\u6709\u9650\u5dee\u5206\u65f6\u57df\u6cd5\uff08FDTD\uff09 \u662f\u8ba1\u7b97\u7535\u78c1\u6ce2\u4f20\u64ad\u6700\u8457\u540d\u7684\u6570\u503c\u7b97\u6cd5\u4e4b\u4e00\u3002FDTD \u65b9\u6cd5\u53ef\u4ee5\u6a21\u62df\u4efb\u610f\u5f62\u72b6\u7684\u7ed3\u6784\u3001\u975e\u7ebf\u6027\u4ecb\u8d28\uff0c\u5e76\u4e14\u80fd\u591f\u8ba1\u7b97\u5bbd\u9891\u5e26\u7535\u78c1\u6ce2\u7684\u4f20\u64ad\u3002NS-FDTD \u901a\u8fc7\u5bf9\u5355\u4e00\u9891\u7387\u6ce2\u8ba1\u7b97\u7684\u4f18\u5316\u51cf\u5c11\u4e86\u8ba1\u7b97\u8d44\u6e90\uff0c\u4f7f\u5f97\u5728\u76f8\u540c\u8d44\u6e90\u6d88\u8017\u7684\u60c5\u51b5\u4e0b\u53ef\u4ee5\u8ba1\u7b97\u66f4\u9ad8\u7cbe\u5ea6\u7684\u7ed3\u6784\u3002\u5229\u7528 NS-FDTD \u65b9\u6cd5\uff0c\u7814\u7a76\u8005\u4eec\u5df2\u7ecf\u6210\u529f\u51c6\u786e\u6a21\u62df\u4e86\u4e00\u7c7b\u7279\u6b8a\u7684\u7535\u78c1\u6a21\u5f0f\u2014\u2014\u8033\u8bed\u56de\u5eca\u6a21\u5f0f\uff08Whispering Gallery Modes, WGM\uff09\u3002<\/p>\n\n\n\n
\n\u8033\u8bed\u56de\u5eca\u6a21\u5f0f\u6307\u7684\u662f\u7535\u78c1\u6ce2\u6216\u58f0\u6ce2\u5728\u4e00\u4e2a\u5706\u5f62\u3001\u7403\u5f62\u6216\u73af\u5f62\u7ed3\u6784\u7684\u5185\u58c1\u9644\u8fd1\u4f20\u64ad\uff0c\u5e76\u5728\u5176\u5468\u56f4\u7ed5\u884c\u591a\u6b21\u800c\u4e0d\u6613\u6563\u5c04\u5230\u5916\u90e8\u7684\u73b0\u8c61\u3002<\/p>\n\n\n\n
\u4e00\u4e2a\u76f4\u89c2\u7684\u4f8b\u5b50\u662f\uff0c\u5728\u4f26\u6566\u5723\u4fdd\u7f57\u5927\u6559\u5802\u7684\u5706\u5f62\u7a79\u9876\u4e0b\uff0c\u5982\u679c\u4e00\u4e2a\u4eba\u5728\u4e00\u4fa7\u8f7b\u58f0\u8033\u8bed\uff0c\u53e6\u4e00\u4e2a\u4eba\u5728\u8fdc\u79bb\u6570\u5341\u7c73\u7684\u53e6\u4e00\u4fa7\u4ecd\u7136\u53ef\u4ee5\u542c\u5230\u3002\u8fd9\u662f\u56e0\u4e3a\u58f0\u97f3\u6ce2\u6cbf\u7740\u5706\u5f62\u5899\u58c1\u4f20\u64ad\uff0c\u5e76\u4fdd\u6301\u5728\u7279\u5b9a\u7684\u8def\u5f84\u4e0a\uff0c\u4ece\u800c\u51cf\u5c11\u4e86\u80fd\u91cf\u635f\u5931\u3002<\/p>\n<\/blockquote>\n\n\n\n
\u4f20\u7edf\u7684 FDTD \u65b9\u6cd5\u5728\u8ba1\u7b97 WGM \u7684\u65f6\u5019\u5f80\u5f80\u8bef\u5dee\u8f83\u5927\uff0c\u4f46\u662fNS-FDTD \u65b9\u6cd5\u7cbe\u5ea6\u66f4\u9ad8\uff0c\u56e0\u6b64\u8ba1\u7b97\u7ed3\u679c\u66f4\u63a5\u8fd1\u4e8e\u7406\u8bba\u503c\uff0c\u5982\u4e0a\u56fe\u6240\u793a\u3002<\/p>\n\n\n\n
(a) \u56fe\uff1aMie \u7406\u8bba\u7684\u89e3\u6790\u89e3\uff0c\u4f5c\u4e3a\u53c2\u8003\u6807\u51c6\u3002<\/p>\n\n\n\n
(b) \u56fe\uff1a\u4f7f\u7528 \u4f20\u7edf FDTD \u65b9\u6cd5\u5728\u7c97\u7f51\u683c\u4e0a\u8fdb\u884c\u7684\u6a21\u62df\uff0c\u7ed3\u679c\u4e0e\u7406\u8bba\u503c\u504f\u5dee\u8f83\u5927\u3002<\/p>\n\n\n\n
(c) \u56fe\uff1a\u4f7f\u7528 NS-FDTD \u65b9\u6cd5\u5728\u76f8\u540c\u7c97\u7f51\u683c\u4e0a\u7684\u6a21\u62df\uff0c\u7ed3\u679c\u4e0e Mie \u7406\u8bba \u9ad8\u5ea6\u543b\u5408\uff0c\u660e\u663e\u4f18\u4e8e\u4f20\u7edf FDTD \u8ba1\u7b97\u7ed3\u679c\u3002<\/p>\n\n\n\n
\u521d\u7ea7\u7bc7<\/h2>\n\n\n\n
\u9996\u5148\u4ecb\u7ecd\u4e00\u7ef4\u7684\u6807\u51c6\/\u975e\u6807\u51c6 FDTD \u7406\u8bba\u3002\u5728\u8ba1\u7b97\u673a\u6a21\u62df\u4e2d\uff0c\u9700\u8981\u989d\u5916\u8003\u8651\u7684\u662f\u6570\u503c\u7a33\u5b9a\u548c\u8fb9\u754c\u6761\u4ef6\u3002<\/p>\n\n\n\n
