We present a fourth-order finite difference scheme for the Helmholtz equation in polar coordinates. We employ the finite difference format in the interior of the region and derive a nine-point fourth-order scheme. Specially, ghost points outside the region are applied to obtain the approximation for the Neumann boundary condition. We obtain the matrix form of the linear system and the sparsity of the coefficient matrix is favorable for the computation of the Helmholtz equation. The feasibility and accuracy of the method are validated by two test examples which have exact solutions.
Helmholtz equation has attracted much attention in many fields such as electromagnetic cavity scattering problems [
In general Cartesian coordinates, high-order methods for solving the Helmholtz equation have been well developed since the accuracy is related to the amount of the grid points with the wave number increases. Manohar et al. [
In polar coordinates, the symmetric problem can be described more concisely. A finite difference method was proposed to solve the parabolic equation in polar coordinates in [
The existed works didn’t discuss the Neumann boundary condition and the sparsity of the coefficient matrix. In this paper, we develop a fourth-order scheme for the Helmholtz equation in polar coordinates with Dirichlet and Neumann boundary conditions. The sparse matrix form of the linear system is derived. With the help of the sparsity of the coefficient matrix of the linear system, the computational cost is remarkably reduced. Moreover, we give the approximation for the Neumann boundary condition. The feasibility and the order of method are verified by the Helmholtz equation with Dirichlet and Neumann boundary condition.
The paper is outlined as follows. In Section 2, a fourth-order finite difference scheme and the sparse matrix form for the Helmholtz equation in polar coordinates are derived. In Section 3, the approximation for the boundary condition is obtained. Two numerical experiments of the high-order algorithm are presented in Section 4. The paper is concluded in Section 5.
We consider the following two-dimensional Helmholtz equation in polar coordinates
1 r ∂ ∂ r ( r ∂ u ∂ r ) + 1 r 2 ∂ 2 u ∂ θ 2 + k 2 u = f , ( r , θ ) ∈ D , (1)
where k is the wave number, D is a given region and f is a known function. For convenience, we rewrite the above equation as
1 r ∂ ∂ r ( r ∂ u ∂ r ) = f r = f − k 2 u − 1 r 2 ∂ 2 u ∂ θ 2 , (2)
1 r 2 ∂ 2 u ∂ θ 2 = f θ = f − k 2 u − 1 r ∂ ∂ r ( r ∂ u ∂ r ) . (3)
Define a uniform mesh on the region D = ( r a , r b ) × ( θ c , θ d ) . r i = r a + ( i − 1 ) h r , i = 0 , 1 , ⋯ , M + 1 , θ j = θ c + ( j − 1 ) h θ , j = 0 , 1 , ⋯ , N + 1 . u ( r i , θ j ) denotes the fourth-order solution of u at point ( r i , θ j ) .
First, we employ a fourth-order approximation on the left side of Equation (2) and obtain
1 r ∂ ∂ r ( r ∂ u ∂ r ) | i , j = 1 r i 1 h r ( r i + 1 2 u i + 1 , j − u i , j h r − r i − 1 2 u i , j − u i − 1 , j h r ) − h r 2 12 ( ∂ 2 f r ∂ r 2 + 1 r ∂ f r ∂ r + 1 r 2 f r − 2 r 3 ∂ u ∂ r ) | i , j + O ( h r 4 ) , (4)
where
∂ f r ∂ r = ∂ f ∂ r − k 2 ∂ u ∂ r − ∂ ∂ r ( 1 r 2 ∂ 2 u ∂ θ 2 ) ,
∂ 2 f r ∂ r 2 = ∂ 2 f ∂ r 2 − k 2 ∂ 2 u ∂ r 2 − ∂ 2 ∂ r 2 ( 1 r 2 ∂ 2 u ∂ θ 2 ) .
