^{1}

^{*}

^{2}

By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.

High order and high efficiency numerical computation for partial differential equations is very important in many scientific and engineering modeling problems. Compared to low order (second order) methods, high order methods can achieve satisfactory errors on much coarser grids and thus greatly curtail the computational cost [

Recently, there has been growing interest in developing sixth order compact finite difference schemes. By using Taylor series expansion, Soptz and Carey [

1) derivatives of source term appeared in the right-hand side which require analytical forms or approximations for the derivatives with certain order accuracy;

2) not complete compact schemes which may cause problems near to the boundaries;

3) complicated resulting linear systems which increase the difficulty of choosing effective iterative solvers.

Another category of sixth order compact approximations is based on Richardson extrapolation which is a technique introduced by Lewis Fry Richardson in the early of 20th century [

In this paper, our goal is to analyze and compare these Richardson extrapolation-based sixth order methods in computational accuracy. Therefore, we first describe these three Richardson extrapolation-based sixth order methods in Section 2. Then, we give a detailed analysis on their truncation errors in Section 3. After that, we use numerical experiments to verify our theoretical analysis in Section 4. At last, the conclusion and comments are given in Section 5.

Consider a 2D Poisson equation in the form of

u x x + u y y = f ( x , y ) , ( x , y ) ∈ Ω , (1)

where Ω is a rectangular domain with suitable boundary conditions defined on ∂ Ω . The solution u and the forcing term f ( x , y ) are assumed to be sufficiently smooth and have necessary continuous partial derivatives. Ω can be discretized with uniform meshsize h in the x and y coordinate directions. The mesh points are ( x i , y j ) with x i = i h and y j = j h .

Assume we have pth order accurate approximate solutions u i , j 2 h and u i , j h on the Ω 2 h grid and Ω h grid. A (p + 2)th order accurate solution u ˜ i , j 2 h on the Ω 2 h grid can be obtained by the general Richardson extrapolation, which can be written as [

u ˜ i , j 2 h = 2 p u 2 i , 2 j h − u i , j 2 h 2 p − 1 . (2)

To get a sixth order solution, FOC schemes are first used to compute fourth order accurate solutions u i , j 2 h and u i , j h on the coarse grid Ω 2 h and fine grid Ω h , respectively. Then the Richardson extrapolation formula as

u ˜ i , j 2 h = 16 u 2 i , 2 j h − u i , j 2 h 15 (3)

is used to compute the sixth order accurate solution u ˜ i , j 2 h on the coarse grid Ω 2 h [

One method using Richardson extrapolation to compute a sixth order accurate solution on the fine grid is to use an operator based interpolation scheme, proposed by Wang and Zhang [

u ˜ i , j h = − 1 20 [ F i , j − 4 ( u i + 1, j h + u i − 1, j h + u i , j + 1 h + u i , j − 1 h ) − ( u i + 1, j + 1 h + u i + 1, j − 1 h + u i − 1, j + 1 h + u i − 1, j − 1 h ) ] . (4)

The second Richardson extrapolation sixth order computation is to use multiple coarse grids. For a 2D problem, the fine grid can be coarsened into four coarse grids, each of which is composed of one subset of fine grid points from: (even, even) fine grid points, (even, odd) fine grid points, (odd, even) fine grid points, and (odd, odd) fine grid points. The sixth order solution for (even, even) fine grid points comes from Richardson extrapolation. Dai et al. [

only one step of operator interpolation is needed to reach the sixth order solution. Computation details can be found in [

The third approach of using Richardson extrapolation for computing sixth order solutions is Completed Richardson extrapolation. Completed Richardson extrapolation was developed by Roache and Knupp [

Assume u i , j ∗ be the exact solution at node ( i , j ) on the fine grid. With the definition of fourth order accurate solution, we have

u i , j ∗ = u i , j h + A i , j h 4 + O ( h 6 ) , (5)

where As are the coefficients of the leading error term.

For the (even, even) fine grid points, we could directly inject extrapolated coarse grid solution to get sixth order solution. The correction between the fourth order solution and the extrapolated sixth order solution, say c i , j h , can be used to approximate the leading error term A i , j h 4 . By using Equation (3), we can deduce

A i , j h 4 = c i , j h = 1 15 ( u i , j h − u i / 2 , j / 2 2 h ) , i = even , j = even (6)

Then, we could use the correction (6) to approximate corrections for other three subsets of fine grid points, and thus compute sixth order solutions for them.