1. \u6709\u9650\u5dee\u5206\u6a21\u578b\uff08Finite Difference Model\uff09<\/h3>\n\n\n\n
\u5f88\u591a\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u6ca1\u6709\u89e3\u6790\u89e3\uff0c\u56e0\u6b64\u53ea\u80fd\u4f9d\u8d56\u6570\u503c\u6a21\u62df\u3002\u6709\u9650\u5dee\u5206\uff08FDM\uff09\u662f\u6700\u5e38\u89c1\u7684\u6570\u503c\u8ba1\u7b97\u65b9\u6cd5\uff0c\u57fa\u672c\u601d\u60f3\u662f\u7528\u5dee\u5206\u8868\u8fbe\u5f0f\u6765\u8fd1\u4f3c\u6c42\u89e3\u5fae\u5206\u65b9\u7a0b\u3002<\/p>\n\n\n\n
1.1 \u524d\u5411\u5dee\u5206\uff08Forward Finite Difference, FFD\uff09<\/h4>\n\n\n\n
\u6c42\u89e3\u4e00\u4e2a\u4e00\u7ef4\u51fd\u6570\u7684\u5bfc\u6570\uff0c\u9996\u5148\u4f7f\u7528\u6cf0\u52d2\u5c55\u5f00\uff08Taylor Series Expansion\uff09\uff0c\u5bf9\u51fd\u6570 $f(x)$ \u5728 $x+\\Delta x$ \u5904\u7684\u5c55\u5f00\uff1a
$$
f(x + \\Delta x) = f(x) + \\Delta x \\frac{df(x)}{dx} + \\frac{\\Delta x^2}{2!} \\frac{d^2 f(x)}{dx^2} + \\cdots ,
\\tag{1}
$$
\u5f53 $\\Delta x$ \u8db3\u591f\u5c0f\u7684\u65f6\u5019\uff0c\u53ef\u4ee5\u5ffd\u7565\u9ad8\u9636\u9879\uff08\u5373 $ \\frac{\\Delta x^2}{2!} \\frac{d^2 f(x)}{dx^2} + \\cdots $ \uff09\uff0c\u53ea\u4fdd\u7559\u7b2c\u4e00\u9636\u5bfc\u6570\u9879\uff0c\u6700\u540e\u6574\u7406\u53ef\u4ee5\u5f97\u5230\uff1a
$$
\\frac{df(x)}{dx} \\approx \\frac{f(x+\\Delta x) – f(x)}{\\Delta x} .
\\tag{2}
$$
\u516c\u5f0f\uff082\uff09\u79f0\u4e3a\u524d\u5411\u6709\u9650\u5dee\u5206\uff08Forward Finite Difference, FFD\uff09\u8fd1\u4f3c\u3002<\/p>\n\n\n\n
\u5728\u6cf0\u52d2\u5c55\u5f00\u4e2d\u5ffd\u7565\u4e86 $\\frac{\\Delta x^2}{2!} \\frac{d^2 f(x)}{dx^2} + \\cdots$ \u8fd9\u4e00\u9879\uff0c\u56e0\u6b64\u622a\u65ad\u8bef\u5dee\u4e3a\u4e00\u9636\u8bef\u5dee\uff0c\u5373 $O(\\Delta x) $ \u3002<\/p>\n\n\n\n
1.2 \u540e\u5411\u5dee\u5206\uff08Backward Finite Difference, BFD\uff09<\/h4>\n\n\n\n
\u7c7b\u4f3c\u5730\uff0c\u5bf9\u51fd\u6570 $f(x)$ \u5728 $x – \\Delta x$ \u5904\u7684\u5c55\u5f00\uff1a
$$
f(x – \\Delta x) = f(x) – \\Delta x \\frac{df(x)}{dx} + \\frac{\\Delta x^2}{2!} \\frac{d^2 f(x)}{dx^2} + \\cdots .
\\tag{3}
$$
\u6613\u5f97\uff1a
$$
\\frac{df(x)}{dx} \\approx \\frac{f(x) – f(x- \\Delta x)}{\\Delta x} .
\\tag{4}
$$
\u8bef\u5dee\u540c\u4e0a\u3002<\/p>\n\n\n\n
1.3 \u4e2d\u5fc3\u5dee\u5206\uff08Central Finite Difference, CFD\uff09<\/h4>\n\n\n\n
\u524d\u9762\u4e24\u4e2a\u7b97\u6cd5\u90fd\u53ea\u7528\u4e86\u5f53\u524d\u70b9\u4e0e\u4e00\u4e2a\u76f8\u90bb\u70b9\u8ba1\u7b97\uff0c\u56e0\u6b64\u7cbe\u5ea6\u8f83\u4f4e\u3002\u4e2d\u5fc3\u5dee\u5206\u5219\u662f\u4f7f\u7528\u524d\u540e\u4e24\u4e2a\u76f8\u90bb\u70b9\uff0c\u63d0\u9ad8\u4e86\u7cbe\u5ea6\u3002<\/p>\n\n\n\n
\u5bf9\u4e8e\u51fd\u6570 $f(x)$ \uff0c\u5c06\u516c\u5f0f\uff081\uff09\u4e0e\u516c\u5f0f\uff083\uff09\u76f8\u51cf\uff0c\u6d88\u53bb\u4e8c\u9636\u5bfc\u6570\u9879\uff0c\u5f97\u5230\uff1a
$$
f(x + \\Delta x) – f(x – \\Delta x) = 2\\Delta x \\frac{df}{dx} + O(\\Delta x^3) .