Then approximating the derivatives of u with the second-order accuracy by central differences, we have
f r | i , j = f i , j − k 2 u i , j − 1 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 + O ( h r 2 ) , ∂ f r ∂ r | i , j = ∂ f ∂ r | i , j − k 2 u i + 1 , j − u i − 1 , j 2 h r − 1 2 h r ( 1 r i + 1 2 u i + 1 , j + 1 − 2 u i + 1 , j + u i + 1 , j − 1 h θ 2 − 1 r i − 1 2 u i − 1 , j + 1 − 2 u i − 1 , j + u i − 1 , j − 1 h θ 2 ) + O ( h r 2 ) , ∂ 2 f r ∂ r 2 | i , j = ∂ 2 f ∂ r 2 | i , j − k 2 u i + 1 , j − 2 u i , j + u i − 1 , j h r 2 − 1 h r 2 ( 1 r i + 1 2 u i + 1 , j + 1 − 2 u i + 1 , j + u i + 1 , j − 1 h θ 2 − 2 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 + 1 r i − 1 2 u i − 1 , j + 1 − 2 u i − 1 , j + u i − 1 , j − 1 h θ 2 ) + O ( h r 2 ) . (5)
By substituting Equation (5) into Equation (4), we can obtain a nine-point stencil approximation for 1 r ∂ ∂ r ( r ∂ u ∂ r ) | i , j .
We now turn to the fourth-order approximation for the left side of Equation (3). Applying a standard central difference operator to the left item of Equation (3), we have
1 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 = f θ | i , j , (6)
and the error
1 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 = 1 r ∂ 2 u ∂ θ 2 + 1 r h θ 2 12 ∂ 4 u ∂ θ 4 + O ( h θ 4 ) . (7)
Differentiating both sides of (3) with respect to θ and combing (7), we have
1 r 2 ∂ 2 u ∂ θ 2 | i , j = 1 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ − h θ 2 12 ∂ 2 f θ ∂ θ 2 | i , j + O ( h θ 4 ) . (8)
Similarly, all derivatives of f θ with respect to θ can be approximated as follows
f ″ θ | i , j = ∂ 2 f ∂ θ 2 | i , j − k 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 − 1 r i 1 h θ 2 h r 2 { r i + 1 2 ( u i + 1 , j + 1 − u i , j + 1 ) − r i − 1 2 ( u i , j + 1 − u i − 1 , j + 1 ) − 2 [ r i + 1 2 ( u i + 1 , j − u i , j ) − r i − 1 2 ( u i , j − u i − 1 , j − 1 ) ] + r i + 1 2 ( u i + 1 , j − 1 − u i , j − 1 ) − r i − 1 2 ( u i , j − 1 − u i − 1 , j − 1 ) } + O ( h θ 2 ) . (9)
By substituting Equation (9) into Equation (8), the fourth-order finite difference scheme for ∂ 2 u ∂ θ 2 is obtained.
Therefore, combing Equation (4) and Equation (8), we have the overall fourth-order finite difference form for the Helmholtz equation in polar coordinates
1 r ∂ ∂ r ( r ∂ u ∂ r ) + ∂ 2 u ∂ θ 2 + k 2 u = 1 r i 1 h r ( r i + 1 2 u i + 1 , j − u i , j h r − r i − 1 2 u i , j − u i − 1 , j h r ) + 1 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 − h r 2 12 ( ∂ 2 f r ∂ r 2 + 1 r ∂ f r ∂ r + 1 r 2 f r − 2 r 3 ∂ u ∂ r ) | i , j − h θ 2 12 ∂ 2 f θ ∂ θ 2 | i , j + k 2 u i , j = f i , j (10)
