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使用有限差分计算一阶和二阶导数时，边界处存在较大误差

Jagannath Mahapatra 发表于 Dev
18
贾加纳斯·马哈帕特拉（Jagannath Mahapatra）

我正在使用有限差分计算存储在3d数组a（n1，n2，n3）中的函数a（x，y，z）的一阶和二阶导数。在此，边界处的功能值为零。这是Fortran中的代码：

implicit none
integer i1,i2,i3

integer, parameter :: n1 = 33
integer, parameter :: n2 = 33
integer, parameter :: n3 = 32

real*8 pi, a(n1,n2,n3), a2(n1,n2,n3), z(n3),x(n1),y(n2),h,a1(n1,n2,n3)
real*8 num(n1,n2,n3),deno(n1,n2,n3),diff(n1,n2,1),A0,dx,dy

pi=3.14159265358979323846d0
dx=2.0d0*pi/(n1-1)
dy=4.0d0*pi/(n2-1)

do i1=1,n1
  x(i1)=-pi+(i1-1)*dx
  do i2=1,n2
    y(i2)=-2.0d0*pi+(i2-1)*dy
    do i3=1,n3
      z(i3)=(i3-1)*2.0d0*pi/n3
      a(i1,i2,1)= dcos(x(i1)/2.0d0) * dcos(y(i2)/4.0d0)  !input array
      a1(i1,i2,1)= - 0.25d0*dcos(x(i1)/2.0d0) * dsin(y(i2)/4.0d0)  !analytical expression of first order y-derivative
    enddo
  enddo
enddo

do i1=1,n1
  do i2=1,n2
    write(20,*)x(i1),y(i2),a(i1,i2,1)
  enddo
enddo
call d1y(n1,n2,n3,a,a2)
do i1=1,n1
  do i2=1,n2
     num(i1,i2,1)=(a2(i1,i2,1)-a1(i1,i2,1))  !numerator of error calculation
     deno(i1,i2,1)=a2(i1,i2,1)               !denomenator of error calculation
     if (dabs(deno(i1,i2,1)) .lt. 1e-10)deno(i1,i2,1)=1.0d0
     diff(i1,i2,1)=dabs(num(i1,i2,1))/dabs(deno(i1,i2,1))  !relative error in 1st order derivative calculation
    write(21,*)x(i1),y(i2),a(i1,i2,1),a2(i1,i2,1),diff(i1,i2,1),a1(i1,i2,1)
    write(21,*)
  enddo
enddo
end

subroutine d1y(n1,n2,n3,a,a2)
implicit none
integer n1, n2, n3, i1, i2, i3
real*8 pi, a(n1,n2,n3), a2(n1,n2,n3), z(n3),x(n1),y(n2),h,a1(n1,n2,n3)
pi=3.14159265358979323846d0
h=4.0d0*pi/(n2-1)

do i1=1,n1
   do i3=1,n3
      do i2=1,n2
         if(i2 .eq. 1)then
            a2(i1,i2,i3)=( -3.0d0*a(i1,i2,i3) + 4.0d0*a(i1,i2+1,i3) - a(i1,i2+2,i3) )/ (2.0d0*h)
          else if(i2 .eq. n2)then
            a2(i1,i2,i3)=( 3.0d0*a(i1,i2,i3) - 4.0d0*a(i1,i2-1,i3) + a(i1,i2-2,i3) )/ (2.0d0*h)
          else
            a2(i1,i2,i3)=( a(i1,i2+1,i3) - a(i1,i2-1,i3) )/ (2.0d0*h)
          endif
       enddo
    enddo
 enddo     
end subroutine

我的输入函数a(i1,i2,1)= dcos(x(i1)/2.0d0) * dcos(y(i2)/4.0d0)，因此是示例输入数据（对于17 * 17 * 16的网格）

