130 lines
4.0 KiB
Go
130 lines
4.0 KiB
Go
// PDEDiffParabolicI
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/*
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------------------------------------------------------
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作者 : Black Ghost
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日期 : 2018-12-14
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版本 : 0.0.0
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------------------------------------------------------
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求解抛物型偏微分方程的差分解法（隐式）
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理论：
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对于抛物型偏微分方程：
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du d^2u
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---- = A ------ + B
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dt dx^2
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u(x, 0) = p(x)
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u(0, t) = u1(t), u(L, t) = u2(t)
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0 < x < L, 0 < t < T
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则古典隐式差分格式为，x分为m等份，t分为n等份
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Au_(j+1) = uj + F_(j+1)
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|1+2l -l |
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|-l 1+2l -l |
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A = | .......... |
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| -l 1+2l -l |
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| -l 1+2l|
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u_(j+1) = [u_(1,j+1),u_(2,j+1),...,u_(m-1,j+1)]'
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F_(j+1) = [lu1((j+1)*tau)+B*tau,B*tau,B*tau,...,B*tau,lu2((j+1)*tau)+B*tau]'
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V_(j+1) = uj + F_(j+1)
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j = 0,1,...,n-1
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u0 = [u_(1,0),u_(2,0),...,u_(m-1,0)]'
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= [p(h),p(2h),...,p((m-1)h)]'
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参考 李信真, 车刚明, 欧阳洁, 等. 计算方法. 西北工业大学
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出版社, 2000, pp 214-215.
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------------------------------------------------------
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输入 :
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funp, funu1, funu2 边界函数
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x0 求解范围，2x2
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A, B 常系数
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m, n 网格数量
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输出 :
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sol 解矩阵
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err 解出标志：false-未解出或达到步数上限；
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true-全部解出
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------------------------------------------------------
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*/
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package goNum
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// PDEDiffParabolicI 求解抛物型偏微分方程的差分解法（隐式）
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func PDEDiffParabolicI(funp, funu1, funu2 func(float64) float64, x0 Matrix, A, B float64, m, n int) (Matrix, bool) {
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/*
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求解抛物型偏微分方程的差分解法（隐式）
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输入 :
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funp, funu1, funu2 边界函数
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x0 求解范围，2x2
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A, B 常系数
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m, n 网格数量
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输出 :
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sol 解矩阵
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err 解出标志：false-未解出或达到步数上限；
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true-全部解出
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*/
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//判断网格数量
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if (m < 1) || (n < 1) {
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panic("Error in goNum.PDEDiffParabolicI: Grid numbers error")
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}
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var err bool = false
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sol := ZeroMatrix(m+1, n+1)
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hx := (x0.GetFromMatrix(1, 0) - x0.GetFromMatrix(0, 0)) / float64(m) //x方向步长
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ht := (x0.GetFromMatrix(1, 1) - x0.GetFromMatrix(0, 1)) / float64(n) //t方向步长
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//1. 计算t第零层上的值u_(i,0) i=0,1,...,m
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for i := 0; i < m+1; i++ {
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sol.SetMatrix(i, 0, funp(x0.GetFromMatrix(0, 0)+float64(i)*hx))
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}
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//2. 计算左右边界上的节点u_(0,j)和u_(m,j) j=1,2,...,n
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for j := 1; j < n+1; j++ {
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sol.SetMatrix(0, j, funu1(x0.GetFromMatrix(0, 1)+float64(j)*ht)) //左边界
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sol.SetMatrix(m, j, funu2(x0.GetFromMatrix(0, 1)+float64(j)*ht)) //右边界
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}
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l := A * ht / (hx * hx)
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//稳定性判断
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if l <= 0 {
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panic("Error in goNum.PDEDiffParabolicS: lambda less than or equal to zero")
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}
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//A赋值
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AA := ZeroMatrix(m-1, m-1)
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ui := ZeroMatrix(m-1, 1)
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Fi := ZeroMatrix(m-1, 1)
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AA.SetMatrix(0, 0, 1.0+2.0*l) //第零行
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AA.SetMatrix(0, 1, -1.0*l)
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ui.Data[0] = sol.GetFromMatrix(1, 0)
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for i := 1; i < m-2; i++ {
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AA.SetMatrix(i, i-1, -1.0*l)
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AA.SetMatrix(i, i, 1.0+2.0*l)
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AA.SetMatrix(i, i+1, -1.0*l)
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ui.Data[i] = sol.GetFromMatrix(i+1, 0)
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Fi.Data[i] = B * ht
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}
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AA.SetMatrix(m-2, m-3, -1.0*l) //第零行
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AA.SetMatrix(m-2, m-2, 1.0+2.0*l)
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ui.Data[m-2] = sol.GetFromMatrix(m-1, 0)
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//内部节点循环求解
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for j := 0; j < n; j++ {
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//F，每一步需要重新计算第一项和最后一项
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Fi.Data[0] = l*funu1(float64(j+1)*ht) + B*ht
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Fi.Data[m-2] = l*funu2(float64(j+1)*ht) + B*ht
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//
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ui1, errtemp := LEs_Chasing(AA, AddMatrix(ui, Fi))
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if errtemp != true {
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panic("Error in goNum.PDEDiffParabolicI: Chasing solved error")
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}
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for i := 0; i < m-1; i++ {
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ui.Data[i] = ui1.Data[i]
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sol.SetMatrix(i+1, j+1, ui1.Data[i])
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}
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}
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err = true
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return sol, err
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}
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