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nuknal
191
vendor/github.com/nuknal/goNum/PDEDiffEllipticalP5.go
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191
vendor/github.com/nuknal/goNum/PDEDiffEllipticalP5.go
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// PDEDiffEllipticalP5
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/*
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------------------------------------------------------
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作者 : Black Ghost
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日期 : 2019-01-08
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版本 : 0.0.0
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------------------------------------------------------
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求解椭圆型偏微分方程（Poisson）的差分解法（五点格式）
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理论：
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对于椭圆型偏微分方程（Poisson方程）：
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d^2u d^2u
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------ + ------ = g(x, y)
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dx^2 dy^2
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u(x, 0) = fy0(x), u(x, b) = fyb(x)
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u(0, y) = fx0(y), u(a, y) = fxa(y)
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0 < x < a, 0 < y < b
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x分为n等份，y分为m等份
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hy^2[u_(i+1,j) + u_(i-1,j) - 2u_(i,j)] +
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hx^2[u_(i,j+1) + u_(i,j-1) - 2u_(i,j)] - g_(i,j)*hx^2*hy^2 = 0
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解以上方程组可得解
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参考 John H. Mathews and Kurtis D. Fink. Numerical
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methods using MATLAB, 4th ed. Pearson
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Education, 2004. ss 10.3.
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------------------------------------------------------
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输入 :
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funy0, funyb, funx0, funxa, fung 边界函数及g(x, y)
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x0 求解范围，2x2
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n, m 网格数量, 对应x和y
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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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// PDEDiffEllipticalP5 求解椭圆型偏微分方程（Poisson）的差分解法（五点格式）
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func PDEDiffEllipticalP5(funy0, funyb, funx0, funxa func(float64) float64,
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fung func(float64, float64) float64, x0 Matrix, n, m int) (Matrix, bool) {
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/*
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求解椭圆型偏微分方程（Poisson）的差分解法（五点格式）
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输入 :
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funy0, funyb, funx0, funxa, fung 边界函数及g(x, y)
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x0 求解范围，2x2
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n, m 网格数量, 对应x和y
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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.PDEDiffEllipticalP5: Grid numbers error")
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}
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//判断初值维数
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if (x0.Rows < 2) || (x0.Columns < 2) {
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panic("Error in goNum.PDEDiffEllipticalP5: Initial values error")
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}
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var err bool = false
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sol := ZeroMatrix(m+1, n+1) //行y变化，列x变化
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hx := (x0.GetFromMatrix(1, 0) - x0.GetFromMatrix(0, 0)) / float64(n) //x方向步长
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hy := (x0.GetFromMatrix(1, 1) - x0.GetFromMatrix(0, 1)) / float64(m) //y方向步长
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hx2 := hx * hx
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hy2 := hy * hy
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hxhy2 := hx2 * hy2
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//边界框的解
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//第一行的值和最后一行的值，不包括第一个和最后一个
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for i := 1; i < n; i++ {
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sol.SetMatrix(0, i, funy0(x0.GetFromMatrix(0, 0)+hx*float64(i)))
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sol.SetMatrix(m, i, funyb(x0.GetFromMatrix(0, 0)+hx*float64(i)))
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}
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//第一列的值和最后一列的值，包括第一个和最后一个
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for j := 0; j < m+1; j++ {
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sol.SetMatrix(j, 0, funx0(x0.GetFromMatrix(0, 1)+hy*float64(j)))
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sol.SetMatrix(j, n, funxa(x0.GetFromMatrix(0, 1)+hy*float64(j)))
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}
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//求解中间点，主对角占优矩阵解法，利用高斯消去方法
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AA := ZeroMatrix((n-1)*(m-1), (n-1)*(m-1)) //系数矩阵A
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BA := ZeroMatrix((n-1)*(m-1), 1) //值矩阵B
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//赋值系数矩阵和值矩阵
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//第一行, j = 1
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//第一个
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AA.SetMatrix(0, 0, -2.0*hx2-2.0*hy2)
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AA.SetMatrix(0, 1, hy2)
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AA.SetMatrix(0, n-1, hx2)
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tempBA := hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*1.0, x0.GetFromMatrix(0, 1)+hy*1.0)
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tempBA = tempBA - hy2*funx0(x0.GetFromMatrix(0, 1)+hy*1.0)
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tempBA = tempBA - hx2*funy0(x0.GetFromMatrix(0, 0)+hx*1.0)
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BA.SetMatrix(0, 0, tempBA)
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for i := 2; i < n-1; i++ {
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AA.SetMatrix(i-1, i-2, hy2)
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AA.SetMatrix(i-1, i-1, -2.0*hx2-2.0*hy2)
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AA.SetMatrix(i-1, i, hy2)
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AA.SetMatrix(i-1, (n-1)*1+i-1, hx2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*float64(i), x0.GetFromMatrix(0, 1)+hy*1.0)
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tempBA = tempBA - hx2*funy0(x0.GetFromMatrix(0, 0)+hx*float64(i))
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BA.SetMatrix((n-1)*0+i-1, 0, tempBA)