\\tag{5}
$$
\u8fdb\u4e00\u6b65\u6574\u7406\u5f97\u5230\uff1a
$$
\\frac{df}{dx} \\approx \\frac{f(x + \\Delta x) – f(x – \\Delta x)}{2\\Delta x} .
\\tag{6}
$$
\u6709\u4e9b\u6559\u6750\u4f1a\u7528 $ \\Delta x\/2 $ \u66ff\u6362 $ \\Delta x $ \u5f97\u5230\u516c\u5f0f\uff087\uff09\uff0c\u4e24\u8005\u5176\u5b9e\u662f\u7b49\u4ef7\u7684\u3002
$$
\\frac{df}{dx} \\approx \\frac{f(x + \\Delta x\/2) – f(x – \\Delta x\/2)}{\\Delta x} .
\\tag{7}
$$
\u4e0a\u9762\u4e24\u4e2a\u516c\u5f0f\u90fd\u88ab\u79f0\u4e3a\u4e8c\u9636\u4e2d\u5fc3\u6709\u9650\u5dee\u5206\uff08central finite difference\uff09<\/strong>\u516c\u5f0f\u3002<\/p>\n\n\n\n\u53e6\u5916\uff0c\u5bf9\u4e8e\u51fd\u6570 $f(x)$ \uff0c\u5c06\u516c\u5f0f\uff081\uff09\u4e0e\u516c\u5f0f\uff083\uff09\u76f8\u52a0\uff0c\u6d88\u53bb\u4e00\u9636\u5bfc\u6570\u9879\u540e\u5f97\u5230\uff1a
$$
f(x+ \\Delta) + f(x- \\Delta) = 2f(x) + \\Delta x^2 \\frac{d^2 f}{dx^2} + O(\\Delta x^4).
\\tag{8}
$$
\u6574\u7406\u540e\u5f97\u5230\uff1a
$$
\\frac{d^2 f}{dx^2} \\approx \\frac{f(x + \\Delta x) – 2f(x) + f(x – \\Delta x)}{\\Delta x^2}
\\tag{9}
$$<\/p>\n\n\n\n
1.4 \u9ad8\u9636\u6709\u9650\u5dee\u5206\uff08Higher-Order Finite Difference\uff09<\/h4>\n\n\n\n
\u5728\u6709\u9650\u5dee\u5206\u6cd5\u4e2d\uff0c\u901a\u8fc7\u589e\u52a0\u91c7\u6837\u70b9\u6570\u7684\u65b9\u6cd5\u63d0\u9ad8\u8ba1\u7b97\u7cbe\u5ea6\u8fdb\u800c\u5f97\u5230\u66f4\u9ad8\u9636\u7684\u5dee\u5206\u516c\u5f0f\u3002<\/p>\n\n\n\n
\u4e3a\u4e86\u63d0\u9ad8\u7cbe\u5ea6\uff0c\u8fd9\u91cc\u5728\u4e8c\u9636\u4e2d\u5fc3\u6709\u9650\u5dee\u5206\u6cd5\u7684\u57fa\u7840\u4e0a\u989d\u5916\u589e\u52a0\u4e24\u4e2a\u91c7\u6837\u70b9 $x + 2\\Delta x, x – 2\\Delta x$ \uff0c\u5e76\u5728\u6b64\u5904\u8fdb\u884c\u6cf0\u52d2\u5c55\u5f00\uff1a<\/p>\n\n\n\n
\u5bf9\u4e8e\u53f3\u4fa7\u70b9\uff0c\u6cf0\u52d2\u5c55\u5f00\u4e3a\uff1a
$$
f(x+2\\Delta x) = f(x) + 2\\Delta x\\frac{df(x)}{dx} + 2\\Delta x^2 \\frac{d^2f(x)}{dx^2}+O(\\Delta x^3) ,
\\tag{10}
$$
\u7c7b\u4f3c\u5730\uff0c
$$
f(x – 2\\Delta x) = f(x) – 2\\Delta x \\frac{df(x)}{dx} + 2\\Delta x^2 \\frac{d^2 f(x)}{dx^2} + O(\\Delta x^3) .
\\tag{11}
$$
\u901a\u8fc7\u7ebf\u6027\u7ec4\u5408\uff08\u6bd4\u5982\u5c06\u516c\u5f0f\uff0811\uff09\u53d6\u53cd\u518d\u53d6\u534a\uff0c\u4e0a\u9762\u4e24\u5f0f\u76f8\u52a0\uff09\uff0c\u6d88\u9664\u9ad8\u9636\u8bef\u5dee\u9879\uff0c\u5f97\u5230\uff1a
$$
\\frac{d^2 f(x)}{dx^2} \\approx \\frac{1}{\\Delta x^2} \\left[ \\frac{4}{3} (f(x + \\Delta x) + f(x – \\Delta x)) – \\frac{1}{12} (f(x + 2\\Delta x) + f(x – 2\\Delta x)) – \\frac{5}{2} f(x) + \\cdots \\right] .