    -3.1415926535897931       -6.2831853071795862        3.7493994566546440E-033
  -3.1415926535897931       -5.4977871437821380        1.1945836920083898E-017
  -3.1415926535897931       -4.7123889803846897        2.3432602026631496E-017
  -3.1415926535897931       -3.9269908169872414        3.4018865378450254E-017
  -3.1415926535897931       -3.1415926535897931        4.3297802811774670E-017
  -3.1415926535897931       -2.3561944901923448        5.0912829964730140E-017
  -3.1415926535897931       -1.5707963267948966        5.6571305614385013E-017
  -3.1415926535897931      -0.78539816339744828        6.0055777714832775E-017
  -3.1415926535897931        0.0000000000000000        6.1232339957367660E-017
  -3.1415926535897931       0.78539816339744828        6.0055777714832775E-017
  -3.1415926535897931        1.5707963267948966        5.6571305614385013E-017
  -3.1415926535897931        2.3561944901923439        5.0912829964730146E-017
  -3.1415926535897931        3.1415926535897931        4.3297802811774670E-017
  -3.1415926535897931        3.9269908169872423        3.4018865378450242E-017
  -3.1415926535897931        4.7123889803846897        2.3432602026631496E-017
  -3.1415926535897931        5.4977871437821371        1.1945836920083910E-017
  -3.1415926535897931        6.2831853071795862        3.7493994566546440E-033
  -2.7488935718910690       -6.2831853071795862        1.1945836920083898E-017
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  -2.7488935718910690       -4.7123889803846897        7.4657834050342639E-002
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  -2.7488935718910690       -1.5707963267948966       0.18023995550173702     
  -2.7488935718910690      -0.78539816339744828       0.19134171618254495     
  -2.7488935718910690        0.0000000000000000       0.19509032201612833     
  -2.7488935718910690       0.78539816339744828       0.19134171618254495     
  -2.7488935718910690        1.5707963267948966       0.18023995550173702     
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  -2.7488935718910690        4.7123889803846897        7.4657834050342639E-002
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  -2.7488935718910690        6.2831853071795862        1.1945836920083898E-017
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  -2.3561944901923448        5.4977871437821371        7.4657834050342722E-002
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  -1.5707963267948966       -1.5707963267948966       0.65328148243818829     
  -1.5707963267948966      -0.78539816339744828       0.69351992266107376     
  -1.5707963267948966        0.0000000000000000       0.70710678118654757     
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  -1.5707963267948966        1.5707963267948966       0.65328148243818829     
  -1.5707963267948966        2.3561944901923439       0.58793780120967942     
  -1.5707963267948966        3.1415926535897931       0.50000000000000011     
  -1.5707963267948966        3.9269908169872423       0.39284747919355101     
  -1.5707963267948966        4.7123889803846897       0.27059805007309856     
  -1.5707963267948966        5.4977871437821371       0.13794968964147170     
  -1.5707963267948966        6.2831853071795862        4.3297802811774670E-017
  -1.1780972450961724       -6.2831853071795862        5.0912829964730140E-017
  -1.1780972450961724       -5.4977871437821380       0.16221167441072892     
  -1.1780972450961724       -4.7123889803846897       0.31818964514320852     
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  -1.1780972450961724       -3.1415926535897931       0.58793780120967942     
  -1.1780972450961724       -2.3561944901923448       0.69134171618254492     
  -1.1780972450961724       -1.5707963267948966       0.76817775671141630     
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  -1.1780972450961724        2.3561944901923439       0.69134171618254503     
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  -1.1780972450961724        6.2831853071795862        5.0912829964730140E-017
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 -0.78539816339744828        1.5707963267948966       0.85355339059327373     
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 -0.78539816339744828        5.4977871437821371       0.18023995550173721     
 -0.78539816339744828        6.2831853071795862        5.6571305614385013E-017
 -0.39269908169872414       -6.2831853071795862        6.0055777714832775E-017
 -0.39269908169872414       -5.4977871437821380       0.19134171618254495     
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 -0.39269908169872414       -3.9269908169872414       0.54489510677581865     
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   0.0000000000000000       -6.2831853071795862        6.1232339957367660E-017
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   1.1780972450961720        3.1415926535897931       0.58793780120967942     
   1.1780972450961720        3.9269908169872423       0.46193976625564331     
   1.1780972450961720        4.7123889803846897       0.31818964514320858     
   1.1780972450961720        5.4977871437821371       0.16221167441072912     
   1.1780972450961720        6.2831853071795862        5.0912829964730146E-017
   1.5707963267948966       -6.2831853071795862        4.3297802811774670E-017
   1.5707963267948966       -5.4977871437821380       0.13794968964147156     
   1.5707963267948966       -4.7123889803846897       0.27059805007309856     
   1.5707963267948966       -3.9269908169872414       0.39284747919355117     
   1.5707963267948966       -3.1415926535897931       0.50000000000000011     
   1.5707963267948966       -2.3561944901923448       0.58793780120967942     
   1.5707963267948966       -1.5707963267948966       0.65328148243818829     
   1.5707963267948966      -0.78539816339744828       0.69351992266107376     
   1.5707963267948966        0.0000000000000000       0.70710678118654757     
