\u8fd9\u79cd\u8fed\u4ee3\u7684\u672c\u8d28\u601d\u60f3\uff0c\u662f\u5c06\u5fae\u5206\u65b9\u7a0b\uff0c\u8f6c\u6362\u4e3a\u5dee\u5206\u65b9\u7a0b\u7684\u6570\u5b66\u65b9\u6cd5\uff0c\u901a\u8fc7\u53d8\u4e3a\u5dee\u5206\u95ee\u9898\uff0c\u4f7f\u5f97PBD\u8fed\u4ee3\u6709\u5e8f\uff08\u6216\u8005\u8bf4\u6709\u68af\u5ea6\uff09\u8d8b\u5411\u4e8e\u76ee\u6807\u4f4d\u7f6e\uff0c\u672c\u6587\u7b80\u5355\u4ecb\u7ecd\u4e00\u4e0bPBD\u7684\u5e38\u7528\u8fed\u4ee3\u8ba1\u7b97\u65b9\u6cd5\u3002<\/p>\n
\u6b27\u62c9\u79ef\u5206\u4e5f\u662f\u73b0\u5728\u5e38\u7528\u7684\u6e38\u620f\u5f15\u64ce\u7684\u9ed8\u8ba4\u8fed\u4ee3\u65b9\u6cd5\uff0c\u5176\u7279\u70b9\u5c31\u662f\u7b97\u7684\u5feb\uff0c\u800c\u4e14\u7406\u89e3\u7b80\u5355\uff0c\u8ba1\u7b97\u5feb\u901f\u5728\u6e38\u620f\u884c\u4e1a\u975e\u5e38\u91cd\u8981\uff0c\u4f46\u662f\u7f3a\u70b9\u5c31\u662f\u727a\u7272\u4e86\u4e00\u5b9a\u7684\u7a33\u5b9a\u6027\uff0c\u6216\u8005\u8bf4\u662f\u6b27\u62c9\u79ef\u5206\u7684\u4e0d\u7a33\u5b9a\u6027\u662f\u7d2f\u52a0<\/strong>\u7684\uff0c\u8fd9\u4e2a\u7279\u6027\u51b3\u5b9a\u4e86\u5728\u7269\u7406\u4eff\u771f\u5c24\u5176\u662f\u7cbe\u786e\u7269\u7406\u4eff\u771f\uff0c\u9700\u8981\u614e\u91cd\u4f7f\u7528\u8fd9\u4e2a\u65b9\u6cd5<\/p>\n \u5728\u663e\u5f0f\u6b27\u62c9\u4e2d\uff0c\\(<\/span>x_n\\)<\/span>\u548c\\(<\/span>y_n\\)<\/span>\u5206\u522b\u7528\u4e8e\u5904\u7406\u8868\u8fbe\u5f53\u524d\u6b65\u9aa4\u7684\u6c42\u548c\u6c42\u89e3<\/p>\n \\( y_{n+1} = y_n + h \\cdot f(x_n, y_n)\\) <\/span><\/p>\n \\( y_{n+1} = y_n + h \\cdot f(x_{n+1}, y_{n+1})\\)<\/span><\/p>\n \u663e\u5f0f\u6b27\u62c9\u65b9\u6cd5\u4e0d\u540c\u7684\u662f\uff0c\u9690\u5f0f\u6b27\u62c9\u65b9\u6cd5\u5728\u66f4\u65b0\u89e3\u65f6\u9700\u8981\u901a\u8fc7\u8fed\u4ee3\u6765\u6c42\u89e3 \\(y_{n+1}\\)<\/span>\u3002\u901a\u5e38\uff0c\u53ef\u4ee5\u4f7f\u7528\u725b\u987f\u8fed\u4ee3\u6cd5\u7b49\u65b9\u6cd5\u6765\u627e\u5230 \\(y_{n+1}\\)<\/span>\u3002\u8fd9\u4f7f\u5f97\u9690\u5f0f\u6b27\u62c9\u65b9\u6cd5\u66f4\u7a33\u5b9a\uff0c\u9002\u7528\u4e8e\u4e00\u4e9b\u663e\u5f0f\u65b9\u6cd5\u53ef\u80fd\u5931\u6548\u7684\u60c5\u51b5\uff0c\u4f46\u6bcf\u4e2a\u6b65\u9aa4\u7684\u8fed\u4ee3\u8fc7\u7a0b\u4f1a\u589e\u52a0\u8ba1\u7b97\u590d\u6742\u6027\u3002<\/span><\/p>\n \u534a\u9690\u5f0f\u6b27\u62c9\u65b9\u6cd5\uff08Semi-Implicit Euler method\uff09\u7ed3\u5408\u4e86\u663e\u5f0f\u548c\u9690\u5f0f\u6b27\u62c9\u65b9\u6cd5\u7684\u7279\u70b9\u3002\u5176\u4e2d\uff0c\u4e00\u90e8\u5206\u7684\u66f4\u65b0\u662f\u663e\u5f0f\u7684\uff0c\u800c\u53e6\u4e00\u90e8\u5206\u5219\u662f\u9690\u5f0f\u7684<\/span><\/p>\n \\begin{align*} \u5728\u8fd9\u91cc\uff0c\u9996\u5148\u8fdb\u884c\u663e\u5f0f\u6b65\u9aa4\uff0c\u4f7f\u7528\u5f53\u524d\u6b65\u9aa4\u7684\u89e3\\(x_n\\)\u548c\\(y_n\\)<\/span>\u66f4\u65b0\u4e00\u4e2a\u4e2d\u95f4\u503c \\(y^*_{n+1}\\)<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u7136\u540e\uff0c\u5728\u9690\u5f0f\u6b65\u9aa4\u4e2d\uff0c\u4f7f\u7528\u4e2d\u95f4\u503c \\(y^*_{n+1}\\)\u200b <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c\u4e0b\u4e00\u4e2a\u6b65\u9aa4\u7684\u4f4d\u7f6e \\(x_{n+1}\\)<\/span>\u00a0\u6765\u66f4\u65b0\u6700\u7ec8\u7684\u89e3 \\(y_{n+1}\\)<\/span>\u3002\u8fd9\u79cd\u65b9\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u63d0\u9ad8\u6570\u503c\u7a33\u5b9a\u6027\uff0c\u5e76\u51cf\u5c11\u5bf9\u6c42\u89e3\u5668\u7684\u4f9d\u8d56\uff0c\u4f46\u4ecd\u7136\u4fdd\u6301\u4e86\u4e00\u5b9a\u7684\u8ba1\u7b97\u6548\u7387\u3002<\/span><\/p>\n Verlet\u79ef\u5206\u662f\u4e00\u79cd\u7528\u4e8e\u6a21\u62df\u5206\u5b50\u52a8\u529b\u5b66\u548c\u5176\u4ed6\u6a21\u62df\u4e2d\u7684\u6570\u503c\u79ef\u5206\u65b9\u6cd5\uff0c\u7279\u522b\u9002\u7528\u4e8e\u4fdd\u6301\u7269\u7406\u7cfb\u7edf\u80fd\u91cf\u5b88\u6052\u7684\u95ee\u9898\uff0c\u5728PBD\u7b97\u6cd5\u4e2d\uff0c\u66f4\u52a0\u9002\u5408\u7528\u4e8e\u505a\u6bcf\u4e00\u6b65\u7684\u8fed\u4ee3\u8ba1\u7b97\uff0c\u548c\u6b27\u62c9\u79ef\u5206\u53ea\u8003\u8651\u5f53\u524d\u548c\u4e0b\u4e00\u65f6\u523b\uff08\u6216\u8005\u4e0a\u4e00\u65f6\u523b\u548c\u5f53\u524d\u65f6\u523b\uff09\u4e0d\u540c\uff0cVerlet\u79ef\u5206\u540c\u65f6\u8003\u8651\u4e0a\u4e00\u65f6\u523b\u3001\u5f53\u524d\u65f6\u523b\u3001\u4e0b\u4e00\u65f6\u523b\uff0c\u4e5f\u662f\u6211\u4eec\u4f7f\u7528\u6a21\u62df\u7684\u9996\u9009\u3002<\/span><\/p>\n \\begin{align*} &\\text{\u521d\u59cb\u5316:} \\\\ &\\quad y_0, v_0 \\quad \\text{(\u521d\u59cb\u4f4d\u7f6e\u548c\u901f\u5ea6)} \\\\ &\\text{\u65f6\u95f4\u6b65:} \\\\ &\\quad y_{n+1} = 2y_n – y_{n-1} + h^2 \\cdot a_n \\quad \\text{(\u66f4\u65b0\u4f4d\u7f6e)} \\\\ &\\quad v_{n+1} = \\frac{y_{n+1} – y_{n-1}}{2h} \\quad \\text{(\u66f4\u65b0\u901f\u5ea6)} \\\\ &\\quad a_n = f(y_n) \\quad \\text{(\u8ba1\u7b97\u52a0\u901f\u5ea6)} \\\\ \\end{align*}<\/span><\/p>\n \u6ce8\u610f yn<\/span><\/span><\/span>\u00a0\u662f\u4f4d\u7f6e\uff0cvn<\/span><\/span><\/span><\/span>\u00a0\u662f\u901f\u5ea6\uff0can<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u52a0\u901f\u5ea6\uff0ch <\/span><\/span><\/span><\/span>\u662f\u65f6\u95f4\u6b65\u957f<\/span><\/p><\/blockquote>\n \u53e6\u5916\u6709\u4e00\u79cd\u5199\u6cd5\u662f\u8fd9\u6837\u7684\uff1a\\( y_{n+1} = y_n + (y_n – y_{n-1}) + h^2 \\cdot a_n \\) \u5176\u672c\u8d28\u662f\u4e00\u6837\u7684<\/span><\/p>\n \u8865\u5145velocity-verlet<\/span>\uff1a<\/p>\n \\begin{align*} \u6ce8\u610f yn<\/span><\/span><\/span>\u00a0\u662f\u4f4d\u7f6e\uff0cvn<\/span><\/span><\/span><\/span>\u00a0\u662f\u901f\u5ea6\uff0can<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u52a0\u901f\u5ea6\uff0ch <\/span><\/span><\/span><\/span>\u662f\u65f6\u95f4\u6b65\u957f\uff0cVelocity Verlet\u7b97\u6cd5\u901a\u8fc7\u5bf9\u4f4d\u7f6e\u548c\u901f\u5ea6\u7684\u591a\u4e2a\u6b65\u9aa4\u8fdb\u884c\u8fed\u4ee3\uff0c\u540c\u65f6\u4fdd\u6301\u4e86\u8f83\u597d\u7684\u6570\u503c\u7a33\u5b9a\u6027\u548c\u80fd\u91cf\u5b88\u6052\u6027<\/span><\/p><\/blockquote>\n \u4f18\u52bf\uff1a<\/strong><\/p>\n \u52a3\u52bf\uff1a<\/strong><\/p>\n Runge-Kutta\u5e38\u5e38\u79f0\u4e4b\u4e3a\u9f99\u683c-\u5e93\u5854\u7b97\u6cd5\u6216RK4\uff084\u8868\u793a\u9636\u5c42\uff0c\u4e00\u822c\u6765\u8bf44\u9636\u5c31\u53ef\u4ee5\u4e86\uff09\uff0c\u662f\u4e00\u79cd\u5229\u7528\u6cf0\u52d2\u5c55\u5f00\u601d\u60f3\u7684\u7b97\u6cd5\u53bb\u9ad8\u7cbe\u5ea6\u8fed\u4ee3\u903c\u8fd1\u76ee\u6807\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\uff0c\u5176\u76f8\u6bd4\u6b27\u62c9\u548cVerlet\u79ef\u5206\u800c\u8a00\uff0c\u7cbe\u786e\u5ea6\u66f4\u9ad8\uff0c\u7ed3\u679c\u66f4\u52a0\u51c6\u786e\uff0c\u5355\u4e5f\u4f1a\u727a\u7272\u4e00\u90e8\u5206\u7cbe\u5ea6\u53bb\u5b9e\u73b0\uff0c\u5176\u672c\u8d28\u662f\u6cf0\u52d2\u7684\u5c55\u5f00\u7684\u56db\u9636\u516c\u5f0f\u3002<\/p>\n \\begin{align*} k_1 &= h \\cdot f(x_n, y_n) \\\\ k_2 &= h \\cdot f(x_n + \\frac{h}{2}, y_n + \\frac{k_1}{2}) \\\\ k_3 &= h \\cdot f(x_n + \\frac{h}{2}, y_n + \\frac{k_2}{2}) \\\\ k_4 &= h \\cdot f(x_n + h, y_n + k_3) \\\\ y_{n+1} &= y_n + \\frac{1}{6} \\cdot (k_1 + 2k_2 + 2k_3 + k_4) \\end{align*}<\/span><\/p>\n \u4f18\u52bf\uff1a<\/strong><\/p>\n \u52a3\u52bf\uff1a<\/strong><\/p>\n \u8bf7\u590d\u4e60\uff0c\u6cf0\u52d2\u5c55\u5f00\u548c\u4e00\u4e9b\u57fa\u672c\u5fae\u79ef\u5206\u77e5\u8bc6\uff0c\u540c\u65f6\u590d\u4e60\u725b\u987f\u4e09\u5927\u5b9a\u5f8b\uff0c\u80e1\u514b\u5b9a\u5f8b\u7b49\u57fa\u672c\u529b\u5b66\u5b9a\u7406<\/p>\n \u9009\u62e9\u4f9d\u636e\uff1a<\/strong> \u9009\u62e9\u6570\u503c\u79ef\u5206\u65b9\u6cd5\u5e94\u57fa\u4e8e\u5177\u4f53\u95ee\u9898\u7684\u7269\u7406\u7279\u6027\u3001\u6570\u503c\u7a33\u5b9a\u6027\u9700\u6c42\u548c\u8ba1\u7b97\u6027\u80fd\u9700\u6c42\u3002Verlet\u79ef\u5206\u901a\u5e38\u7528\u4e8e\u4fdd\u5b88\u7cfb\u7edf\uff0cRK4\u7528\u4e8e\u9ad8\u7cbe\u5ea6\u6c42\u89e3\uff0c\u800c\u6b27\u62c9\u65b9\u6cd5\u5219\u5728\u7b80\u5355\u6027\u548c\u4e00\u4e9b\u7279\u5b9a\u573a\u666f\u4e0b\u53ef\u80fd\u66f4\u5b9e\u7528\u3002<\/span><\/p>\n","protected":false},"excerpt":{"rendered":" PBD\u8ba1\u7b97\u8fed\u4ee3\u7684\u51e0\u79cd\u7b97\u6cd5\u8bc4\u4f30 \u8fd9\u79cd\u8fed\u4ee3\u7684\u672c\u8d28\u601d\u60f3\uff0c\u662f\u5c06\u5fae\u5206\u65b9\u7a0b\uff0c\u8f6c\u6362\u4e3a\u5dee\u5206\u65b9\u7a0b\u7684\u6570\u5b66\u65b9\u6cd5\uff0c\u901a\u8fc7\u53d8\u4e3a\u5dee […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,118],"tags":[123,116,6],"class_list":["post-1490","post","type-post","status-publish","format-standard","hentry","category-algorithm","category-unity3d","tag-pbd","tag-unity3d","tag-algorithm"],"views":2217,"_links":{"self":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts\/1490","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1490"}],"version-history":[{"count":32,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts\/1490\/revisions"}],"predecessor-version":[{"id":1540,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts\/1490\/revisions\/1540"}],"wp:attachment":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1490"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1490"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1490"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\u663e\u5f0f\u6b27\u62c9\u79ef\u5206\uff08\u5411\u524d\u6b27\u62c9\uff09<\/h6>\n
\u9690\u5f0f\u6b27\u62c9\u79ef\u5206\uff08\u5411\u540e\u6b27\u62c9\uff09<\/h6>\n
\u534a\u9690\u5f0f\u6b27\u62c9\u79ef\u5206<\/h6>\n
\n&\\text{Explicit step: } \\\\
\n&y_{n+1}^* = y_n + h \\cdot f(x_n, y_n) \\\\
\n&\\text{Implicit step: } \\\\
\n&y_{n+1} = y_n + h \\cdot f(x_{n+1}, y_{n+1}^*)
\n\\end{align*}<\/p>\nVerlet\u79ef\u5206<\/h3>\n
\n&\\text{\u521d\u59cb\u5316:} \\\\
\n&\\quad x_0, v_0, a_0 \\quad \\text{(\u521d\u59cb\u4f4d\u7f6e\u3001\u901f\u5ea6\u548c\u52a0\u901f\u5ea6)} \\\\
\n&\\text{\u65f6\u95f4\u6b65:} \\\\
\n&\\quad x_{n+1} = x_n + v_n h + \\frac{1}{2} a_n h^2 \\quad \\text{(\u66f4\u65b0\u4f4d\u7f6e)} \\\\
\n&\\quad a_{n+1} = f(x_{n+1}) \\quad \\text{(\u8ba1\u7b97\u65b0\u4f4d\u7f6e\u7684\u52a0\u901f\u5ea6)} \\\\
\n&\\quad v_{n+1} = v_n + \\frac{1}{2}(a_n + a_{n+1}) h \\quad \\text{(\u66f4\u65b0\u901f\u5ea6)} \\\\
\n\\end{align*}<\/p>\n\n
\n
Runge\u2013Kutta methods<\/h3>\n
\n
\n
\u8be6\u7ec6\u7406\u89e3<\/h2>\n
\u65b9\u6cd5\u9009\u62e9<\/h2>\n