) 0 + h 2 ⁢ ( ∂ 4 ⁡ ϕ ∂ ⁡ x 4 + ∂ 4 ⁡ ϕ ∂ ⁡ y 4 + ∂ 4 ⁡ ϕ ∂ ⁡ z 4 ) 0 + h 4 12 ⁢ ( ∂ 6 ⁡ ϕ ∂ ⁡ x 6 + ∂ 6 ⁡ ϕ ∂ ⁡ y 6 + ∂ 6 ⁡ ϕ ∂ ⁡ z 6 ) 0 + h 6 360 ⁢ ( ∂ 8 ⁡ ϕ ∂ ⁡ x 8 + ∂ 8 ⁡ ϕ ∂ ⁡ y 8 + ∂ 8 ⁡ ϕ ∂ ⁡ z 8 ) 0 + h 8 20160 ⁢ ( ∂ 10 ⁡ ϕ ∂ ⁡ x 10 + ∂ 10 ⁡ ϕ ∂ ⁡ y 10 + ∂ 10 ⁡ ϕ ∂ ⁡ z 10 ) 0 + O ⁢ ( h 10 ) .
Substituting (B.1)–(B.4) into (2.13), we obtain

( λ 2 - 64 15 ⁢ h 2 ) ⁢ u + ( 1 + λ 2 ⁢ h 2 12 + λ 4 ⁢ h 4 240 ) ⁢ ( ∂ 2 ∂ ⁡ x 2 + ∂ 2 ∂ ⁡ y 2 + ∂ 2 ∂ ⁡ z 2 ) ⁢ u + ( h 2 6 + 7 ⁢ λ 2 ⁢ h 4 360 ) ⁢ ( ∂ 4 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 + ∂ 4 ∂ ⁡ x 2 ⁢ ∂ ⁡ z 2 + ∂ 4 ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 ) ⁢ u + ( h 2 12 + λ 2 ⁢ h 4 144 + λ 4 ⁢ h 6 2880 ) ⁢ ( ∂ 4 ∂ ⁡ x 4 + ∂ 4 ∂ ⁡ y 4 + ∂ 4 ∂ ⁡ z 4 ) ⁢ u + ( h 4 72 + 7 ⁢ λ 2 ⁢ h 6 4320 ) ⁢ ( ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 + ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ y 4 ) ⁢ u + ( h 4 360 + λ 2 ⁢ h 6 4320 ) ⁢ ( ∂ 6 ∂ ⁡ x 6 + ∂ 6 ∂ ⁡ y 6 + ∂ 6 ∂ ⁡ z 6 ) ⁢ u + h 6 2160 ⁢ ( ∂ 8 ⁡ ϕ ∂ ⁡ x 6 ⁢ ∂ ⁡ y 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 6 + ∂ 8 ⁡ ϕ ∂ ⁡ y 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 6 + ∂ 8 ⁡ ϕ ∂ ⁡ x 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 6 ) ⁢ u + h 4 30 ⁢ ∂ 8 ⁡ u ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 + h 6 20160 ⁢ ( ∂ 8 ∂ ⁡ x 8 + ∂ 8 ∂ ⁡ y 8 + ∂ 8 ∂ ⁡ z 8 ) ⁢ u 0 + h 6 360 ⁢ ( ∂ 8 ∂ ⁡ x 4 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 + ∂ 8 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 ⁢ ∂ ⁡ z 2 + ∂ 8 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 ) ⁢ u + h 6 864 ⁢ ( ∂ 8 ∂ ⁡ x 4 ⁢ ∂ ⁡ y 4 + ∂ 8 ∂ ⁡ x 4 ⁢ ∂ ⁡ z 4 + ∂ 8 ∂ ⁡ y 4 ⁢ ∂ ⁡ z 4 ) ⁢ u = f + ( h 2 12 + λ 2 ⁢ h 4 240 ) ⁢ ( ∂ 2 ∂ ⁡ x 2 + ∂ 2 ∂ ⁡ y 2 + ∂ 2 ∂ ⁡ z 2 ) ⁢ f + ( h 4 360 + λ 2 ⁢ h 6 2880 ) ⁢ ( ∂ 4 ∂ ⁡ x 4 + ∂ 4 ∂ ⁡ y 4 + ∂ 4 ∂ ⁡ z 4 ) ⁢ f + h 4 90 ⁢ ( ∂ 4 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 + ∂ 4 ∂ ⁡ x 2 ⁢ ∂ ⁡ z 2 + ∂ 4 ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 ) ⁢ f - h 6 8640 ⁢ ( ∂ 6 ∂ ⁡ x 6 + ∂ 6 ∂ ⁡ y 6 + ∂ 6 ∂ ⁡ z 6 ) ⁢ f + h 6 1080 ⁢ ( ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 + ∂ 6 ⁡ ϕ ∂ ⁡ x 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ⁡ ϕ ∂ ⁡ z 2 ⁢ ∂ ⁡ y 4 ) ⁢ f + O ⁢ ( h 8 ) .