For the (odd, odd) fine grid points, the rotated grid interpolation, as

A i , j = 1 4 ( A i + 1 , j + 1 + A i + 1 , j − 1 + A i − 1 , j + 1 + A i − 1 , j − 1 ) + O ( h 2 ) ; i = odd , j = odd . (7)

is used to obtain the formula for sixth order solution computation as

u ˜ i , j h = u i , j h + c i , j h , i = o d d , j = o d d ; (8)

where

c i , j h = 1 4 ( c i + 1 , j + 1 h + c i + 1 , j − 1 h + c i − 1 , j + 1 h + c i − 1 , j − 1 h ) .

For the (odd, even) and (even, odd) fine grid points, the standard grid interpolation, as

A i , j = 1 4 ( A i + 1 , j + A i − 1 , j + A i , j + 1 + A i , j − 1 ) + O ( h 2 ) , i = odd , j = even ; i = even , j = odd . (9)

is used to obtain the formula for sixth order solution computation as

u ˜ i , j h = u i , j h + c i , j h , i = odd , j = even ; i = even , j = odd ; (10)

where

c i , j h = 1 4 ( c i + 1 , j h + c i − 1 , j h + c i , j + 1 h + c i , j − 1 h ) .

In this section, we will give an analysis of truncation errors to compare the accuracy of three Richardson extrapolation-based sixth order methods described in Section 2. All of these methods need to use FOC schemes to get the fourth order solutions on fine and coarse grids. We first analyze the truncation error of the FOC schemes. For more general applications, we will derive the truncation error for the FOC scheme with unequal meshsizes [

Denote Δ x and Δ y to be the meshsizes in the x and y coordinate directions, respectively. The standard second order central difference operators are

δ x 2 u i , j = u i + 1 , j − 2 u i , j + u i − 1 , j Δ x 2 , δ y 2 u i , j = u i , j + 1 − 2 u i , j + u i , j − 1 Δ y 2

By using Taylor series, we have

δ x 2 u i , j = u x x + Δ x 2 12 u x 4 + Δ x 4 360 u x 6 + Δ x 6 20160 u x 8 + O ( Δ x 8 ) , (11)

and

δ y 2 u i , j = u y y + Δ y 2 12 u y 4 + Δ y 4 360 u y 6 + Δ y 6 20160 u y 8 + O ( Δ y 8 ) . (12)

From Equations (11) and (12) we can discretize Equation (1) at the grid point x i , j as

δ x 2 u i , j + δ y 2 u i , j = f i , j + 1 12 ( Δ x 2 u x 4 + Δ y 2 u y 4 ) + 1 360 ( Δ x 4 u x 6 + Δ y 4 u y 6 ) + 1 20160 ( Δ x 6 u x 8 + Δ y 6 u y 8 ) + O ( Δ 8 ) . (13)

By taking two times partial derivatives of x and y on both sides of Equation (1), respectively, we have

u x 4 = f x x − u y y x x , (14)

and

u y 4 = f x x − u x x y y . (15)

Using central difference operators and Taylor series in Equations (14) and (15) gives

( u x 4 ) i , j = δ x 2 f i , j − 1 Δ y 2 ( δ x 2 u i , j + 1 − 2 δ x 2 u i , j + δ x 2 u i , j − 1 ) − Δ x 2 12 f x 4 − Δ x 4 360 f x 6 − 1 Δ y 2 ( − Δ x 2 12 ( Δ y 2 u x 4 y 2 + Δ y 4 12 u x 4 y 4 ) − Δ x 4 360 Δ y 2 u x 6 y 2 ) + Δ y 2 12 u x 2 y 4 + Δ y 4 360 u x 2 y 6 + O ( Δ 6 ) , (16)

and

( u y 4 ) i , j = δ y 2 f i , j − 1 Δ x 2 ( δ y 2 u i + 1 , j − 2 δ y 2 u i , j + δ y 2 u i − 1 , j ) − Δ y 2 12 f y 4 − Δ y 4 360 f y 6 − 1 Δ x 2 ( − Δ y 2 12 ( Δ x 2 u x 2 y 4 + Δ x 4 12 u x 4 y 4 ) − Δ y 4 360 Δ x 2 u x 2 y 6 ) + Δ x 2 12 u x 4 y 2 + Δ x 4 360 u x 6 y 2 + O ( Δ 6 ) . (17)

By continuously taking partial derivatives of x on both sides of Equation (14), we get

f x 4 = u x 6 + u x 4 y 2 (18)

f x 6 = u x 8 + u x 6 y 2 . (19)