Replacing the derivatives of f r and f θ by Equations (5) and (9), we can derive
1 r i 1 h r ( r i + 1 2 u i + 1 , j − u i , j h r − r i − 1 2 u i , j − u i − 1 , j h r ) + 1 r i 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 − h r 2 12 [ ∂ 2 f ∂ r 2 | i , j − k 2 u i + 1 , j − 2 u i , j + u i − 1 , j h r 2 ] + 1 12 h θ 2 [ 1 r i + 1 2 ( u i + 1 , j + 1 − 2 u i + 1 , j + u i + 1 , j − 1 ) − 2 r i 2 ( u i , j + 1 − 2 u i , j + u i , j − 1 ) + 1 r i − 1 2 ( u i − 1 , j + 1 − 2 u i − 1 , j + u i − 1 , j − 1 ) ]
− h r 2 12 r i [ ∂ f ∂ r | i , j − k 2 u i + 1 , j − u i − 1 , j 2 h r − 1 2 h r h θ 2 ( 1 r i + 1 2 ( u i + 1 , j + 1 − 2 u i + 1 , j + u i + 1 , j − 1 ) − 1 r i − 1 2 ( u i − 1 , j + 1 − 2 u i − 1 , j + u i − 1 , j − 1 ) ) ] − h r 2 12 r i 2 ( f i , j − k 2 u i , j − 1 r i 2 h θ 2 ( u i , j + 1 − 2 u i , j + u i , j − 1 ) ) + h r 12 r i 3 ( u i + 1 , j − u i − 1 , j ) − h θ 2 12 ( ∂ 2 f ∂ θ 2 | i , j − k 2 u i , j + 1 − 2 u i , j + u i , j − 1 h θ 2 )
+ 1 12 h r 2 r i [ r i + 1 2 ( u i + 1 , j + 1 − u i , j + 1 ) − r i − 1 2 ( u i , j + 1 − u i − 1 , j + 1 ) − 2 ( r i + 1 2 ( u i + 1 , j − u i , j ) − r i − 1 2 ( u i , j − u i − 1 , j ) ) + r i + 1 2 ( u i + 1 , j − 1 − u i , j − 1 ) − r i − 1 2 ( u i , j − 1 − u i − 1 , j − 1 ) ] + k 2 u i , j = f i , j . (11)
For simplicity, we rewrite the above Equation (11) in the following form
l 1 u i − 1 , j − 1 + l 2 u i − 1 , j + l 3 u i − 1 , j + 1 + l 4 u i , j − 1 + l 5 u i , j + l 6 u i , j + 1 + l 7 u i + 1 , j − 1 + l 8 u i + 1 , j + l 9 u j + 1 , j + 1 = b i , j , (12)
where
l 1 = l 3 = 1 12 h θ 2 r i − 1 2 − h r 24 h θ 2 r i − 1 2 1 r i + r i − 1 2 12 h r 2 1 r i , l 2 = 5 r i − 1 2 6 h r 2 r i + k 2 12 − 1 6 h θ 2 r i − 1 2 + ( h r 12 h θ 2 r i − 1 2 − h r k 2 r i ) 1 12 − h r 12 r i 3 , l 4 = l 6 = 5 6 r i 2 h θ 2 + h r 2 12 r i 4 h θ 2 − r i + 1 2 + r i − 1 2 12 h r 2 r i + k 2 12 , l 5 = − 5 3 h θ 2 r i 2 − 5 ( r i + 1 2 + r i − 1 2 ) 6 h r 2 r i + 2 k 2 3 − ( h r 2 6 r i 2 h θ 2 − h r 2 k 2 r i 2 ) 1 r i 2 ,
l 7 = l 9 = 1 12 h θ 2 r i + 1 2 + h r 24 h θ 2 r i + 1 2 1 r i + r i + 1 2 12 h r 2 1 r i , l 8 = 5 r i + 1 2 6 h r 2 r i + k 2 12 − 1 6 h θ 2 r i + 1 2 − ( h r 12 h θ 2 r i + 1 2 − h r k 2 12 ) 1 r i + h r 12 r i 3 , b i , l = f i , l + h r 2 12 r i 2 f i , l + h r 2 12 r i ∂ f ∂ r + h r 2 12 ∂ 2 f ∂ r 2 + h θ 2 12 ∂ 2 f ∂ θ 2 .