   1.5707963267948966       0.78539816339744828       0.69351992266107376     
   1.5707963267948966        1.5707963267948966       0.65328148243818829     
   1.5707963267948966        2.3561944901923439       0.58793780120967942     
   1.5707963267948966        3.1415926535897931       0.50000000000000011     
   1.5707963267948966        3.9269908169872423       0.39284747919355101     
   1.5707963267948966        4.7123889803846897       0.27059805007309856     
   1.5707963267948966        5.4977871437821371       0.13794968964147170     
   1.5707963267948966        6.2831853071795862        4.3297802811774670E-017
   1.9634954084936211       -6.2831853071795862        3.4018865378450242E-017
   1.9634954084936211       -5.4977871437821380       0.10838637566236962     
   1.9634954084936211       -4.7123889803846897       0.21260752369181410     
   1.9634954084936211       -3.9269908169872414       0.30865828381745508     
   1.9634954084936211       -3.1415926535897931       0.39284747919355101     
   1.9634954084936211       -2.3561944901923448       0.46193976625564326     
   1.9634954084936211       -1.5707963267948966       0.51327996715933655     
   1.9634954084936211      -0.78539816339744828       0.54489510677581843     
   1.9634954084936211        0.0000000000000000       0.55557023301960207     
   1.9634954084936211       0.78539816339744828       0.54489510677581843     
   1.9634954084936211        1.5707963267948966       0.51327996715933655     
   1.9634954084936211        2.3561944901923439       0.46193976625564331     
   1.9634954084936211        3.1415926535897931       0.39284747919355101     
   1.9634954084936211        3.9269908169872423       0.30865828381745491     
   1.9634954084936211        4.7123889803846897       0.21260752369181410     
   1.9634954084936211        5.4977871437821371       0.10838637566236972     
   1.9634954084936211        6.2831853071795862        3.4018865378450242E-017
   2.3561944901923448       -6.2831853071795862        2.3432602026631496E-017
   2.3561944901923448       -5.4977871437821380        7.4657834050342639E-002
   2.3561944901923448       -4.7123889803846897       0.14644660940672630     
   2.3561944901923448       -3.9269908169872414       0.21260752369181418     
   2.3561944901923448       -3.1415926535897931       0.27059805007309856     
   2.3561944901923448       -2.3561944901923448       0.31818964514320852     
   2.3561944901923448       -1.5707963267948966       0.35355339059327384     
   2.3561944901923448      -0.78539816339744828       0.37533027751786530     
   2.3561944901923448        0.0000000000000000       0.38268343236508984     
   2.3561944901923448       0.78539816339744828       0.37533027751786530     
   2.3561944901923448        1.5707963267948966       0.35355339059327384     
   2.3561944901923448        2.3561944901923439       0.31818964514320858     
   2.3561944901923448        3.1415926535897931       0.27059805007309856     
   2.3561944901923448        3.9269908169872423       0.21260752369181410     
   2.3561944901923448        4.7123889803846897       0.14644660940672630     
   2.3561944901923448        5.4977871437821371        7.4657834050342722E-002
   2.3561944901923448        6.2831853071795862        2.3432602026631496E-017
   2.7488935718910685       -6.2831853071795862        1.1945836920083910E-017
   2.7488935718910685       -5.4977871437821380        3.8060233744356686E-002
   2.7488935718910685       -4.7123889803846897        7.4657834050342722E-002
   2.7488935718910685       -3.9269908169872414       0.10838637566236978     
   2.7488935718910685       -3.1415926535897931       0.13794968964147170     
   2.7488935718910685       -2.3561944901923448       0.16221167441072909     
   2.7488935718910685       -1.5707963267948966       0.18023995550173721     
   2.7488935718910685      -0.78539816339744828       0.19134171618254514     
   2.7488935718910685        0.0000000000000000       0.19509032201612853     
   2.7488935718910685       0.78539816339744828       0.19134171618254514     
   2.7488935718910685        1.5707963267948966       0.18023995550173721     
   2.7488935718910685        2.3561944901923439       0.16221167441072912     
   2.7488935718910685        3.1415926535897931       0.13794968964147170     
   2.7488935718910685        3.9269908169872423       0.10838637566236972     
   2.7488935718910685        4.7123889803846897        7.4657834050342722E-002
   2.7488935718910685        5.4977871437821371        3.8060233744356721E-002
   2.7488935718910685        6.2831853071795862        1.1945836920083910E-017
   3.1415926535897931       -6.2831853071795862        3.7493994566546440E-033
   3.1415926535897931       -5.4977871437821380        1.1945836920083898E-017
   3.1415926535897931       -4.7123889803846897        2.3432602026631496E-017
   3.1415926535897931       -3.9269908169872414        3.4018865378450254E-017
   3.1415926535897931       -3.1415926535897931        4.3297802811774670E-017
   3.1415926535897931       -2.3561944901923448        5.0912829964730140E-017
   3.1415926535897931       -1.5707963267948966        5.6571305614385013E-017
   3.1415926535897931      -0.78539816339744828        6.0055777714832775E-017
   3.1415926535897931        0.0000000000000000        6.1232339957367660E-017
   3.1415926535897931       0.78539816339744828        6.0055777714832775E-017
   3.1415926535897931        1.5707963267948966        5.6571305614385013E-017
   3.1415926535897931        2.3561944901923439        5.0912829964730146E-017
   3.1415926535897931        3.1415926535897931        4.3297802811774670E-017
   3.1415926535897931        3.9269908169872423        3.4018865378450242E-017
   3.1415926535897931        4.7123889803846897        2.3432602026631496E-017
   3.1415926535897931        5.4977871437821371        1.1945836920083910E-017
   3.1415926535897931        6.2831853071795862        3.7493994566546440E-033