Similarly, by continuously taking partial derivatives of y on both sides of Equation (15), we get

f y 4 = u y 6 + u x 2 y 4 (20)

f y 6 = u y 8 + u x 2 y 6 . (21)

Substituting Equations (18) and (19) in Equation (16) gives

( u x 4 ) i , j = δ x 2 f i , j − 1 Δ y 2 ( δ x 2 u i , j + 1 − 2 δ x 2 u i , j + δ x 2 u i , j − 1 ) − Δ x 2 12 u x 6 + Δ y 2 12 u x 2 y 4 + Δ x 2 Δ y 2 144 u x 4 y 4 − Δ x 4 360 u x 8 + Δ y 4 360 u x 2 y 6 + O ( Δ 6 ) . (22)

And, substituting Equations (20) and (21) in Equation (17) gives

( u y 4 ) i , j = δ y 2 f i , j − 1 Δ x 2 ( δ y 2 u i + 1 , j − 2 δ y 2 u i , j + δ y 2 u i − 1 , j ) − Δ y 2 12 u y 6 + Δ x 2 12 u x 4 y 2 + Δ x 2 Δ y 2 144 u x 4 y 4 − Δ y 4 360 u y 8 + Δ x 4 360 u x 6 y 2 + O ( Δ 6 ) . (23)

Then, we use Equations (22) and (23) to replace the u x 4 and u y 4 terms in Equation (13) as

δ x 2 u i , j + δ y 2 u i , j = f i , j + 1 12 ( Δ x 2 δ x 2 f i , j + Δ y 2 δ y 2 f i , j ) − 1 12 ( Δ x 2 Δ y 2 ( δ x 2 u i , j + 1 − 2 δ x 2 u i , j + δ x 2 u i , j − 1 ) + Δ y 2 Δ x 2 ( δ y 2 u i + 1 , j − 2 δ y 2 u i , j + δ y 2 u i − 1 , j ) ) + ( τ 4 ) i , j + ( τ 6 ) i , j + O ( Δ 8 ) , (24)

where

( τ 4 ) i , j = 1 144 ( u x 4 y 2 + u x 2 y 4 ) Δ x 2 Δ y 2 − 1 240 ( u x 6 Δ x 4 + u y 6 Δ y 4 ) ,

( τ 6 ) i , j = 1 1728 ( Δ x 4 Δ y 2 + Δ x 2 Δ y 4 ) u x 4 y 4 + 1 4320 ( Δ x 2 Δ y 4 u x 2 y 6 + Δ x 4 Δ y 2 u x 6 y 2 ) − 11 60480 ( Δ x 6 u x 8 + Δ y 6 u y 8 ) .

Let us use the second order central difference operators in Equation (24),

multiply 6 Δ x 2 on both sides, and denote the mesh aspect ratio λ = Δ x Δ y , we can

get a general FOC scheme like the one presented in [

m 1 ( u i + 1 , j + 1 + u i + 1 , j − 1 + u i − 1 , j + 1 + u i − 1 , j − 1 ) + m 2 ( u i , j + 1 + u i , j − 1 ) + m 3 ( u i + 1 , j + u i − 1 , j ) − m 4 u i , j = Δ x 2 2 ( 8 f i , j + f i + 1 , j + f i − 1 , j + f i , j + 1 + f i , j − 1 ) , (25)

where the coefficients are

m 1 = ( 1 + λ 2 ) / 2 , m 2 = 5 λ 2 − 1 , m 3 = 5 − λ 2 , m 4 = 10 ( 1 + λ 2 ) .

The fourth order truncation error of the FOC scheme (25) is

τ ˜ 4 = { 1 24 λ 2 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 λ 4 ) } Δ x 4 . (26)

And, the sixth order truncation error of the FOC scheme (25) is

τ ˜ 6 = { 1 288 ( 1 λ 2 + 1 λ 4 ) u x 4 y 4 + 1 720 ( u x 2 y 6 λ 4 + u x 6 y 2 λ 2 ) − 11 10080 ( u x 8 + u y 8 λ 6 ) } Δ x 6 . (27)

In a special case with Δ x = Δ y = h , the FOC scheme has the form as

u i + 1 , j + 1 + u i + 1 , j − 1 + u i − 1 , j + 1 + u i − 1 , j − 1 + 4 ( u i , j + 1 + u i , j − 1 + u i + 1 , j + u i − 1 , j ) − 20 u i , j = h 2 2 ( 8 f i , j + f i + 1 , j + f i − 1 , j + f i , j + 1 + f i , j − 1 ) . (28)