The relationship between the nine points can be illustrated in three parts in matrix form A X , A Y , A D , which represent the horizontal, vertical, diagonal relationships respectively,
( A X + A Y + A D ) U = F , (13)
where
U = ( u 1 , 1 , u 1 , 2 , ⋯ , u 1 , N , u 2 , 1 , u 2 , 2 , ⋯ , u 2 , N , ⋯ , u M , 1 , u M , 2 , ⋯ , u M , N ) T , A X = A x ⊗ I N , A Y = diag { A Y ( 1 ) , A Y ( 2 ) , ⋯ , A Y ( M ) } , A Y ( i ) = tridiag { l 6 ( i ) , l 5 ( i ) 4 , l 4 ( i ) } , A D = A D 1 ⊗ I D 1 + A D 2 ⊗ I D 2 + A D 3 ⊗ I N , A x = ( l 5 ( 1 ) 4 l 8 ( 1 ) l 2 ( 2 ) l 5 ( 2 ) 4 l 8 ( 2 ) ⋱ l 2 ( M ) l 5 ( M ) 4 ) , A D 1 = ( 0 l 7 ( 1 ) l 1 ( 2 ) 0 l 7 ( 2 ) ⋱ ⋱ l 1 ( M − 1 ) 0 l 7 ( M − 1 ) l 1 ( M ) 0 ) ,
I r 1 = ( 0 1 0 1 ⋱ ⋱ 0 1 0 ) , A D 2 = ( 0 l 9 ( 1 ) l 3 ( 2 ) 0 l 9 ( 2 ) ⋱ ⋱ l 3 ( M − 1 ) 0 l 9 ( M − 1 ) l 3 ( M ) 0 ) , I r 2 = ( 0 1 0 ⋱ ⋱ 1 0 1 0 ) , A D 3 = ( l 5 ( 1 ) 2 l 5 ( 2 ) 2 ⋱ l 5 ( M ) 2 )
and ⊗ denotes the Kronecker product, I N is the N × N identity matrix. F is the right side item containing b i , j and the boundary conditions which will be discussed in the following section.
Implementation of Dirichlet boundary conditions at r = r a , r b , θ = θ c , θ d is straightforward. And the right side item of Equation (13) can be written as
F = f ˜ + U B l + U B r + U B b + U B t , (14)
where
f ˜ = ( b 11 , b 12 , ⋯ , b 1 N , b 21 , b 22 , ⋯ , b 2 N , ⋯ , b M 1 , b M 2 , ⋯ , b M N ) T , U B l = ( B 1 u 0 , : , 0 , 0 , ⋯ , 0 , 0 , 0 , ⋯ , 0 , 0 , 0 , ⋯ , 0 ) T , u 0 , : = ( u 0 , 0 , u 0 , 1 , ⋯ , u 0 , N + 2 ) T , U B r = ( B 2 u M + 1 , : , 0 , 0 , ⋯ , 0 , 0 , 0 ⋯ , 0 , 0 , 0 , ⋯ , 0 ) T , u M , : = ( u M , 0 , u M , 1 , ⋯ , u M , N + 2 ) T , U B b = B 3 u : , 0 ⊗ a N , U B b = B 4 u : , N + 1 ⊗ b N , u : , 0 = ( u 1 , 0 , u 2 , 0 , ⋯ , u M , 0 ) T , u : , N + 1 = ( u 1 , N + 1 , u 2 , N + 1 , ⋯ , u M , N + 1 ) T ,
B 1 = ( l 3 ( 1 ) l 2 ( 1 ) l 1 ( 1 ) l 3 ( 1 ) l 2 ( 1 ) l 1 ( 1 ) ⋱ ⋱ ⋱ l 3 ( 1 ) l 2 ( 1 ) l 1 ( 1 ) ) , B 2 = ( l 9 ( M ) l 8 ( M ) l 7 ( M ) l 9 ( M ) l 8 ( M ) l 7 ( M ) ⋱ ⋱ ⋱ l 9 ( M ) l 8 ( M ) l 7 ( M ) ) , B 3 = B 4 = ( l 4 ( 1 ) l 7 ( 1 ) l 1 ( 2 ) l 4 ( 2 ) l 7 ( 2 ) ⋱ ⋱ ⋱ l 1 ( M − 1 ) l 4 ( M − 1 ) l 7 ( M − 1 ) l 4 ( M ) l 7 ( M ) ) , a N = ( 1 0 ⋮ 0 ) , b N = ( 0 0 ⋮ 1 ) ,
and a N , b N are vectors in N × 1 dimensions.