Substituting (2.6), (2.10), and (3.3) into above equation and after rearrangement, we rewrite the truncation error of scheme (2.13) as follows:

R 3 , h ( 1 ) = λ 4 ⁢ h 6 2880 ⁢ ( ∂ 4 ∂ ⁡ x 4 + ∂ 4 ∂ ⁡ y 4 + ∂ 4 ∂ ⁡ z 4 ) ⁢ u + 7 ⁢ λ 2 ⁢ h 6 4320 ⁢ ( ∂ 6 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 + ∂ 6 ∂ ⁡ x 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ∂ ⁡ y 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ∂ ⁡ z 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ∂ ⁡ z 2 ⁢ ∂ ⁡ y 4 ) ⁢ u + λ 2 ⁢ h 6 4320 ⁢ ( ∂ 6 ∂ ⁡ x 6 + ∂ 6 ∂ ⁡ y 6 + ∂ 6 ∂ ⁡ z 6 ) ⁢ u + h 6 360 ⁢ ( ∂ 8 ∂ ⁡ x 4 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 2 + ∂ 8 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 ⁢ ∂ ⁡ z 2 + ∂ 8 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 ) ⁢ u + h 6 20160 ⁢ ( ∂ 8 ∂ ⁡ x 8 + ∂ 8 ∂ ⁡ y 8 + ∂ 8 ∂ ⁡ z 8 ) ⁢ u + h 6 2160 ⁢ ( ∂ 8 ∂ ⁡ x 6 ⁢ ∂ ⁡ y 2 + ∂ 8 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 6 + ∂ 8 ∂ ⁡ y 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ∂ ⁡ y 2 ⁢ ∂ ⁡ z 6 + ∂ 8 ∂ ⁡ x 6 ⁢ ∂ ⁡ z 2 + ∂ 8 ∂ ⁡ x 2 ⁢ ∂ ⁡ z 6 ) ⁢ u + h 6 864 ⁢ ( ∂ 8 ∂ ⁡ x 4 ⁢ ∂ ⁡ y 4 + ∂ 8 ∂ ⁡ x 4 ⁢ ∂ ⁡ z 4 + ∂ 8 ∂ ⁡ y 4 ⁢ ∂ ⁡ z 4 ) ⁢ u - λ 2 ⁢ h 6 2880 ⁢ ( ∂ 4 ∂ ⁡ x 4 + ∂ 4 ∂ ⁡ y 4 + ∂ 4 ∂ ⁡ z 4 ) ⁢ f - h 6 1080 ⁢ ( ∂ 6 ∂ ⁡ x 2 ⁢ ∂ ⁡ y 4 + ∂ 6 ∂ ⁡ x 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ∂ ⁡ y 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ∂ ⁡ y 2 ⁢ ∂ ⁡ z 4 + ∂ 6 ∂ ⁡ z 2 ⁢ ∂ ⁡ x 4 + ∂ 6 ∂ ⁡ z 2 ⁢ ∂ ⁡ y 4 ) ⁢ f + h 6 8640 ⁢ ( ∂ 6 ∂ ⁡ x 6 + ∂ 6 ∂ ⁡ y 6 + ∂ 6 ∂ ⁡ z 6 ) ⁢ f + O ⁢ ( h 8 ) .

Acknowledgements

The authors are grateful to the anonymous referees for their constructive comments which improved this paper a lot.

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Received: 2022-01-10
Revised: 2022-09-19
Accepted: 2022-09-26
Published Online: 2022-11-08
Published in Print: 2023-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Recent Advances in Boundary Element Methods
  3. Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)
  4. Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case
  5. CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries
  6. Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements
  7. Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems
  8. BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering
  9. Force Computation for Dielectrics Using Shape Calculus
  10. A Time-Adaptive Space-Time FMM for the Heat Equation
  11. Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
  12. High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations
  13. A Posteriori Error Estimates for Darcy–Forchheimer’s Problem
  14. Convergence of a Finite Difference Scheme for a Flame/Smoldering-Front Evolution Equation and Its Application to Wavenumber Selection
https://doi.org/10.1515/cmam-2022-0002
Keywords for this article
Helmholtz Equations; Compact Finite Difference Method; Sixth-Order Accuracy; Truncation Error

Articles in the same Issue

  1. Frontmatter
  2. Recent Advances in Boundary Element Methods
  3. Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)
  4. Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case
  5. CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries
  6. Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements
  7. Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems
  8. BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering
  9. Force Computation for Dielectrics Using Shape Calculus
  10. A Time-Adaptive Space-Time FMM for the Heat Equation
  11. Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
  12. High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations
  13. A Posteriori Error Estimates for Darcy–Forchheimer’s Problem
  14. Convergence of a Finite Difference Scheme for a Flame/Smoldering-Front Evolution Equation and Its Application to Wavenumber Selection
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