The fourth order and sixth order truncation errors of the FOC scheme (28) are

τ FOC4 = { 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 ) } h 4 , (29)

τ FOC6 = { 1 144 u x 4 y 4 + 1 720 ( u x 2 y 6 + u x 6 y 2 ) − 11 10080 ( u x 8 + u y 8 ) } h 6 . (30)

Now we can take a look at the truncation error after applying Richardson extrapolation. From the definition of the fourth order accurate solutions on the fine and coarse grids, we have

u h ∗ = u h 4 + τ FOC4 + τ FOC6 , (31)

u 2 h ∗ = u 2 h 4 + 16 τ FOC4 + 64 τ FOC6 . (32)

Using the Richardson extrapolation Formula (3) gives

u 2 h ∗ = u 2 h 6 − 16 5 τ FOC6 . (33)

Thus, the sixth order truncation error after applying Richardson extrapolation has the form as

τ Extrapo = − 16 5 τ FOC6 . (34)

For all Richardson extrapolation-based sixth order compact approximations, the truncation error of (even, even) fine grid points is τ Extrapo because the corresponding sixth order solution is injected from the extrapolated solution of the standard coarse grid. For other three subsets of fine grid points, three computational strategies (operator based interpolation, multiple coarse grid computation, and completed Richardson extrapolation) could be used to obtain sixth order solutions. In the following part, the truncation error analysis for these methods are given respectively.

The operator based interpolation scheme is from the 9-point FOC scheme (28), which can be iteratively used to approach sixth order solutions for (odd, odd), (odd, even) and (even, odd) fine grid points. It generates a sixth order error in the form as

τ o p = τ FOC4 h 2 = { 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 ) } h 6 . (35)

The operator based interpolation Equation (4) can be written as

u i , j ∗ = u ˜ i , j + τ o p = − 1 20 [ F i , j − 4 ( u i + 1 , j + u i − 1 , j + u i , j + 1 + u i , j − 1 ) − ( u i + 1 , j + 1 + u i + 1 , j − 1 + u i − 1 , j + 1 + u i − 1 , j − 1 ) ] + τ o p . (36)

For any linear equation A u = F , there is a corresponding residual equation A e = r , where e is the error and r is the residual. According to Equation (36), we could get the operator based interpolation on error as

0 = e ˜ i , j + τ o p = − 1 20 [ r i , j − 4 ( e i + 1 , j + e i − 1 , j + e i , j + 1 + e i , j − 1 ) − ( e i + 1 , j + 1 + e i + 1 , j − 1 + e i − 1 , j + 1 + e i − 1 , j − 1 ) ] + τ o p . (37)

When the iterative process of using operator based interpolation converges, the residual r tends to 0. Then Equation (37) becomes to

0 = 1 20 [ 4 ( e i + 1 , j + e i − 1 , j + e i , j + 1 + e i , j − 1 ) + ( e i + 1 , j + 1 + e i + 1 , j − 1 + e i − 1 , j + 1 + e i − 1 , j − 1 ) ] + τ o p . (38)

At this time, the main component of the error is the truncation error.

Assume the truncation errors on (odd, odd), (odd, even) and (even, odd) fine grid points as α o p , β o p and γ o p , respectively. A system of equations on the truncation error for three subsets of fine grid points is generated from Equation (38) as

{ 4 × τ Extrapo + 4 × ( β o p + β o p ) + 4 × ( β o p + β o p ) − 20 α o p = − τ o p , i = odd , j = odd ; 4 × β o p + 4 × ( τ Extrapo + τ Extrapo ) + 4 × ( α o p + α o p ) − 20 β o p = − τ o p , i = odd , j = even ; 4 × γ o p + 4 × ( α o p + α o p ) + 4 × ( τ Extrapo + τ Extrapo ) − 20 γ o p = − τ o p , i = even , j = odd . (39)

From Equation (39), we get

{ α o p = τ Extrapo + 1 6 τ o p , i = odd , j = odd ; β o p = τ Extrapo + 7 48 τ o p , i = odd , j = even ; γ o p = τ Extrapo + 7 48 τ o p , i = even , j = odd . (40)

After applying Richardson extrapolation to get sixth order solutions for (even, even) fine grid points, in the multiple coarse grid computation, X-odd grid view and Y-odd grid view are constructed to compute sixth order solutions for (odd, even) and (even, odd) fine grid points, respectively [

τ x -odd = { 4 × 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + 16 × u y 6 ) } h 6 , λ x -odd = 1 2 ; (41)

τ y -odd = 4 × { 4 × 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( 16 × u x 6 + u y 6 ) } h 6 , λ y -odd = 2. (42)

As for the computation of (odd, even) fine grid points on the X-odd grid view

with the mesh aspect ratio λ x -odd = 1 2 , the coefficients in Equation (25) are set as

m 1 = 5 8 , m 2 = 1 4 , m 3 = 19 4 , m 4 = 50 4 .