Implementation of the Dirichlet boundary conditions at r = r a , r b , θ = θ c , θ d is more complicated. We consider the following Neumann boundary condition and it can be extended to the general cases
∂ u ∂ θ | θ = θ t o p = α u + g ( r , θ ) , (15)
where α is a constant, g ( r , θ ) is a given function.
The ghost points are utilized to derive the difference scheme on the boundary can be written as
∂ u ∂ θ | i , N + 1 = u i , N + 2 − u i , N 2 h θ + O ( h 2 ) . (16)
Therefore, the Neumann boundary condition can be approximated as
α u i , N + 1 + g ( r i , θ N + 1 ) = u i , N + 2 − u i , N 2 h θ , i = 1 , 2 , ⋯ , M . (17)
We can obtain
u : , N + 2 − u : , N = 2 h θ g : , N + 1 + 2 h θ α u : , N + 1 , (18)
where
u : , N = ( u 1 , N , u 2 , N , ⋯ , u M , N ) T , u : , N + 1 = ( u 1 , N + 1 , u 2 , N + 1 , ⋯ , u M , N + 1 ) T , u : , N + 2 = ( u 1 , N + 2 , u 2 , N + 2 , ⋯ , u M , N + 2 ) T , g : , N + 1 = ( g 1 , N + 1 , g 2 , N + 1 , ⋯ , g M , N + 1 ) T . (19)
Substituting j for N + 1 in Equation (12) gives
A u : , N + 2 + B u : N + 1 + A u : , N = b : , N + 1 , (20)
where
A = ( l 4 ( 1 ) l 7 ( 1 ) l 1 ( 2 ) l 4 ( 2 ) l 7 ( 2 ) ⋱ ⋱ ⋱ l 1 ( M − 1 ) l 4 ( M − 1 ) l 7 ( M − 1 ) l 4 ( M ) l 7 ( M ) ) , B = ( l 5 ( 1 ) l 8 ( 1 ) l 2 ( 2 ) l 5 ( 2 ) l 8 ( 2 ) ⋱ ⋱ ⋱ l 2 ( M − 1 ) l 5 ( M − 1 ) l 8 ( M − 1 ) l 5 ( M ) l 8 ( M ) ) , b : , N + 1 = ( b 1 , N + 1 , b 2 , N + 1 , ⋯ , b M , N + 1 ) T .
Combining Equations (18) and (20), we can eliminate u : , N + 2 and obtain
2 A u : , N + ( 2 h θ α A + B ) u : , N + 1 = b : , N + 1 − 2 h θ A g : , N + 1 . (21)
Therefore, collaborating Equations (13) and (21), the global system can be written as follows
( A 11 A 12 A 21 A 22 ) ( U u : , N + 1 ) = ( b 1 b 2 ) , (22)
where
A 11 = A X + A Y + A D , A 12 = B 4 , A 21 = 2 A ⊗ a N , A 22 = 2 h θ α A + B , b 1 = f ˜ + U B l + U B r + U B b , b 2 = b : , N + 1 − 2 h θ A g : , N + 1 .
We can observe the sparsity of the global system of the Helmholtz equation with the Neumann boundary condition in
We first consider the problem with Dirichlet boundary condition in polar coordinates as follows
1 r ∂ ∂ r ( r ∂ u ∂ r ) + 1 r 2 ∂ 2 u ∂ θ 2 + k 2 u = 0 , ( r , θ ) ∈ [ 1 , 2 ] × [ 0 , 2 π ] u | r = 1 = sin ( θ ) , u | r = 2 = 2 sin ( θ ) , u | θ = 0 = u | θ = 2 π = 0. (23)
and the exact solution is u ( r , θ ) = r cos ( θ ) .
