数值求解波动方程 [5]

2022-09-05 23:10 作者:nyasyamorina 0人读过 | 我要投稿

二维 PML

在二维空间里有两个空间轴 $x_1%2Cx_2$ , 那么 PML 变换也应该分别施加在这两个轴上. 对于二维 PML 变换应有 $%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmapsto%5Cfrac%5Cpartial%7B%5Cpartial%5Ctilde%20x_1%7D%3D%5Cfrac1%7B1%2B%5Cfrac%7B%5Csigma_1(x_1)%7Ds%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D$ 和 $%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmapsto%5Cfrac%5Cpartial%7B%5Cpartial%5Ctilde%20x_2%7D%3D%5Cfrac1%7B1%2B%5Cfrac%7B%5Csigma_2(x_2)%7Ds%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D$ .

对二维波动方程施加拉普拉斯变换得 $s%5E2%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_1%5E2%7D%5Cmathcal%20Lu%2Bc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_2%5E2%7D%5Cmathcal%20Lu$ . 然后施加 PML 变换, 为了简便, 记 $%5Cgamma_k%3D1%2B%5Cfrac%7B%5Csigma_k%7D%7Bs%7D$ , 则得 $s%5E2%5Cmathcal%20Lu%3D%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac1%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2B%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac1%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)$ . 因为 $%5Cgamma_1$ 只与 $x_1$ 有关而与 $x_2$ 无关, 所以有 $%5Cgamma_1%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x_2%7D%5Ccdot%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x_2%7D%5Cgamma_1%5Ccdot$ , 同理 $%5Cgamma_2$ 也有类似的结论. 那么上式左右同乘 $%5Cgamma_1%5Cgamma_2$ 得 $s%5E2%5Cgamma_1%5Cgamma_2%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac%7B%5Cgamma_2%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2Bc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac%7B%5Cgamma_1%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)$ . 计算可得: $%5Cfrac%7B%5Cgamma_2%7D%7B%5Cgamma_1%7D%3D1%2B%5Cfrac%7B%5Csigma_2-%5Csigma_1%7D%7Bs%2B%5Csigma_1%7D$ , $%5Cfrac%7B%5Cgamma_1%7D%7B%5Cgamma_2%7D%3D1%2B%5Cfrac%7B%5Csigma_1-%5Csigma_2%7D%7Bs%2B%5Csigma_2%7D$ 和 $s%5E2%5Cgamma_1%5Cgamma_2%3Ds%5E2%2Bs(%5Csigma_1%2B%5Csigma_2)%2B%5Csigma_1%5Csigma_2$ , 代入上式得:

$s%5E2%5Cmathcal%20Lu%2B(%5Csigma_1%2B%5Csigma_2)s%5Cmathcal%20Lu%2B%5Csigma_1%5Csigma_2%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_1%5E2%7D%5Cmathcal%20Lu%2Bc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_2%5E2%7D%5Cmathcal%20Lu%2Bc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac%7B%5Csigma_2-%5Csigma_1%7D%7Bs%2B%5Csigma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2Bc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac%7B%5Csigma_1-%5Csigma_2%7D%7Bs%2B%5Csigma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)$

引入新量 $v_1%2Cv_2$ , 使得 $%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cmathcal%20Lv_1%3A%3D%5Cfrac%7B%5Csigma_2-%5Csigma_1%7D%7Bs%2B%5Csigma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5C%5C%5Cmathcal%20Lv_2%3A%3D%5Cfrac%7B%5Csigma_1-%5Csigma_2%7D%7Bs%2B%5Csigma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cend%7Bmatrix%7D%5Cright.$ , 上式右边变为 $c%5E2%5Cnabla%5E2%5Cmathcal%20Lu%2Bc%5E2%5Cvec%5Cnabla%5Ccdot%5Cmathcal%20L%5Cbegin%7Bbmatrix%7Dv_1%5C%5Cv_2%5Cend%7Bbmatrix%7D$ , 并且由定义有 $%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Ds%5Cmathcal%20Lv_1%2B%5Csigma_1%5Cmathcal%20Lv_1%3D(%5Csigma_2-%5Csigma_1)%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5C%5Cs%5Cmathcal%20Lv_2%2B%5Csigma_2%5Cmathcal%20Lv_2%3D(%5Csigma_1-%5Csigma_2)%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cend%7Bmatrix%7D%5Cright.$ . 把全部东西从 s 域变回时域即得到:

$%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20t%5E2%7D%2B(%5Csigma_1%2B%5Csigma_2)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20t%7D%3Dc%5E2%5Cleft(%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_1%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_2%5E2%7D%2B%5Cfrac%7B%5Cpartial%20v_1%7D%7B%5Cpartial%20x_1%7D%2B%5Cfrac%7B%5Cpartial%20v_2%7D%7B%5Cpartial%20x_2%7D%5Cright)-%5Csigma_1%5Csigma_2u$

$%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cfrac%7B%5Cpartial%20v_1%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_2-%5Csigma_1)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_1%7D-%5Csigma_1v_1%5C%5C%5Cfrac%7B%5Cpartial%20v_2%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_1-%5Csigma_2)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_2%7D-%5Csigma_2v_2%5Cend%7Bmatrix%7D%5Cright.$

这就是二维的 PML 波动方程了.

在写出离散化前, 为了简便先进行一下符号约定: 记 $%5Cmathbb%7Bfield%7D_%7B%5Ccdots%2C%5Cmathbb%7Baxis%5Cpm1%7D%2C%5Ccdots%7D$ 为 $%5Cmathbb%7Bfield_%7B%5Cpm%20axis%7D%7D$ , 用这个记法可以很方便地描述与任意一个元素相邻的其他元素, 比如说 $u_%7B%2Bx_1%7D%3Du_%7Bx_1%2B1%2Cx_2%2Ct%7D$ , 还有 $v_%7B1%2C-t%7D%3D%7Bv_1%7D_%7Bx_1%2Cx_2%2Ct-1%7D$ , 根据这个记法可以有 $u%3A%3Du_%7Bx_1%2Cx_2%2Ct%7D$ , 虽然在一定程度上会产生歧义, 但是因为离散化后的式子只会有指定索引的元素, 而没有一整个场, 所以将就着用吧.

使用上述记法, 并使用 $%5Cfrac%7Bv_%7Bk%2C%2Bt%7D-v_%7Bk%7D%7D%7B%5CDelta%20t%7D%5Capprox%5Cfrac%7B%5Cpartial%20v_k%7D%7B%5Cpartial%20t%7D$ 以节省内存, 二维 PML 波动方程离散化为

$u_%7B%2Bt%7D%3Dc_1(u_%7B%2Bx_1%7D%2Bu_%7B-x_1%7D)%2Bc_2(u_%7B%2Bx_2%7D%2Bu_%7B-x_2%7D)%2Bc_3(v_%7B1%2C%2Bx_1%7D-v_%7B1%2C-x_1%7D)%2Bc_4(v_%7B2%2C%2Bx_2%7D-v_%7B2%2C-x_2%7D)%2Bc_5u_%7B-t%7D%2Bc_6u$

$%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dv_%7B1%2C%2Bt%7D%3D%5Cfrac%7B(%5Csigma_2-%5Csigma_1)%5CDelta%20t%7D%7B2%5CDelta%20x_1%7D(u_%7B%2Bx_1%7D-u_%7B-x_1%7D)%2B(1-%5Csigma_1%5CDelta%20t)v_1%5C%5Cv_%7B2%2C%2Bt%7D%3D%5Cfrac%7B(%5Csigma_1-%5Csigma_2)%5CDelta%20t%7D%7B2%5CDelta%20x_2%7D(u_%7B%2Bx_2%7D-u_%7B-x_2%7D)%2B(1-%5Csigma_2%5CDelta%20t)v_2%5Cend%7Bmatrix%7D%5Cright.$

其中 $c_1%3D%5Cfrac%7Bc%5E2%5CDelta%20t%5E2%7D%7B%5CDelta%20x_1%5E2c_u%7D%3B%5C%3Bc_2%3D%5Cfrac%7Bc%5E2%5CDelta%20t%5E2%7D%7B%5CDelta%20x_2%5E2c_u%7D%3B%5C%3Bc_3%3D%5Cfrac%7Bc%5E2%5CDelta%20t%5E2%7D%7B2%5CDelta%20x_1c_u%7D%3B%5C%3Bc_4%3D%5Cfrac%7Bc%5E2%5CDelta%20t%5E2%7D%7B2%5CDelta%20x_2c_u%7D%3B%5C%3B$ $c_5%3D%5Cfrac%7B(%5Csigma_1%2B%5Csigma_2)%5CDelta%20t-1%7D%7Bc_u%7D%3B%5C%3Bc_6%3D%5Cfrac%7B2-%5Csigma_1%5Csigma_2%5CDelta%20t%5E2%7D%7Bc_u%7D-2(c_1%2Bc_2)$ 和 $c_u%3D(%5Csigma_1%2B%5Csigma_2)%5CDelta%20t%2B1$ .