\\tag{12}
$$
\u8fd9\u4e2a\u6a21\u578b\u4f7f\u7528\u4e86\u56db\u4e2a\u70b9\uff0c\u5e76\u4e14\u901a\u8fc7\u52a0\u6743\u5e73\u5747\u51cf\u5c11\u4e86\u8bef\u5dee\u3002\u4f46\u662f\u4f1a\u5e26\u6765\u6570\u503c\u4e0d\u7a33\u5b9a\u7b49\u6f5c\u5728\u95ee\u9898\u3002<\/p>\n\n\n\n
\u5f53\u6211\u4eec\u7528\u9ad8\u9636\u6709\u9650\u5dee\u5206\u6765\u53d6\u4ee3\u5fae\u5206\u65b9\u7a0b\u65f6\uff0c\u4f1a\u5f15\u5165\u989d\u5916\u7684\u6570\u503c\u89e3\uff0c\u5373\u865a\u5047\u89e3\uff08spurious solutions\uff09\u3002\u8fd9\u4e9b\u89e3\u5b9e\u9645\u5e76\u4e0d\u5b58\u5728\uff0c\u662f\u7531\u4e8e\u9ad8\u9636\u5dee\u5206\u65b9\u7a0b\u7684\u79bb\u6563\u5316\u7279\u6027\u5bfc\u81f4\u7684\u3002<\/p>\n\n\n\n
\n\u6bd4\u5982\u4e00\u9636\u5fae\u5206\u65b9\u7a0b\u5982 $ \\frac{df}{dx} = f(x) $ \u901a\u5e38\u6709\u4e00\u4e2a\u72ec\u7acb\u89e3\uff0c\u4e8c\u9636\u5fae\u5206\u65b9\u7a0b\u5982\u6ce2\u52a8\u65b9\u7a0b $\\frac{d^2f}{dx^2}=k^2f(x)$ \u901a\u5e38\u4f1a\u6709\u4e24\u4e2a\u72ec\u7acb\u89e3\uff08\u4e24\u4e2a\u81ea\u7531\u53c2\u6570\uff09\uff0c\u56db\u9636\u5fae\u5206\u65b9\u7a0b\u4f1a\u6709\u56db\u4e2a\u72ec\u7acb\u89e3\u7b49\u7b49\u3002<\/p>\n\n\n\n
\u5728\u6570\u503c\u8ba1\u7b97\u4e2d\uff0c\u82e5\u4f7f\u7528\u56db\u9636\u6709\u9650\u5dee\u5206\u65b9\u6cd5\u6765\u8fd1\u4f3c\u4e00\u4e2a\u539f\u672c\u4e3a\u4e8c\u9636\u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u90a3\u4e48\u5dee\u5206\u65b9\u7a0b\u7684\u7ed3\u675f\u5c31\u4f1a\u5f3a\u884c\u63d0\u9ad8\u4e86\uff0c\u8fdb\u800c\u5bfc\u81f4\u989d\u5916\u7684\u865a\u5047\u89e3\u3002\u8fd9\u4e9b\u591a\u51fa\u6765\u7684\u89e3\u5e76\u4e0d\u5bf9\u5e94\u539f\u672c\u7684\u7269\u7406\u7cfb\u7edf\u3002<\/p>\n<\/blockquote>\n\n\n\n
\u5c24\u5176\u662f\u63cf\u8ff0\u7535\u78c1\u6ce2\u4f20\u64ad\u7684\u4e8c\u9636\u5fae\u5206\u6ce2\u52a8\u65b9\u7a0b\uff0c\u5e94\u8be5\u4f7f\u7528\u4e8c\u9636\u4e2d\u5fc3\u6709\u9650\u5dee\u5206\u6cd5\uff0c\u800c\u4e0d\u662f\u66f4\u9ad8\u9636\u7684\u65b9\u6cd5\u3002<\/p>\n\n\n\n
2. \u4e00\u7ef4\u6ce2\u52a8\u65b9\u7a0b<\/h3>\n\n\n\n
\u5728\u8ba1\u7b97\u673a\u4e0a\u6a21\u62df\u6ce2\u7684\u4f20\u64ad\uff0c\u9700\u8981\u7528\u5230\u6570\u503c\u65b9\u6cd5\u8fd1\u4f3c\u8ba1\u7b97\u3002\u5176\u4e2d\uff0c\u6709\u9650\u5dee\u5206\u65f6\u57df\u6cd5\u5219\u53d1\u6325\u5f3a\u5927\u7684\u4f5c\u7528\u3002<\/p>\n\n\n\n
\u4e00\u7ef4\u6ce2\u52a8\u65b9\u7a0b\u7684\u6570\u5b66\u8868\u8fbe\u4e3a\uff1a
$$
\\frac{\\partial^2 \\psi}{\\partial t^2} = v^2 \\frac{\\partial^2 \\psi}{\\partial x^2} ,
\\tag{1}
$$
\u5176\u4e2d $\\psi(x,t)$ \u8868\u793a\u6ce2\u7684\u632f\u5e45\uff0c $v$ \u662f\u6ce2\u7684\u4f20\u64ad\u901f\u5ea6\uff08\u5149\u5728\u771f\u7a7a\u4e2d\u662f $3 \\times 10^8$ \u7c73\/\u79d2\uff09\u3002<\/p>\n\n\n\n
\u8fd9\u4e2a\u65b9\u7a0b\u7684\u7269\u7406\u610f\u4e49\u662f\uff1a\u6ce2\u7684\u53d8\u5316\u4e0e\u65f6\u95f4\u3001\u7a7a\u95f4\u53d8\u5316\u6709\u5173\u3002<\/p>\n\n\n\n
\u5bf9\u4e8e\u4e00\u4e2a\u65e0\u9650\u957f\u7684\u6ce2\u52a8\u4ecb\u8d28\uff0c\u89e3\u901a\u5e38\u662f\u4e00\u4e2a\u5f62\u5f0f\u975e\u5e38\u7b80\u5355\u7684\u884c\u6ce2\uff1a
$$
\\psi(x,t) = A \\cos(kx – \\omega t) + B \\sin(kx – \\omega t) .
