數(shù)值求解波動方程 [5]

2022-09-05 23:10 作者:nyasyamorina 0人讀過 | 我要投稿

二維 PML

????????在二維空間里有兩個空間軸? $x_1%2Cx_2$ ,? 那么 PML 變換也應該分別施加在這兩個軸上.? 對于二維 PML 變換應有 $%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmapsto%5Cfrac%5Cpartial%7B%5Cpartial%5Ctilde%20x_1%7D%3D%5Cfrac1%7B1%2B%5Cfrac%7B%5Csigma_1(x_1)%7Ds%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D$ ?和 $%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmapsto%5Cfrac%5Cpartial%7B%5Cpartial%5Ctilde%20x_2%7D%3D%5Cfrac1%7B1%2B%5Cfrac%7B%5Csigma_2(x_2)%7Ds%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D$ .

????????對二維波動方程施加拉普拉斯變換得? $s%5E2%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_1%5E2%7D%5Cmathcal%20Lu%2Bc%5E2%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x_2%5E2%7D%5Cmathcal%20Lu$ .? 然后施加 PML 變換,? 為了簡便,? 記 $%5Cgamma_k%3D1%2B%5Cfrac%7B%5Csigma_k%7D%7Bs%7D$ ,? 則得 $s%5E2%5Cmathcal%20Lu%3D%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac1%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2B%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac1%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)$ .? 因為 $%5Cgamma_1$ ?只與 $x_1$ ?有關而與 $x_2$ ?無關,? 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 $%5Cgamma_2$ 也有類似的結論.? 那么上式左右同乘? $%5Cgamma_1%5Cgamma_2$ ?得 $s%5E2%5Cgamma_1%5Cgamma_2%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac%7B%5Cgamma_2%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2Bc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac%7B%5Cgamma_1%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)$ .? 計算可得:?? $%5Cfrac%7B%5Cgamma_2%7D%7B%5Cgamma_1%7D%3D1%2B%5Cfrac%7B%5Csigma_2-%5Csigma_1%7D%7Bs%2B%5Csigma_1%7D$ ,?? $%5Cfrac%7B%5Cgamma_1%7D%7B%5Cgamma_2%7D%3D1%2B%5Cfrac%7B%5Csigma_1-%5Csigma_2%7D%7Bs%2B%5Csigma_2%7D$ ?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 $%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cmathcal%20Lv_1%3A%3D%5Cfrac%7B%5Csigma_2-%5Csigma_1%7D%7Bs%2B%5Csigma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5C%5C%5Cmathcal%20Lv_2%3A%3D%5Cfrac%7B%5Csigma_1-%5Csigma_2%7D%7Bs%2B%5Csigma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cend%7Bmatrix%7D%5Cright.$ ,? 上式右邊變?yōu)? $c%5E2%5Cnabla%5E2%5Cmathcal%20Lu%2Bc%5E2%5Cvec%5Cnabla%5Ccdot%5Cmathcal%20L%5Cbegin%7Bbmatrix%7Dv_1%5C%5Cv_2%5Cend%7Bbmatrix%7D$ ,? 并且由定義有 $%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Ds%5Cmathcal%20Lv_1%2B%5Csigma_1%5Cmathcal%20Lv_1%3D(%5Csigma_2-%5Csigma_1)%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5C%5Cs%5Cmathcal%20Lv_2%2B%5Csigma_2%5Cmathcal%20Lv_2%3D(%5Csigma_1-%5Csigma_2)%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cend%7Bmatrix%7D%5Cright.$ .? 把全部東西從 s 域變回時域即得到:

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

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

這就是二維的 PML 波動方程了.

????????在寫出離散化前,? 為了簡便先進行一下符號約定:? 記? $%5Cmathbb%7Bfield%7D_%7B%5Ccdots%2C%5Cmathbb%7Baxis%5Cpm1%7D%2C%5Ccdots%7D$ ?為 $%5Cmathbb%7Bfield_%7B%5Cpm%20axis%7D%7D$ ,? 用這個記法可以很方便地描述與任意一個元素相鄰的其他元素,? 比如說 $u_%7B%2Bx_1%7D%3Du_%7Bx_1%2B1%2Cx_2%2Ct%7D$ ,? 還有 $v_%7B1%2C-t%7D%3D%7Bv_1%7D_%7Bx_1%2Cx_2%2Ct-1%7D$ ,? 根據(jù)這個記法可以有 $u%3A%3Du_%7Bx_1%2Cx_2%2Ct%7D$ ,? 雖然在一定程度上會產(chǎn)生歧義,? 但是因為離散化后的式子只會有指定索引的元素,? 而沒有一整個場,? 所以將就著用吧.

????????使用上述記法,??并使用 $%5Cfrac%7Bv_%7Bk%2C%2Bt%7D-v_%7Bk%7D%7D%7B%5CDelta%20t%7D%5Capprox%5Cfrac%7B%5Cpartial%20v_k%7D%7B%5Cpartial%20t%7D$ 以節(jié)省內(nèi)存,? 二維 PML 波動方程離散化為

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

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

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 $c_u%3D(%5Csigma_1%2B%5Csigma_2)%5CDelta%20t%2B1$ .

????????離散化后,? 出現(xiàn)了 10 個系數(shù),? 這些系數(shù)的重復度較高,? 所以可以先預計算一部分系數(shù),? 然后在 update! 里再計算其他系數(shù),? 這樣就可以在提升計算速度的同時不占用大量內(nèi)存.? 但是我內(nèi)存大,? 下面實現(xiàn)預計算全部 10?個系數(shù),? 以使計算效率最大化.