离散化后, 出现了 10 个系数, 这些系数的重复度较高, 所以可以先预计算一部分系数, 然后在 update! 里再计算其他系数, 这样就可以在提升计算速度的同时不占用大量内存. 但是我内存大, 下面实现预计算全部 10 个系数, 以使计算效率最大化.

根据上面的式子, 二维 PML 波动方程实现为:

尽管看起来长得有点恐怖, 但是逻辑是跟之前的一样的 (用了很多 julia 特性化简). 模拟成品如下:

另外重写了 WaveRecord 类和方法, 现在产生的中间帧会储存在硬盘上而不是直接放在内存上, 并且暴露里中间把场渲染成图片的方法: `save_frame`.

上面的视频使用以下代码进行模拟:

三维波动方程 & PML 版本

因为受限于内存以及计算速度等因素, 三维波动方程就不打算进行实现了 (主要是懒), 所以干脆在这里把三维的 PML 一起放出了.

三维波动方程为 $%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20t%5E2%7D%3Dc%5E2%5Cleft(%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_1%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_2%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_3%5E2%7D%5Cright)$ . 为三维的加上 PML 与二维的类似:

首先把波动方程从时域变为 s 域得 $s%5E2%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_1%5E2%7D%5Cmathcal%20Lu%2Bc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_2%5E2%7D%5Cmathcal%20Lu%2Bc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_3%5E2%7D%5Cmathcal%20Lu$ .

做 PML 变换得 $s%5E2%5Cmathcal%20Lu%3D%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac1%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2B%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac1%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)%2B%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_3%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cleft(%5Cfrac1%7B%5Cgamma_3%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20Lu%5Cright)$ , 其中 $%5Cgamma_k%3A%3D1%2B%5Cfrac%7B%5Csigma_k(x_k)%7D%7Bs%7D$ .

上式乘以 $%5Cgamma_1%5Cgamma_2%5Cgamma_3$ 得 $s%5E2%5Cgamma_1%5Cgamma_2%5Cgamma_3%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac%7B%5Cgamma_2%5Cgamma_3%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%0A%2Bc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac%7B%5Cgamma_1%5Cgamma_3%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)%20%2B%0Ac%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cleft(%5Cfrac%7B%5Cgamma_1%5Cgamma_2%7D%7B%5Cgamma_3%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20Lu%5Cright)$ .

计算可得: $s%5E2%5Cgamma_1%5Cgamma_2%5Cgamma_3%3Ds%5E2%2B(%5Csigma_1%2B%5Csigma_2%2B%5Csigma_3)s%2B(%5Csigma_1%5Csigma_2%2B%5Csigma_2%5Csigma_3%2B%5Csigma_3%5Csigma_1)%2B%5Cfrac%7B%5Csigma_1%5Csigma_2%5Csigma_3%7Ds$ 以及 $%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%0A%5Cfrac%7B%5Cgamma_2%5Cgamma_3%7D%7B%5Cgamma_1%7D%3D1%2B%5Cfrac%7B(%5Csigma_2%2B%5Csigma_3-%5Csigma_1)s%2B%5Csigma_2%5Csigma_3%7D%7B(s%2B%5Csigma_1)s%7D%5C%5C%0A%5Cfrac%7B%5Cgamma_1%5Cgamma_3%7D%7B%5Cgamma_2%7D%3D1%2B%5Cfrac%7B(%5Csigma_1%2B%5Csigma_3-%5Csigma_2)s%2B%5Csigma_1%5Csigma_3%7D%7B(s%2B%5Csigma_2)s%7D%5C%5C%5Cfrac%7B%5Cgamma_2%5Cgamma_3%7D%7B%5Cgamma_3%7D%3D1%2B%5Cfrac%7B(%5Csigma_1%2B%5Csigma_2-%5Csigma_3)s%2B%5Csigma_1%5Csigma_2%7D%7B(s%2B%5Csigma_3)s%7D%0A%5Cend%7Bmatrix%7D%5Cright.$ .