\\tag{2}
$$
\u8fd9\u79cd\u60c5\u51b5\u9002\u7528\u4e8e\u81ea\u7531\u7a7a\u95f4\u4f20\u64ad\u7684\u7535\u78c1\u6ce2\u3002\u6bd4\u5982\u56fa\u5b9a\u8fb9\u754c\uff0c\u884c\u6ce2\u53ef\u4ee5\u7528\u6b63\u5f26\u51fd\u6570\u5c55\u5f00\uff1a
$$
\\psi(x,t) = \\sum_{n} A_n \\sin(k_n x) \\cos(\\omega_n t).
\\tag{3}
$$
\u4f46\u662f\u5982\u679c\u8fb9\u754c\u4e0d\u89c4\u5219\uff0c\u4f8b\u5982\u7535\u78c1\u6ce2\u5728\u5730\u9762\u4e0a\u53cd\u5c04\u3001\u6c34\u6ce2\u51b2\u51fb\u6d77\u5cb8\u7ebf\u3001\u58f0\u6ce2\u5728\u591a\u4e2a\u623f\u95f4\u4f20\u64ad\u7b49\u60c5\u666f\uff0c\u5219\u65e0\u6cd5\u7b80\u5355\u5730\u5c55\u5f00\u6210\u6b63\u5f26\u6216\u8005\u6307\u6570\u7684\u5f62\u5f0f\uff0c\u89e3\u6790\u6c42\u89e3\u5c06\u6781\u5176\u590d\u6742\u3002\u6bd4\u5982\u58f0\u6ce2\u63a5\u89e6\u4e0d\u540c\u4ecb\u8d28\u7684\u5899\u9762\u4f1a\u6709\u4e0d\u540c\u7684\u53cd\u5c04\u8def\u5f84\uff0c\u5bfc\u81f4\u58f0\u6ce2\u7684\u4f20\u64ad\u8def\u5f84\u53d8\u5f97\u65e0\u6cd5\u76f4\u63a5\u6c42\u89e3\u3002<\/p>\n\n\n\n
\u89e3\u6790\u89e3\u901a\u5e38\u4f1a\u5047\u8bbe\u6ce2\u901f $v$ \u662f\u6052\u5b9a\u7684\uff0c\u4e5f\u5c31\u662f\u6ce2\u5728\u5747\u5300\u4ecb\u8d28\u4e2d\u4f20\u64ad\u3002\u4f46\u662f\u5b9e\u9645\u4e0a\u6ce2\u4f1a\u7a7f\u8fc7\u975e\u5747\u5300\u4ecb\u8d28\uff0c\u5982\u58f0\u6ce2\u5728\u4e0d\u540c\u6e29\u5ea6\u7684\u623f\u95f4\u5177\u6709\u4e0d\u540c\u7684\u4f20\u64ad\u901f\u5ea6\u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6ce2\u52a8\u65b9\u7a0b\u4f1a\u53d8\u6210\u53d8\u7cfb\u6570\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u5982\uff1a
$$
\\frac{\\partial^2 \\psi}{\\partial t^2} = v(x)^2 \\frac{\\partial^2 \\psi}{\\partial x^2} ,
\\tag{4}
$$
\u5176\u4e2d\u901f\u5ea6 $v$ \u4f1a\u968f\u7740\u4f4d\u7f6e $x$ \u53d8\u5316\uff0c\u5bfc\u81f4\u65e0\u6cd5\u76f4\u63a5\u4f7f\u7528\u5085\u7acb\u53f6\u53d8\u6362\u6c42\u51fa\u89e3\u6790\u503c\u3002<\/p>\n\n\n\n
\u53c8\u6216\u8005\u662f\u6ce2\u52a8\u65b9\u7a0b\u53f3\u4fa7\u6709\u6fc0\u52b1\u6e90\uff0c\u5373\u591a\u4e2a\u6ce2\u5f62\u90fd\u6709\u53ef\u80fd\u4e0d\u540c\u7684\u6e90\u76f8\u4e92\u5e72\u6d89\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e5f\u5f88\u96be\u6c42\u51fa\u89e3\u6790\u89e3\u3002<\/p>\n\n\n\n
\u5e76\u4e14\u5728\u771f\u5b9e\u4e16\u754c\u4e2d\uff0c\u80fd\u91cf\u7684\u4f20\u64ad\u4f1a\u6709\u635f\u8017\uff0c\u56e0\u6b64\u9700\u8981\u5bf9\u6ce2\u52a8\u65b9\u7a0b\u5f15\u5165\u635f\u8017\u9879\uff0c\u6b64\u65f6\u6ce2\u52a8\u65b9\u7a0b\u5c31\u4f1a\u53d8\u6210\u975e\u7ebf\u6027\u65b9\u7a0b\u6216\u8005\u9ad8\u9636\u5fae\u5206\u65b9\u7a0b\uff0c\u4f7f\u5f97\u65e0\u6cd5\u76f4\u63a5\u6c42\u89e3\u3002<\/p>\n\n\n\n
\u56e0\u6b64\u6211\u4eec\u5f15\u5165\u6709\u9650\u5dee\u5206\u65f6\u57df\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e9b\u95ee\u9898\u3002<\/p>\n\n\n\n