输入和输出功能如下图所示。由于输出函数与相同，这表明导数计算是正确的。在子例程“ d1y”中，我使用了前向和后向有限差分公式来计算边界处的导数以及两个边界之间的点的中心差。然后，我计算出相对误差。如下图所示，在y轴的边界处有0.003的误差，在边界之间的点处有0.0015的误差。输入函数：<code> dcos（x（i1）/2.0d0）* dcos（y（i2）/4.0d0）</ code>输出：输入函数的y-双导数
- 0.25d0*dcos(x(i1)/2.0d0) * dsin(y(i2)/4.0d0)从y轴查看

我使用相同的技术计算了二阶导数。该子例程如下：

`subroutine d2y(n1,n2,n3,a,a2)
    implicit none
    integer n1, n2, n3, i1, i2, i3
    real*8 pi, a(n1,n2,n3), a2(n1,n2,n3), z(n3),x(n1),y(n2),h
    pi=3.14159265358979323846d0
    h=4.0d0*pi/(n2-1)
    a2=0.0d0
    i3 =1
      do i1=1,n1
        do i2=1,n2
          if(i2 == 1)then
             a2(i1,i2,i3)=( 2.0d0*a(i1,i2,i3) - 5.0d0*a(i1,i2+1,i3) + 4.0d0*a(i1,i2+2,i3) - a(i1,i2+3,i3))/(h*h)
          else if( i2 == n2)then
             a2(i1,i2,i3)=( 2.0d0*a(i1,i2,i3) - 5.0d0*a(i1,i2-1,i3) + 4.0d0*a(i1,i2-2,i3) - a(i1,i2-3,i3))/(h*h)
          else
             a2(i1,i2,i3)= ( a(i1,i2+1,i3) - 2.0d0*a(i1,i2,i3) + a(i1,i2-1,i3) )/(h*h)
          endif
        enddo
      enddo
   ! enddo 

    end subroutine

这里还误差在边界处的大，实际上，相对于一阶导数的误差是非常大的，如图：yz平面的视图。
为什么边界这么大？请解释。

`

Dan Sp.

First, I stand corrected. The names 'Forward Difference' and 'Backward Difference' usually refer to the standard 1st order formulas. You do have correct 2nd order, One-sided 1st derivative formulas.

The 2nd order, 1st derivative, Forward difference formula is derived with a Taylor series expression at x+h and at x+2h as follows:

f(x+h) = f(x) + h*f'(x) + h^2*f''(x)/2! + h^3*f'''(x)/3! ....

f(x+2h) = f(x) + 2h*f'(x) + 4h^2*f''(x)/2! + 8h^3*f'''(x)/3! ....

Now, take 4 times the 1st series and subtract the 2nd series and solve for f'(x). Note that the second derivative term disappears.

f'(x) = 3h/2 * f(x) - 2/h *f(x+h) + 1/(2h) * f(x+2h) + 0 - 4h^2*f'''(x)/3!

This is the formula you are using and it is 2nd order accurate. This does not mean that the accuracy is identical to the central difference accuracy, only that the order of the accuracy is the same.

If you look at the derivation of central difference while keeping the error term, you will find that the error term looks like:

-h^2*f'''(x)/6

Did you note that the error term in the forward difference formula has a constant in front of it. In general, the error will always be larger. But that is not what it means to be 2nd order accurate. The order of accuracy tells you how the error changes when you change h.

例如，如果将h减半，则应获得约4倍的精度。这意味着在上面的示例中，中心差误差将从大约0.0015变为大约0.0004，边界误差将从大约0.003变为大约0.0008。

最终要点：您的程序正确，错误正确！

本文收集自互联网，转载请注明来源。

如有侵权，请联系 [email protected] 删除。

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