引入 4 个新量: $%5Cbegin%7Bmatrix%7D%0A%5Cmathcal%20L%20f%3A%3D%5Cfrac1s%5Cmathcal%20L%20u%5C%5C%0A%5Cmathcal%20L%20v_1%3A%3D%5Cfrac%7B(%5Csigma_2%2B%5Csigma_3-%5Csigma_1)s%2B%5Csigma_2%20%5Csigma_3%7D%7B(s%2B%5Csigma_1)s%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20L%20u%5C%5C%0A%5Cmathcal%20L%20v_2%3A%3D%5Cfrac%7B(%5Csigma_1%2B%5Csigma_3-%5Csigma_2)s%2B%5Csigma_1%20%5Csigma_3%7D%7B(s%2B%5Csigma_2)s%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20L%20u%5C%5C%5Cmathcal%20L%20v_3%3A%3D%5Cfrac%7B(%5Csigma_1%2B%5Csigma_2-%5Csigma_3)s%2B%5Csigma_1%20%5Csigma_2%7D%7B(s%2B%5Csigma_3)s%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20L%20u%0A%5Cend%7Bmatrix%7D$ , 即 $%5Cbegin%7Bmatrix%7D%0As%5Cmathcal%20L%20f%3D%5Cmathcal%20L%20u%5C%5C%0A(s%2B%5Csigma_1)%5Cmathcal%20L%20v_1%3D(%5Csigma_2%2B%5Csigma_3-%5Csigma_1)%5Cfrac%20%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%2B%5Csigma_2%5Csigma_3%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%20%5Cfrac1s%5Cmathcal%20Lu%5C%5C%0A(s%2B%5Csigma_2)%5Cmathcal%20L%20v_2%3D(%5Csigma_1%2B%5Csigma_3-%5Csigma_2)%5Cfrac%20%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%2B%5Csigma_1%5Csigma_3%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%20%5Cfrac1s%5Cmathcal%20Lu%5C%5C%0A(s%2B%5Csigma_3)%5Cmathcal%20L%20v_3%3D(%5Csigma_1%2B%5Csigma_2-%5Csigma_3)%5Cfrac%20%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20Lu%2B%5Csigma_1%5Csigma_1%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%20%5Cfrac1s%5Cmathcal%20Lu%0A%5Cend%7Bmatrix%7D$ .

把新量代入, 再转回时域得到 $%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20t%5E2%7D%2B(%5Csigma_1%2B%5Csigma_2%2B%5Csigma_3)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20t%7D%3Dc%5E2%5Cleft(%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_1%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2%20u%7D%7B%5Cpartial%20x_2%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2%20u%7D%7B%5Cpartial%20x_3%5E2%7D%2B%5Cfrac%7B%5Cpartial%20v_1%7D%7B%5Cpartial%20x_1%7D%2B%5Cfrac%7B%5Cpartial%20v_2%7D%7B%5Cpartial%20x_2%7D%2B%5Cfrac%7B%5Cpartial%20v_3%7D%7B%5Cpartial%20x_3%7D%5Cright)-(%5Csigma_1%5Csigma_2%2B%5Csigma_2%5Csigma_3%2B%5Csigma_3%5Csigma_1)u-%5Csigma_1%5Csigma_2%5Csigma_3f$ $%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20t%7D%3Du$ , $%5Cbegin%7Bmatrix%7D%0A%5Cfrac%7B%5Cpartial%20v_1%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_2%2B%5Csigma_3-%5Csigma_1)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_1%7D%2B%5Csigma_2%5Csigma_3%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x_1%7D-%5Csigma_1v_1%5C%5C%0A%5Cfrac%7B%5Cpartial%20v_2%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_1%2B%5Csigma_3-%5Csigma_2)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_2%7D%2B%5Csigma_1%5Csigma_3%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x_2%7D-%5Csigma_2v_2%5C%5C%0A%5Cfrac%7B%5Cpartial%20v_3%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_1%2B%5Csigma_2-%5Csigma_3)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_3%7D%2B%5Csigma_1%5Csigma_2%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x_3%7D-%5Csigma_3v_3%0A%5Cend%7Bmatrix%7D$

这就是三维 PML 波动方程了, 虽然原理都是一样的, 但是这个真的太长了, debug 十万年, 所以就原谅我不去实现了 (

有可能这就是数值模拟波动方程的最后一篇专栏了, 虽然最关键的还有一个麦克斯韦方程组还没搞, 但是上面的标量三维 PML 波动方程都这么复杂了, 更何况三维 PML 麦克斯韦方程组.

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