2.1 \u6807\u51c6 FDTD \u7b97\u6cd5<\/h4>\n\n\n\n
\u7531\u4e8e\u8ba1\u7b97\u673a\u4e0d\u80fd\u76f4\u63a5\u8ba1\u7b97\u8fde\u7eed\u7684\u6570\u5b66\u65b9\u7a0b\uff0c\u56e0\u6b64\u9700\u8981\u79bb\u6563\u5316<\/strong>\uff0c\u5373\u5c06\u7a7a\u95f4\u4e0e\u65f6\u95f4\u5212\u5206\u4e3a\u7f51\u683c\uff0c\u7136\u540e\u8ba9\u8ba1\u7b97\u673a\u9010\u6b65\u8ba1\u7b97\u6bcf\u4e2a\u683c\u5b50\u7684\u6ce2\u52a8\u60c5\u51b5\u3002<\/p>\n\n\n\n\u5728\u8fde\u7eed\u7684\u5f62\u5f0f\u4e0b\uff0c\u4e00\u7ef4\u6ce2\u52a8\u65b9\u7a0b
$$
\\frac{\\partial^2 \\psi}{\\partial t^2} = v^2 \\frac{\\partial^2 \\psi}{\\partial x^2} ,
\\tag{1*}
$$
\u7684\u7b49\u4ef7\u5f62\u5f0f\u662f
$$
\\left( \\frac{\\partial^2}{\\partial t^2} – v^2 \\frac{\\partial^2}{\\partial x^2} \\right) \\psi(x,t) = 0.
\\tag{5}
$$
\u56e0\u6b64\uff0c\u7528\u95f4\u9694 $\\Delta x$ \u79bb\u6563\u7a7a\u95f4\uff0c\u6bcf\u4e2a\u70b9\u7528\u7d22\u5f15 $i$ \u6765\u8868\u793a $x = i\\Delta x$ \uff1b\u7528\u95f4\u9694 $\\Delta t$ \u79bb\u6563\u65f6\u95f4\uff0c\u6bcf\u4e2a\u65f6\u523b\u7528\u7f29\u5f71 $n$ \u6765\u8868\u793a $t = n\\Delta t$ \uff0c\u5176\u4e2d $n$ \u5747\u4e3a\u6574\u6570\u3002\u6240\u4ee5\u628a\u6ce2\u51fd\u6570\u5199\u4e3a\uff1a
$$
\\psi_i^n = \\psi(i\\Delta x, n\\Delta t) .
\\tag{6}
$$
\u7136\u540e\u7528\u4e2d\u5fc3\u6709\u9650\u5dee\u5206\uff08central finite difference\uff09\u65b9\u6cd5\u8ba1\u7b97\u6ce2\u51fd\u6570\u7684\u5fae\u5206\u3002<\/p>\n\n\n\n
\u65f6\u95f4\u4e0e\u7a7a\u95f4\u65b9\u5411\u7684\u4e8c\u9636\u504f\u5bfc\u51fd\u6570\u7684\u4e2d\u5fc3\u6709\u9650\u5dee\u5206\u8fd1\u4f3c\u5206\u522b\u662f
$$
\\frac{\\partial^2 \\psi}{\\partial t^2} \\approx \\frac{\\psi_i^{n+1} – 2\\psi_i^n + \\psi_i^{n-1}}{\\Delta t^2} ,
\\tag{7}
$$<\/p>\n\n\n\n
$$
\\frac{\\partial^2 \\psi}{\\partial x^2} \\approx \\frac{\\psi_{i+1}^{n} – 2\\psi_i^n + \\psi_{i-1}^{n}}{\\Delta x^2} .
\\tag{8}
$$<\/p>\n\n\n\n
\u5176\u4e2d $\\psi_{i+1}^{n}$ \u662f\u53f3\u4fa7\u76f8\u90bb\u7f51\u683c\u70b9\u7684\u6ce2\u52a8\u503c\u3002<\/p>\n\n\n\n
\u5c06\u516c\u5f0f\uff087\uff09\u548c\u516c\u5f0f\uff088\uff09\u4ee3\u5165\u6ce2\u52a8\u65b9\u7a0b\uff08\u516c\u5f0f\uff081\uff09\uff09\uff0c\u5f97\u5230\u516c\u5f0f\uff089\uff09\u3002
$$
\\frac{\\psi_i^{n+1} – 2\\psi_i^n + \\psi_i^{n-1}}{\\Delta t^2} = v^2 \\frac{\\psi_{i+1}^{n} – 2\\psi_i^n + \\psi_{i-1}^{n}}{\\Delta x^2} ,
\\tag{9}
$$
\u6574\u7406\u540e\u4fbf\u5f97\u5230\u6807\u51c6\u76841D FDTD\u8ba1\u7b97\u516c\u5f0f\uff0c1D standard finite difference time domain (FDTD) algorithm<\/strong>\uff1a
$$
\\psi_i^{n+1} = 2\\psi_i^n – \\psi_i^{n-1} + \\left( \\frac{v^2 \\Delta t^2}{\\Delta x^2} \\right) (\\psi_{i+1}^{n} – 2\\psi_i^n + \\psi_{i-1}^{n}) .
