139 lines
4.5 KiB
Go
139 lines
4.5 KiB
Go
// PDEDiffParabolicS
|
||
/*
|
||
------------------------------------------------------
|
||
作者 : Black Ghost
|
||
日期 : 2018-12-14
|
||
版本 : 0.0.0
|
||
------------------------------------------------------
|
||
求解抛物型偏微分方程的差分解法（六点对称格式）
|
||
理论：
|
||
对于抛物型偏微分方程：
|
||
du d^2u
|
||
---- = A ------ + B
|
||
dt dx^2
|
||
|
||
u(x, 0) = p(x)
|
||
u(0, t) = u1(t), u(L, t) = u2(t)
|
||
|
||
0 < x < L, 0 < t < T
|
||
|
||
则古典隐式差分格式为，x分为m等份，t分为n等份
|
||
Au_(j+1) = B_(j+1)
|
||
|
||
|2(1+l) -l |
|
||
|-l 2(1+l) -l |
|
||
A = | .......... |
|
||
| -l 2(1+l) -l |
|
||
| -l 2(1+l)|
|
||
|
||
u_(j+1) = [u_(1,j+1),u_(2,j+1),...,u_(m-1,j+1)]'
|
||
F_(j+1) = [lu_(0,j)+2(1-l)u_(1,j)+lu_(2,j)+lu_(0,j+1)+2B*tau,
|
||
lu_(1,j)+2(1-l)u_(2,j)+lu_(3,j)+2B*tau,
|
||
...
|
||
lu_(m-3,j)+2(1-l)u_(m-2,j)+lu_(m-1,j)+2B*tau,
|
||
lu_(m-2,j)+2(1-l)u_(m-1,j)+lu_(m,j)+lu_(m,j+1)+2B*tau]'
|
||
j = 0,1,...,n-1
|
||
|
||
u0 = [u_(1,0),u_(2,0),...,u_(m-1,0)]'
|
||
= [p(h),p(2h),...,p((m-1)h)]'
|
||
|
||
|
||
参考 John H. Mathews and Kurtis D. Fink. Numerical
|
||
methods using MATLAB, 4th ed. Pearson
|
||
Education, 2004. ss 10.2.3.
|
||
------------------------------------------------------
|
||
输入 :
|
||
funp, funu1, funu2 边界函数
|
||
x0 求解范围，2x2
|
||
A, B 常系数
|
||
m, n 网格数量
|
||
输出 :
|
||
sol 解矩阵
|
||
err 解出标志：false-未解出或达到步数上限；
|
||
true-全部解出
|
||
------------------------------------------------------
|
||
*/
|
||
|
||
package goNum
|
||
|
||
// PDEDiffParabolicS 求解抛物型偏微分方程的差分解法（隐式）
|
||
func PDEDiffParabolicS(funp, funu1, funu2 func(float64) float64, x0 Matrix, A, B float64, m, n int) (Matrix, bool) {
|
||
/*
|
||
求解抛物型偏微分方程的差分解法（隐式）
|
||
输入 :
|
||
funp, funu1, funu2 边界函数
|
||
x0 求解范围，2x2
|
||
A, B 常系数
|
||
m, n 网格数量
|
||
输出 :
|
||
sol 解矩阵
|
||
err 解出标志：false-未解出或达到步数上限；
|
||
true-全部解出
|
||
*/
|
||
//判断网格数量
|
||
if (m < 1) || (n < 1) {
|
||
panic("Error in goNum.PDEDiffParabolicS: Grid numbers error")
|
||
}
|
||
|
||
var err bool = false
|
||
sol := ZeroMatrix(m+1, n+1)
|
||
hx := (x0.GetFromMatrix(1, 0) - x0.GetFromMatrix(0, 0)) / float64(m) //x方向步长
|
||
ht := (x0.GetFromMatrix(1, 1) - x0.GetFromMatrix(0, 1)) / float64(n) //t方向步长
|
||
|
||
//1. 计算t第零层上的值u_(i,0) i=0,1,...,m
|
||
for i := 0; i < m+1; i++ {
|
||
sol.SetMatrix(i, 0, funp(x0.GetFromMatrix(0, 0)+float64(i)*hx))
|
||
}
|
||
//2. 计算左右边界上的节点u_(0,j)和u_(m,j) j=1,2,...,n
|
||
for j := 1; j < n+1; j++ {
|
||
sol.SetMatrix(0, j, funu1(x0.GetFromMatrix(0, 1)+float64(j)*ht)) //左边界
|
||
sol.SetMatrix(m, j, funu2(x0.GetFromMatrix(0, 1)+float64(j)*ht)) //右边界
|
||
}
|
||
|
||
l := A * ht / (hx * hx)
|
||
//稳定性判断
|
||
if l <= 0 {
|
||
panic("Error in goNum.PDEDiffParabolicS: lambda less than or equal to zero")
|
||
}
|
||
//A赋值
|
||
AA := ZeroMatrix(m-1, m-1)
|
||
ui := ZeroMatrix(m+1, 1)
|
||
AA.SetMatrix(0, 0, 2.0*(1.0+l)) //第零行
|
||
AA.SetMatrix(0, 1, -1.0*l)
|
||
ui.Data[0] = sol.GetFromMatrix(0, 0)
|
||
for i := 1; i < m-2; i++ {
|
||
AA.SetMatrix(i, i-1, -1.0*l)
|
||
AA.SetMatrix(i, i, 2.0*(1.0+l))
|
||
AA.SetMatrix(i, i+1, -1.0*l)
|
||
ui.Data[i] = sol.GetFromMatrix(i, 0)
|
||
}
|
||
AA.SetMatrix(m-2, m-3, -1.0*l) //第零行
|
||
AA.SetMatrix(m-2, m-2, 2.0*(1.0+l))
|
||
ui.Data[m-2] = sol.GetFromMatrix(m-2, 0)
|
||
ui.Data[m-1] = sol.GetFromMatrix(m-1, 0)
|
||
ui.Data[m] = sol.GetFromMatrix(m, 0)
|
||
//内部节点循环求解
|
||
for j := 0; j < n; j++ {
|
||
Fi := ZeroMatrix(m-1, 1)
|
||
Fi.Data[0] = l*ui.Data[0] + 2.0*(1.0-l)*ui.Data[1] + l*ui.Data[2] + l*sol.GetFromMatrix(0, j+1) + 2.0*B*ht
|
||
for i := 1; i < m-2; i++ {
|
||
Fi.Data[i] = l*ui.Data[i] + 2.0*(1.0-l)*ui.Data[i+1] + l*ui.Data[i+2] + 2.0*B*ht
|
||
}
|
||
Fi.Data[m-2] = l*ui.Data[m-2] + 2.0*(1.0-l)*ui.Data[m-1] + l*ui.Data[m] + l*sol.GetFromMatrix(m, j+1) + 2.0*B*ht
|
||
//ui1为m-1行，ui为m+1行
|
||
ui1, errtemp := LEs_Chasing(AA, Fi)
|
||
if errtemp != true {
|
||
panic("Error in goNum.PDEDiffParabolicS: Chasing solved error")
|
||
}
|
||
ui.Data[0] = sol.GetFromMatrix(0, j+1)
|
||
for i := 1; i < m; i++ {
|
||
ui.Data[i] = ui1.Data[i-1]
|
||
sol.SetMatrix(i, j+1, ui1.Data[i-1])
|
||
}
|
||
ui.Data[m] = sol.GetFromMatrix(m, j+1)
|
||
}
|
||
|
||
err = true
|
||
return sol, err
|
||
}
|