Assume the truncation error of (odd, even) fine grid points to be α m c g , an equation upon the error of X-odd grid view is generated from Equation (25) with the above coefficients as

5 8 × 4 τ Extrapo + 1 4 × ( α m c g + α m c g ) + 19 4 × ( τ Extrapo + τ Extrapo ) − 50 4 α = − τ x -odd , (43)

which gives

α m c g = τ Extrapo + τ x -odd 12 . (44)

As for the computation of (even, odd) fine grid points on the Y-odd grid view with the mesh aspect ratio λ y -odd = 2 , the coefficients in Equation (25) are set as

m 1 = 5 2 , m 2 = 19 , m 3 = 1 , m 4 = 50.

Assume the truncation error of (even, odd) fine grid points to be β m c g , an equation upon the error of Y-odd grid view is generated from Equation (25) with the above coefficients as

5 2 × 4 τ Extrapo + 19 × ( τ Extrapo + τ Extrapo ) + 1 × ( β m c g + β m c g ) − 50 β m c g = − τ y -odd , (45)

which gives

β m c g = τ Extrapo + τ y -odd 48 . (46)

The update for (odd, odd) fine grid points uses the operator based interpolation Equation (4) and updated sixth order solutions of (even, even), (odd, even) and (even, odd) fine grid points. Assume the truncation error of (odd, odd) fine grid points to be γ m c g , an equation upon the error of the fine grid points is generated by Equation (36) as

4 τ Extrapo + 4 × ( α m c g + α m c g ) + 4 × ( β m c g + β m c g ) − 20 γ m c g = − τ o p , (47)

which gives

γ m c g = τ Extrapo + τ x -odd 30 + τ y -odd 120 + τ o p 20 . (48)

The completed Richardson extrapolation uses two kinds of interpolation on the correction Equation (6) of (even, even) fine grid points to approximate corrections for other fine grid points. As we all know, interpolation brings error. Next, we will analyze how much of the error.

The A coefficients in Equations (7) and (9) can be viewed as a function of u which has the form of A ( u ) = τ FOC4 / h 4 . Based on the Taylor series expansion, the O ( h 2 ) term in Equation (7) has an explicit form as the

h 2 2 ( ∂ 2 ( τ FOC4 / h 4 ) ∂ x 2 + ∂ 2 ( τ FOC4 / h 4 ) ∂ y 2 ) ; while the O ( h 2 ) term in Equation (9) has an explicit form as h 2 4 ( ∂ 2 ( τ FOC4 / h 4 ) ∂ x 2 + ∂ 2 ( τ FOC4 / h 4 ) ∂ y 2 ) .

The second order truncation error for Equation (7), which uses rotated grid interpolation to approximate (odd, odd) fine grid points, has the form as

τ RotateInter = { 1 24 u x 4 y 4 + 1 120 ( u x 6 y 2 + u x 2 y 6 ) − 1 80 ( u x 8 + u y 8 ) } h 2 ; (49)

while the second order truncation error for Equation (9), which uses standard grid interpolation to approximate (odd, even) and (even, odd) fine grid points, has the form as

τ StandInter = { 1 48 u x 4 y 4 + 1 240 ( u x 6 y 2 + u x 2 y 6 ) − 1 160 ( u x 8 + u y 8 ) } h 2 . (50)

We find that τ RotateInter = 2 τ StandInter .

Consider (odd, odd) fine grid points at first. Equation (7) can be re-written as

A i , j = 1 4 ( A i + 1, j + 1 + A i + 1, j − 1 + A i − 1, j + 1 + A i − 1, j − 1 ) + τ RotateInter , i = odd , j = odd . (51)

The sixth order computation for the (odd, odd) fine grid points is only related to (even, even) fine grid points. For the (even, even) fine grid points, by using definition of fourth order solutions obtained from the FOC scheme, we have

A even,even = 1 h 4 [ u even,even ∗ − u even,even 4 − τ FOC6 ] . (52)

After injecting the extrapolated coarse grid solution into the fine grid, we have

u even,even ∗ = u even,even 6 + τ Extrapo . (53)