\\tag{10}
$$
\u8fd9\u4e2a\u516c\u5f0f\u7684\u4f9d\u8d56\u65f6\u95f4\u3001\u7a7a\u95f4\u4e2d\u524d\u540e\u4e24\u4e2a\u6b65\u8fdb\u70b9\u8ba1\u7b97\u7684\u3002\u4e5f\u5c31\u662f\u8bf4\u5f53\u524d\u70b9\u7684\u72b6\u6001\u6536\u5230\u76f8\u90bb\u70b9\u7684\u5f71\u54cd\u3002<\/p>\n\n\n\n2.2 \u975e\u6807\u51c6 FDTD \u7b97\u6cd5<\/h4>\n\n\n\n
\u5728\u6807\u51c6 FDTD \u65b9\u6cd5\u4e2d\uff0c\u4f7f\u7528\u4e2d\u5fc3\u5dee\u5206\u8fd1\u4f3c\u79bb\u6563\u6ce2\u52a8\u65b9\u7a0b\u3002\u4f46\u662f\u8fd9\u4e2a\u65b9\u6cd5\u5b58\u5728\u6570\u503c\u8272\u6563\uff08numerical dispersion\uff09<\/strong>\u95ee\u9898\u3002\u5728\u8f83\u7c97\u7f51\u683c\u7684\u65f6\u5019\u4f1a\u6709\u5f88\u5927\u7684\u8bef\u5dee\u3002<\/p>\n\n\n\n\u975e\u6807\u51c6 FDTD \u7b97\u6cd5\u5219\u89e3\u51b3\u4e86\u4e0a\u8ff0\u95ee\u9898\u3002NS-FDTD \u5bf9\u5355\u8272\u6ce2\u505a\u7279\u522b\u5904\u7406\uff0c\u5373\u4f7f\u6570\u503c\u79bb\u6563\u4e5f\u80fd\u786e\u4fdd\u7cbe\u5ea6\u3002<\/p>\n\n\n\n
\u9996\u5148\u5047\u8bbe\u6211\u4eec\u7814\u7a76\u7684\u662f\u67d0\u79cd\u5355\u8272\u6ce2
$$
\\psi(x,t) \\;=\\; e^{\\,i(kx – \\omega t)},
\\tag{11}
$$
\u5176\u4e2d $k$ \u4e3a\u6ce2\u6570\uff08$k = 2\\pi \/ \\lambda$\uff09\uff0c $\\omega$ \u4e3a\u89d2\u9891\u7387\uff08$\\omega = 2\\pi f$\uff09\u3002<\/p>\n\n\n\n
\u516c\u5f0f\uff0811\uff09\u8fde\u7eed\u60c5\u51b5\u4e0b\u7684\u7a7a\u95f4\u5fae\u5206\u662f
$$
\\frac{\\partial \\psi}{\\partial x} \\;=\\; i\\,k\\, \\psi(x,t).
\\tag{12}
$$
\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5982\u679c\u76f4\u63a5\u7528\u6807\u51c6\u5dee\u5206\u6765\u7b97 $\\Delta_x \\psi(x,t)$ \uff0c\u4f1a\u5f97\u5230
$$
\\Delta_x \\psi(x,t)
= \\psi(x+\\Delta x,t) – \\psi(x,t)
= e^{\\,i(kx – \\omega t)}\\bigl(e^{\\,i\\,k\\,\\Delta x} – 1\\bigr) ,
\\tag{13}
$$
\u548c\u771f\u5b9e\u7684\u5fae\u5206\u5e76\u4e0d\u4e00\u81f4\u3002\u53ea\u6709\u5f53 $\\Delta x$ \u8db3\u591f\u5c0f\u65f6\uff0c\u624d\u80fd\u8fd1\u4f3c\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u7f51\u683c\u8f83\u7c97\uff0c\u90a3\u4e48\u5c31\u4f1a\u4ea7\u751f\u6240\u8c13\u7684\u8bef\u5dee\u3002<\/p>\n\n\n\n
NS-FDTD \u5f15\u5165\u4fee\u6b63\u56e0\u5b50 $s(\\Delta x)$ \u8ba9\u79bb\u6563\u7684\u7b97\u7b26\u5c3d\u53ef\u80fd\u51cf\u5c11\u8bef\u5dee
$$
\\Delta_x \\psi(x,t)
\\approx
s(\\Delta x)\\,\\frac{\\partial \\psi(x,t)}{\\partial x}.
\\tag{14}
$$
\u8fd9\u4e2a $s(\\Delta x)$ \u7531 $\\Delta x$ \u5904\u7684\u76f8\u4f4d\u53d8\u5316\u91cf\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f
$$
\\Delta_x \\psi(x,t) = \\psi(x+\\Delta x,t) – \\psi(x,t) .
\\tag{15}
$$
\u90a3\u4e48\u8bef\u5dee\u56e0\u5b50\u7684\u5927\u5c0f\u7531\u516c\u5f0f\uff0812\uff09\uff0813\uff09\uff0814\uff09\uff0815\uff09\u5171\u540c\u63a8\u5bfc\u51fa
$$
s(\\Delta x)<\/p>\n\n\n\n
=\\frac{\\Delta_x \\psi(x,t)}{\\partial_x \\psi(x,t)}<\/h1>\n\n\n\n
\\frac{e^{\\,i\\,k\\,\\Delta x} – 1}{i\\,k\\,\\Delta x}<\/p>\n\n\n\n
\\times \\Delta x<\/h1>\n\n\n\n
\\frac{e^{ik\\Delta x} – 1}{ik}.