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}
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//最后一个
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AA.SetMatrix(n-1-1, n-1-2, hy2)
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AA.SetMatrix(n-1-1, n-1-1, -2.0*hx2-2.0*hy2)
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AA.SetMatrix(n-1-1, (n-1)*1+n-1-1, hx2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*float64(n-1), x0.GetFromMatrix(0, 1)+hy*1.0)
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tempBA = tempBA - hy2*funxa(x0.GetFromMatrix(0, 1)+hy*1.0)
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tempBA = tempBA - hx2*funy0(x0.GetFromMatrix(0, 0)+hx*float64(n-1))
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BA.SetMatrix(n-1-1, 0, tempBA)
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//中间行, 2 <= j <= m-2
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for j := 2; j < m-1; j++ {
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//第一个
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AA.SetMatrix((n-1)*(j-1), (n-1)*(j-1-1), hx2)
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AA.SetMatrix((n-1)*(j-1), (n-1)*(j-1), -2.0*hx2-2.0*hy2)
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AA.SetMatrix((n-1)*(j-1), (n-1)*(j-1)+1, hy2)
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AA.SetMatrix((n-1)*(j-1), (n-1)*(j-1)+n-1, hx2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*1.0, x0.GetFromMatrix(0, 1)+hy*float64(j))
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tempBA = tempBA - hy2*funx0(x0.GetFromMatrix(0, 1)+hy*float64(j))
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BA.SetMatrix((n-1)*(j-1), 0, tempBA)
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for i := 2; i < n-1; i++ {
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AA.SetMatrix((n-1)*(j-1)+i-1, (n-1)*(j-1-1)+i-1, hx2)
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AA.SetMatrix((n-1)*(j-1)+i-1, (n-1)*(j-1)+i-2, hy2)
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AA.SetMatrix((n-1)*(j-1)+i-1, (n-1)*(j-1)+i-1, -2.0*hx2-2.0*hy2)
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AA.SetMatrix((n-1)*(j-1)+i-1, (n-1)*(j-1)+i, hy2)
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AA.SetMatrix((n-1)*(j-1)+i-1, (n-1)*(j-1+1)+i-1, hx2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*float64(i), x0.GetFromMatrix(0, 1)+hy*float64(j))
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BA.SetMatrix((n-1)*(j-1)+i-1, 0, tempBA)
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}
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//最后一个
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AA.SetMatrix((n-1)*(j-1)+n-1-1, (n-1)*(j-1-1)+n-1-1, hx2)
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AA.SetMatrix((n-1)*(j-1)+n-1-1, (n-1)*(j-1)+n-1-2, hy2)
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AA.SetMatrix((n-1)*(j-1)+n-1-1, (n-1)*(j-1)+n-1-1, -2.0*hx2-2.0*hy2)
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AA.SetMatrix((n-1)*(j-1)+n-1-1, (n-1)*(j-1+1)+n-1-1, hx2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*float64(n-1), x0.GetFromMatrix(0, 1)+hy*float64(j))
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tempBA = tempBA - hy2*funxa(x0.GetFromMatrix(0, 1)+hy*float64(j))
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BA.SetMatrix((n-1)*(j-1)+n-1-1, 0, tempBA)
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}
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//最后一行, j = m-1
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//第一个
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AA.SetMatrix((n-1)*(m-1-1), (n-1)*(m-1-1-1), hx2)
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AA.SetMatrix((n-1)*(m-1-1), (n-1)*(m-1-1), -2.0*hx2-2.0*hy2)
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AA.SetMatrix((n-1)*(m-1-1), (n-1)*(m-1-1)+1, hy2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*1.0, x0.GetFromMatrix(0, 1)+hy*float64(m-1))
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tempBA = tempBA - hy2*funx0(x0.GetFromMatrix(0, 1)+hy*float64(m-1))
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tempBA = tempBA - hx2*funyb(x0.GetFromMatrix(0, 0)+hx*1.0)
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BA.SetMatrix((n-1)*(m-1-1), 0, tempBA)
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for i := 2; i < n-1; i++ {
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AA.SetMatrix((n-1)*(m-1-1)+i-1, (n-1)*(m-1-1-1)+i-1, hx2)
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AA.SetMatrix((n-1)*(m-1-1)+i-1, (n-1)*(m-1-1)+i-2, hy2)
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AA.SetMatrix((n-1)*(m-1-1)+i-1, (n-1)*(m-1-1)+i-1, -2.0*hx2-2.0*hy2)
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AA.SetMatrix((n-1)*(m-1-1)+i-1, (n-1)*(m-1-1)+i, hy2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*float64(i), x0.GetFromMatrix(0, 1)+hy*float64(m-1))
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tempBA = tempBA - hx2*funyb(x0.GetFromMatrix(0, 0)+hx*float64(i))
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BA.SetMatrix((n-1)*(m-1-1)+i-1, 0, tempBA)
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}
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//最后一个
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AA.SetMatrix((n-1)*(m-1-1)+n-1-1, (n-1)*(m-1-1-1)+n-1-1, hx2)
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AA.SetMatrix((n-1)*(m-1-1)+n-1-1, (n-1)*(m-1-1)+n-1-2, hy2)
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AA.SetMatrix((n-1)*(m-1-1)+n-1-1, (n-1)*(m-1-1)+n-1-1, -2.0*hx2-2.0*hy2)
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tempBA = hxhy2 * fung(x0.GetFromMatrix(0, 0)+hx*float64(n-1), x0.GetFromMatrix(0, 1)+hy*float64(m-1))
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tempBA = tempBA - hy2*funxa(x0.GetFromMatrix(0, 1)+hy*float64(m-1))
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tempBA = tempBA - hx2*funyb(x0.GetFromMatrix(0, 0)+hx*float64(n-1))
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BA.SetMatrix((n-1)*(m-1-1)+n-1-1, 0, tempBA)
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//求解矩阵方程
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tempp, temperr := LEs_ECPE(Matrix2ToSlices(AA), Matrix1ToSlices(BA))
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if temperr != true {
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panic("Error in goNum.PDEDiffEllipticalP5: Solve error")
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}
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//解赋予sol
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ii := 1
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jj := 1
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for i := 0; i < len(tempp); i++ {
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sol.SetMatrix(ii, jj, tempp[i])
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if jj == n-1 {
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ii++
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jj = 0
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}
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jj++
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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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