Substituting Equation (53) into Equation (52) gives

A even,even = 1 h 4 [ u even,even 6 − u even,even 4 − τ FOC6 + τ Extrapo ] = 1 h 4 [ c even,even − τ FOC6 + τ Extrapo ] . (54)

By using Equations (8), (51) and (54), we get the truncation error of the (odd, odd) fine grid points as

τ CompEx1 = u i , j ∗ − u i , j 6 = ( u i , j 4 + A i , j h 4 + τ FOC6 ) − ( u i , j 4 + 1 4 ( c i + 1 , j + 1 + c i + 1 , j − 1 + c i − 1 , j + 1 + c i − 1 , j − 1 ) ) = ( u i , j 4 + A i , j h 4 + τ FOC6 ) − ( u i , j 4 + 1 4 ( A i + 1 , j + 1 + A i + 1 , j − 1 + A i − 1 , j + 1 + A i − 1 , j − 1 ) h 4 + τ FOC6 − τ Extrapo ) = τ RotateInter h 4 + τ Extrapo , i = odd , j = odd . (55)

Then consider (odd, even) and (even, odd) fine grid points. Equation (9) can be re-written as

A i , j = 1 4 ( A i + 1 , j + A i − 1 , j + A i , j + 1 + A i , j − 1 ) + τ StandInter , i = odd , j = even ; i = even , j = odd . (56)

The sixth order computation for the (odd, even) and (even, odd) fine grid points are related to both (even, even) and (odd, odd) fine grid points. For the updated (odd, odd) fine grid points, we have

u odd,odd ∗ = u odd,odd 6 + τ CompEx1 = u odd,odd 6 + τ RotateInter h 4 + τ Extrapo . (57)

By using the definition of fourth order solutions obtained from the FOC scheme, we have

A odd,odd = 1 h 4 [ u odd,odd ∗ − u odd,odd 4 − τ FOC6 ] . (58)

Substituting Equation (57) into Equation (58) gives

A odd,odd = 1 h 4 [ u odd,odd 6 − u odd,odd 4 − τ FOC6 + τ RotateInter h 4 + τ Extrapo ] = 1 h 4 [ c odd,odd − τ FOC6 + τ RotateInter h 4 + τ Extrapo ] . (59)

By using Equations (10), (54), (56) and (59), we get the truncation errors of the (even, odd) and (odd, even) fine grid points as

τ CompEx2 = u i , j ∗ − u i , j 6 = ( u i , j 4 + A i , j h 4 + τ FOC6 ) − ( u i , j 4 + 1 4 ( c i + 1 , j + c i − 1 , j + c i , j + 1 + c i , j − 1 ) ) = ( u i , j 4 + A i , j h 4 + τ FOC6 ) − ( u i , j 4 + 1 4 ( A i + 1 , j + A i − 1 , j + A i , j + 1 + A i , j − 1 ) h 4 + τ FOC6 − 1 2 τ RotateInter h 4 − τ Extrapo ) = τ StandInter h 4 + 1 2 τ RotateInter h 4 + τ Extrapo = τ RotateInter h 4 + τ Extrapo , i = odd , j = even ; i = even , j = odd . (60)

We find that the truncation errors of (odd, odd), (odd, even) and (even, odd) fine grid points have the same form as ( τ RotateInter h 4 + τ Extrapo ), which is larger than the truncation error of (even, even) fine grid points ( τ Extrapo ) generated from Richardson extrapolation as we expect. It is because another interpolation is involved, i.e., Equation (55) or Equation (60).

In summary, these three Richardson extrapolation-based methods are able to compute the sixth order accurate solution on the entire fine grid. For (even, even) fine grid points, all methods use Richardson extrapolation to get the sixth order solution with truncation error τ Extrapo . For other three subsets of fine grid points, different computational strategies are used to obtain sixth order solutions, which add errors of different magnitude on the truncation error τ Extrapo .

Richardson extrapolation with operator based interpolation | |
---|---|

(even, even) points | τ Extrapo |

(odd, even) points | τ Extrapo + 7 48 [ 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 ) ] h 6 |

(even, odd) points | τ Extrapo + 7 48 [ 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 ) ] h 6 |

(odd, odd) points | τ Extrapo + 8 48 [ 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 ) ] h 6 |

Richardson extrapolation with multiple coarse grid computation | |

(even, even) points | τ Extrapo |

(odd, even) points | τ Extrapo + 1 12 { 4 × 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + 16 × u y 6 ) } h 6 |

(even, odd) points | τ Extrapo + 1 12 { 4 × 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( 16 × u x 6 + u y 6 ) } h 6 |