As we can see from
order = log ( error ( M 1 ) error ( M 2 ) ) / log ( M 2 M 1 ) .
It can be clearly seen from
| k = 1 | k = 5 | ||||
|---|---|---|---|---|---|
| Mesh | Error | Order | Mesh | Error | Order |
| 40 × 40 | 0.0022 | 40 × 40 | 0.0031 | ||
| 80 × 80 | 3.0955e−04 | 3.9638 | 80 × 80 | 3.2997e−04 | 3.4655 |
| 160 × 160 | 4.4832e−05 | 3.9914 | 160 × 160 | 2.7878e−05 | 3.6936 |
| 320 × 320 | 7.0240e−06 | 3.9982 | 320 × 320 | 1.9181e−06 | 3.9310 |
| 640 × 640 | 4.1684e−09 | 3.9997 | 640 × 640 | 1.2417e−07 | 3.9849 |
| k = 10 | k = 20 | ||||
|---|---|---|---|---|---|
| Mesh | Error | Order | Mesh | Error | Order |
| 40 × 40 | 0.0108 | 40 × 40 | 0.0029 | ||
| 80 × 80 | 8.1990e−04 | 3.9831 | 80 × 80 | 3.0197e−04 | 3.5145 |
| 160 × 160 | 6.3034e−05 | 3.8346 | 160 × 160 | 3.4628e−05 | 3.2369 |
| 320 × 320 | 3.9823e−06 | 4.0563 | 320 × 320 | 6.9882e−06 | 2.3506 |
| 640 × 640 | 2.5763e−07 | 3.9859 | 640 × 640 | 2.3137e−07 | 4.9610 |
| Mesh | Method I | Method II |
|---|---|---|
| 64 × 64 | 0.7810 | 0.1519 |
| 128 × 128 | 19.7907 | 0.4309 |
| 512 × 512 | 884.6326 | 2.3043 |
gives the comparison of the computational time(s) for solving the Helmholtz equation in polar coordinates, where Method I and Method II denote methods with and without the utilization of the sparsity of the coefficient matrix of the linear system. It can be observed that with the help of the sparsity of the coefficient matrix of the linear system, the computational cost is remarkably reduced.
This example is a Helmholtz equation in polar coordinates with Neumann boundary condition
1 r ∂ ∂ r ( r ∂ u ∂ r ) + 1 r 2 ∂ 2 u ∂ θ 2 + k 2 u = 0 , ( r , θ ) ∈ [ 1 , 2 ] × [ 0 , 2 π ] ∂ u ∂ θ = r cos ( θ ) , θ = 2 π , u | r = 1 = sin ( θ ) , u | r = 2 = 2 sin ( θ ) , u | θ = 0 = 0 , (24)
and the exact solution is u = r sin ( θ ) . As we can see from Figures 4-6 that the
numerical solution is well agreed with the exact solution with k = 5 and M = N = 128 .
In this paper, we propose a high-order fast algorithm for solving the two-dimensional Helmholtz equation with Dirichlet and Neumann boundary conditions in polar coordinates. We develop a fourth-order accurate compact finite difference approximation to the Helmholtz equation. The sparse matrix form for the Helmholtz equation in polar coordinates is obtained which improves the efficiency for the computation process. Two numerical experiments have demonstrated the validity of the fourth-order algorithm.
This research was supported by the Fundamental Research Funds for the Central Universities (No. 2018MS129).
The authors declare no conflicts of interest regarding the publication of this paper.
Zhu, N. and Zhao, M.L. (2019) High-Order Finite Difference Method for Helmholtz Equation in Polar Coordinates. American Journal of Computational Mathematics, 9, 174-186. https://doi.org/10.4236/ajcm.2019.93013