\\tag{16}
$$
\u901a\u8fc7\u6b27\u62c9\u516c\u5f0f\uff0c\u5c06\u8bef\u5dee\u56e0\u5b50\u7b80\u5316\u4e3a
$$
s(\\Delta x)
= \\frac{2}{k}\\,\\sin!\\Bigl(\\frac{k\\,\\Delta x}{2}\\Bigr)\\,e^{\\,i\\,\\frac{k\\,\\Delta x}{2}} .
\\tag{17}
$$
\u6ce8\u610f\u5230\uff0c\u5f53 $\\Delta x \\to 0$ \u65f6\uff0c\u975e\u6807\u51c6\u5dee\u5206\u5c31\u9000\u5316\u56de\u201c\u6807\u51c6\u4e2d\u5fc3\u5dee\u5206\u201d\u7684\u60c5\u5f62\uff0c\u4e0e\u771f\u6b63\u7684\u5bfc\u6570\u8fd1\u4f3c\u51e0\u4e4e\u4e00\u81f4\u3002\u5176\u5b9e\uff0c\u516c\u5f0f\uff0817\uff09\u5305\u542b\u4e86\u76f8\u4f4d\u56e0\u5b50\u548c\u5e45\u5ea6\u4e24\u4e2a\u90e8\u5206\uff0c\u524d\u8005\u5bf9\u5e94\u6307\u6570\u90e8\u5206\u3002<\/p>\n\n\n\n
\u7c7b\u4f3c\u5730\uff0c\u63a8\u5bfc\u51fa\u65f6\u95f4\u65b9\u5411\u7684\u4fee\u6b63\u51fd\u6570
$$
s(\\Delta t)
= \\frac{2}{\\omega}\\,\\sin\\Bigl(\\frac{\\omega\\,\\Delta t}{2}\\Bigr)e^{-\\,i\\omega\\,\\Delta t\/2}.
\\tag{18}
$$
\u603b\u7ed3\u4e00\u4e0b\uff0c\u6211\u4eec\u4ece\u6ce2\u52a8\u65b9\u7a0b\u5f00\u59cb\uff0c\u5c06\u7a7a\u95f4\u548c\u65f6\u95f4\u7684\u79bb\u6563\u5316\u4e4b\u540e\u7528\u4e2d\u5fc3\u5dee\u5206\u8fd1\u4f3c\uff0c\u7136\u540e\u5f15\u5165\u4fee\u6b63\u51fd\u6570\uff08\u5173\u4e8e\u4fee\u6b63\u51fd\u6570\u7684\u6307\u6570\u9879\u5176\u5b9e\u88ab\u9690\u542b\u5730\u5904\u7406\u4e86\uff09
$$
\\frac{\\psi_i^{n+1} – 2\\psi_i^n + \\psi_i^{n-1}}{\\left(\\frac{2}{\\omega} \\sin(\\omega \\Delta t \/ 2)\\right)^2} =
v^2 \\frac{\\psi_{i+1}^{n} – 2\\psi_i^n + \\psi_{i-1}^{n}}{\\left(\\frac{2}{k} \\sin(k \\Delta x \/ 2)\\right)^2}.
\\tag{19}
$$
\u6574\u7406\u540e\u5f97\u5230
$$
\\psi_i^{n+1} = -\\psi_i^{n-1} + \\left(2 + u_{\\text{NS}}^2 d_x^2 \\right) \\psi_i^n ,
\\tag{20}
$$
\u5176\u4e2d
$$
u_{\\text{NS}} = \\frac{\\sin(\\omega \\Delta t \/ 2)}{\\sin(k \\Delta x \/ 2)} .
\\tag{21}
$$<\/p>\n\n\n\n
3. \u4e00\u7ef4 FDTD \u7a33\u5b9a\u6027<\/h3>\n\n\n\n
\u5728\u505a\u6570\u503c\u6a21\u62df\u65f6\uff0c\u6211\u4eec\u5e0c\u671b\u65f6\u95f4\u6b65\u957f $\\Delta t$ \u5c3d\u91cf\u5927\uff0c\u4ece\u800c\u51cf\u5c11\u603b\u7684\u8ba1\u7b97\u6b65\u6570\uff0c\u4f46\u53c8\u4e0d\u80fd\u8d85\u8fc7\u67d0\u4e2a\u6781\u9650\uff0c\u5426\u5219\u6570\u503c\u89e3\u4f1a\u53d1\u6563\u3002\u8fd9\u4e2a\u6781\u9650\u7a76\u7adf\u662f\u600e\u4e48\u63a8\u5bfc\u51fa\u6765\u7684\uff1f<\/p>\n","protected":false},"excerpt":{"rendered":"
Nonstandard FDTD \u7b14\u8bb0 \u524d\u8a00 \u8be5\u7b14\u8bb0\u5206\u4e3a\u521d\u7ea7\u3001\u4e2d\u7ea7\u548c\u9ad8\u7ea7\u4e09\u4e2a\u90e8\u5206\u3002 \u521d\u7ea7\u7bc7\u4e3b\u8981\u4ecb\u7ecd\u4e00\u7ef4\u7684\u6807\u51c6\u548c […]<\/p>\n","protected":false},"author":1,"featured_media":2368,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[53],"tags":[],"class_list":["post-2367","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tech"],"_links":{"self":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts\/2367","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/comments?post=2367"}],"version-history":[{"count":5,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts\/2367\/revisions"}],"predecessor-version":[{"id":2374,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts\/2367\/revisions\/2374"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/media\/2368"}],"wp:attachment":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/media?parent=2367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/categories?post=2367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/tags?post=2367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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