(odd, odd) points | τ Extrapo + { 1 30 [ 4 × 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + 16 × u y 6 ) ] + 1 30 [ 4 × 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( 16 × u x 6 + u y 6 ) ] + 1 20 [ 1 24 ( u x 4 y 2 + u x 2 y 4 ) − 1 40 ( u x 6 + u y 6 ) ] } h 6 |

Completed Richardson extrapolation | |

(even, even) points | τ Extrapo |

(odd, even) points | τ Extrapo + { 1 24 u x 4 y 4 + 1 120 ( u x 6 y 2 + u x 2 y 6 ) − 1 120 ( u x 8 + u y 8 ) } h 6 |

(even, odd) points | τ Extrapo + { 1 24 u x 4 y 4 + 1 120 ( u x 6 y 2 + u x 2 y 6 ) − 1 120 ( u x 8 + u y 8 ) } h 6 |

(odd, odd) points | τ Extrapo + { 1 24 u x 4 y 4 + 1 120 ( u x 6 y 2 + u x 2 y 6 ) − 1 120 ( u x 8 + u y 8 ) } h 6 |

We tested three Richardson extrapolation-based sixth order methods on two 2D Poisson equations. The 9-point FOC scheme (28) was used to get fourth order solutions on different scale grid levels.

One merit of using Richardson extrapolation for sixth order computation lies in that we can easily choose highly efficient solvers for the resulting large sparse linear systems. In our experiments, we chose a multiscale multigrid (MSMG) computational framework [_{2}-norm of the difference of the successive solutions was reduced by a factor of 10^{10}. The stopping criteria for the iterative operator based interpolation and Gauss-Seidel procedure was 10^{−10}. All reported errors were the maximum absolute errors over the discrete grid of the finest level.

The codes were written in Fortran 77 programming language and run on a PC, which has Intel Core i7-4770 with 3.40 GHz and 16 GB RAM.

Problem 1.

− ∂ 2 u ∂ x 2 − ∂ 2 u ∂ y 2 = α sin ( π b y ) , ( x , y ) ∈ Ω = [ 0 , λ ] × [ 0 , b ] ,

where the boundary conditions are

u ( 0 , y ) = u ( λ , y ) = u ( x , 0 ) = u ( x , b ) = 0.

The parameters are chosen as

α = F π R b , λ = 10 7 m , b = 2 π × 10 6 m , F = 0.3 × 10 − 7 m 2 ⋅ s − 2 , R = 0.6 × 10 − 3 m ⋅ s − 1 .

The analytical solution is

u = − α ( b π ) 2 sin ( π y b ) ( e π x b − 1 ) .

Problem 2.

− ∂ 2 u ∂ x 2 − ∂ 2 u ∂ y 2 = 2 π 2 sin ( π x ) cos ( π y ) , ( x , y ) ∈ Ω = [ 0 , 4 ] × [ 0 , 1 ] ,

which has the Dirichlet boundary condition.

The analytical solution is

u ( x , y ) = sin ( π x ) cos ( π y ) .

We refined the grid from N = 32 to N = 256 for both test problems. For convenience, we used three abbreviations to represent three Richardson extrapolation-based sixth order methods to be compared. “Op-Six” is short for the sixth order method with Richardson extrapolation and operator based interpolation; “MCG-Six” means the sixth order method with Richardson extrapolation and multiple coarse grid computation; “CR-Six” denotes the sixth order method with completed Richardson extrapolation.

For both test problems, we compare the accuracy and efficiency of three methods by computing maximum errors and accuracy order, and recording the CPU time in seconds. The results of Problem 1 are shown in