????????根據(jù)上面的式子, 二維 PML 波動方程實現(xiàn)為:

????????盡管看起來長得有點恐怖,? 但是邏輯是跟之前的一樣的?(用了很多 julia 特性化簡).? 模擬成品如下:

????????另外重寫了 WaveRecord 類和方法,? 現(xiàn)在產(chǎn)生的中間幀會儲存在硬盤上而不是直接放在內(nèi)存上,? 并且暴露里中間把場渲染成圖片的方法: `save_frame`.

上面的視頻使用以下代碼進行模擬:

三維波動方程 & PML 版本

????????因為受限于內(nèi)存以及計算速度等因素,? 三維波動方程就不打算進行實現(xiàn)了 (主要是懶),? 所以干脆在這里把三維的 PML 一起放出了.

????????三維波動方程為? $%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20t%5E2%7D%3Dc%5E2%5Cleft(%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_1%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_2%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_3%5E2%7D%5Cright)$ .? 為三維的加上 PML 與二維的類似:

首先把波動方程從時域變?yōu)?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 變換得 $s%5E2%5Cmathcal%20Lu%3D%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac1%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%2B%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac1%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)%2B%5Cfrac%7Bc%5E2%7D%7B%5Cgamma_3%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cleft(%5Cfrac1%7B%5Cgamma_3%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20Lu%5Cright)$ ,? 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? $s%5E2%5Cgamma_1%5Cgamma_2%5Cgamma_3%5Cmathcal%20Lu%3Dc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cleft(%5Cfrac%7B%5Cgamma_2%5Cgamma_3%7D%7B%5Cgamma_1%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20Lu%5Cright)%0A%2Bc%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cleft(%5Cfrac%7B%5Cgamma_1%5Cgamma_3%7D%7B%5Cgamma_2%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20Lu%5Cright)%20%2B%0Ac%5E2%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cleft(%5Cfrac%7B%5Cgamma_1%5Cgamma_2%7D%7B%5Cgamma_3%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20Lu%5Cright)$ .

計算可得:? $s%5E2%5Cgamma_1%5Cgamma_2%5Cgamma_3%3Ds%5E2%2B(%5Csigma_1%2B%5Csigma_2%2B%5Csigma_3)s%2B(%5Csigma_1%5Csigma_2%2B%5Csigma_2%5Csigma_3%2B%5Csigma_3%5Csigma_1)%2B%5Cfrac%7B%5Csigma_1%5Csigma_2%5Csigma_3%7Ds$ ?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 個新量:? $%5Cbegin%7Bmatrix%7D%0A%5Cmathcal%20L%20f%3A%3D%5Cfrac1s%5Cmathcal%20L%20u%5C%5C%0A%5Cmathcal%20L%20v_1%3A%3D%5Cfrac%7B(%5Csigma_2%2B%5Csigma_3-%5Csigma_1)s%2B%5Csigma_2%20%5Csigma_3%7D%7B(s%2B%5Csigma_1)s%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_1%7D%5Cmathcal%20L%20u%5C%5C%0A%5Cmathcal%20L%20v_2%3A%3D%5Cfrac%7B(%5Csigma_1%2B%5Csigma_3-%5Csigma_2)s%2B%5Csigma_1%20%5Csigma_3%7D%7B(s%2B%5Csigma_2)s%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_2%7D%5Cmathcal%20L%20u%5C%5C%5Cmathcal%20L%20v_3%3A%3D%5Cfrac%7B(%5Csigma_1%2B%5Csigma_2-%5Csigma_3)s%2B%5Csigma_1%20%5Csigma_2%7D%7B(s%2B%5Csigma_3)s%7D%5Cfrac%5Cpartial%7B%5Cpartial%20x_3%7D%5Cmathcal%20L%20u%0A%5Cend%7Bmatrix%7D$ ,? 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,? 再轉(zhuǎn)回時域得到? $%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20t%5E2%7D%2B(%5Csigma_1%2B%5Csigma_2%2B%5Csigma_3)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20t%7D%3Dc%5E2%5Cleft(%5Cfrac%7B%5Cpartial%5E2u%7D%7B%5Cpartial%20x_1%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2%20u%7D%7B%5Cpartial%20x_2%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2%20u%7D%7B%5Cpartial%20x_3%5E2%7D%2B%5Cfrac%7B%5Cpartial%20v_1%7D%7B%5Cpartial%20x_1%7D%2B%5Cfrac%7B%5Cpartial%20v_2%7D%7B%5Cpartial%20x_2%7D%2B%5Cfrac%7B%5Cpartial%20v_3%7D%7B%5Cpartial%20x_3%7D%5Cright)-(%5Csigma_1%5Csigma_2%2B%5Csigma_2%5Csigma_3%2B%5Csigma_3%5Csigma_1)u-%5Csigma_1%5Csigma_2%5Csigma_3f$ ? $%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20t%7D%3Du$ ,?? $%5Cbegin%7Bmatrix%7D%0A%5Cfrac%7B%5Cpartial%20v_1%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_2%2B%5Csigma_3-%5Csigma_1)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_1%7D%2B%5Csigma_2%5Csigma_3%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x_1%7D-%5Csigma_1v_1%5C%5C%0A%5Cfrac%7B%5Cpartial%20v_2%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_1%2B%5Csigma_3-%5Csigma_2)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_2%7D%2B%5Csigma_1%5Csigma_3%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x_2%7D-%5Csigma_2v_2%5C%5C%0A%5Cfrac%7B%5Cpartial%20v_3%7D%7B%5Cpartial%20t%7D%3D(%5Csigma_1%2B%5Csigma_2-%5Csigma_3)%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x_3%7D%2B%5Csigma_1%5Csigma_2%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x_3%7D-%5Csigma_3v_3%0A%5Cend%7Bmatrix%7D$