N | Op-Six | MCG-Six | CR-Six | |||
---|---|---|---|---|---|---|

Error | Order | Error | Order | Error | Order | |

32 | 2.824e−8 | - | 6.237e−8 | - | 1.861e−8 | - |

64 | 4.296e−10 | 6.04 | 9.881e−10 | 5.98 | 2.756e−10 | 6.08 |

128 | 6.785e−12 | 5.98 | 1.716e−11 | 5.85 | 5.016e−12 | 5.78 |

256 | 1.062e−13 | 6.00 | 2.429e−13 | 6.14 | 6.833e−14 | 6.20 |

N | Op-Six CPU | MCG-Six CPU | CR-Six CPU |
---|---|---|---|

32 | 0.0156 | 0.0156 | 0.0156 |

64 | 0.109 | 0.0936 | 0.0936 |

128 | 0.172 | 0.125 | 0.140 |

256 | 0.920 | 0.764 | 0.781 |

N | Op-Six | MCG-Six | CR-Six | |||
---|---|---|---|---|---|---|

Error | Order | Error | Order | Error | Order | |

32 | 2.498e−6 | - | 2.278e−6 | - | 8.927e−7 | - |

64 | 4.582e−8 | 5.77 | 3.624e−8 | 5.97 | 1.362e−8 | 6.03 |

128 | 7.663e−10 | 5.90 | 5.710e−10 | 5.99 | 2.105e−10 | 6.02 |

256 | 1.237e−11 | 5.95 | 8.962e−12 | 5.99 | 3.271e−12 | 6.01 |

N | Op-Six CPU | MCG-Six CPU | CR-Six CPU |
---|---|---|---|

32 | 0.0312 | 0 0156 | 0.0156 |

64 | 0.0936 | 0.0468 | 0.0468 |

128 | 0.250 | 0.140 | 0.172 |

256 | 1.248 | 0.780 | 0.796 |

As for the accuracy, it is clear that all the methods are able to obtain approximate solutions with sixth order accuracy. After comparing the errors of the three methods, we found that, for both test problmes, the solutions solved by the CR-Six method were more accurate than those solved by the Op-Six method and the MCG-Six method. This observation is consistent with our analysis in Section 3. Meanwhile, we notice that the Op-Six computed a little bit more accurate solutions than the MCG-Six method for Problem 1 yet reversed for Problem 2, which shows that there is no fixed conclusion about the accuracy comparison between them.

As for the efficiency, we found that the MCG-Six method and the CR-Six method required less CPU cost than the Op-Six method. The reason is that the Op-Six method involves an iterative refinement procedure with low convergence rate. There is no evident difference between the MCG-Six method and the CR-Six method on CPU cost.

Although the CR-Six method performed better than the other two methods on accuracy and efficiency for the above two test problems, we need to point out that this advantage is for “simple” problems with “good” conditions. Here the “simple” and “good” mean those problems which are not hard to solve (e.g., small Reynolds number) and have smooth solutions, forcing functions and coefficients in the domain. Since the success of CR-Six method relies on the effectiveness of interpolation of the corrections, sufficiently smooth are necessary to guarantee an effective interpolation. We think the CR-Six method may ask for more restrictions on equations than the other two methods for the sixth order approximations on the finest grid. The robustness analysis and numerical experiments on various “difficult” problems is worth exploring in the future.

Compared to the sixth order compact schemes derived by Hermitian polynomial, the Richardson extrapolation-based sixth order compact approximations have many obvious advantages, such as simple stencils, complete compact, easy implementation, suitable for high efficient linear system solvers, etc. We studied three Richardson extrapolation-based sixth order compact computation methods and analyzed the truncation errors of them respectively. All three methods were able to compute the sixth order accurate solution on the fine grid if Richardson extrapolation is applied to the fourth order solutions on fine and coarse grids successfully in the domain. From the truncation error analysis, we got a possible qualitative relationship on the accuracy among these sixth order methods, although it is not completely accurate. We also discussed the multigrid method which is suitable for the Richardson extrapolation-based sixth order methods.

Two 2D Poisson equations are tested in the numerical part. We compared the accuracy and efficiency among three sixth order methods. The test results on accuracy are generally consistent with the observation from the theoretical analysis. The completed Richardson extrapolation method usually performs better in accuracy than the operator based interpolation with Richardson extrapolation method and the multiple coarse grid computation with Richardson extrapolation method. As expected, the efficiency of the operator based interpolation with Richardson extrapolation is lower than the other two sixth order methods.

The exploration of using Richardson extrapolation on sixth order compact computation for high dimensional problems has already begun. In fact, the operator based interpolation with Richardson extrapolation method and the multiple coarse grid computation with Richardson extrapolation method have been used to solve 3D convection-diffusion equations and show exciting performance [

Since Richardson extrapolation-based sixth order methods require two comparable uniform grids (i.e., fine grid with meshsize h and coarse grid with meshsize 2h), the application of this technique to more practical problems in irregular domain with various boundary conditions (i.e., Dirichlet, Neumann, or mixed conditions) has limitations and is not straight forward. One undergoing study is to use the proposed methodology with finite difference ghost-cell technique to obtain high accuracy high efficiency solutions for Poisson equation with mixed boundary conditions. The finite difference ghost-cell technique [

Dai, R.X. and Lin, P.P. (2018) Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation. Journal of Applied Mathematics and Physics, 6, 1139-1159. https://doi.org/10.4236/jamp.2018.66097