Coordinate operators, separated modes and stable updates
GRE Subject Test · GRE Physics Topic 47 6:44 English narration · English + 中文 subtitles burned in
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Writing the Laplacian of a radial function as f″ drops a term; the operator belongs to the coordinates, not the symbol. Use the divergence and Laplacian in the coordinate system selected by the symmetry Solve a simple separated boundary-value mode problem Check convergence, units and stability of an original numerical update
将径向函数的拉普拉斯算子写为f双撇会丢失一项。 该算子属于坐标系，而非符号本身。 请使用由对称性选定的坐标系中的散度与拉普拉斯算子。 求解一个简单的分离边界值模态问题。 检查原创数值更新的收敛性、量纲和稳定性。
Choose coordinates by symmetry before writing operators.
在书写算符之前，请根据对称性选择坐标系。
For a radial function in spherical coordinates, the Laplacian of f equals f double prime plus two divided by r times f prime.
对于球坐标系中的径向函数，f 的拉普拉斯算子等于 f 的二阶导数加上二除以 r 乘以 f 的一阶导数。
The divergence of a radial field A equal to A sub r times r hat equals one divided by r squared, times the derivative with respect to r of r squared times A sub r.
径向场 A 等于 A 下标 r 乘以 r 帽的散度等于一除以 r 平方，乘以 r 平方乘以 A 下标 r 对 r 的导数。
Check: A equal k times r hat divided by r squared has divergence equal to one divided by r squared, times the derivative with respect to r of k, which is zero away from the origin.
验证：A 等于 k 乘以 r 帽除以 r 平方，其散度等于一除以 r 平方，乘以 k 对 r 的导数，在原点以外为零。
And f equal one over r satisfies the Laplacian of f equals f double prime plus two times f prime divided by r, which equals two divided by r cubed minus two divided by r cubed, which is zero away from the origin.
且 f 等于一除以 r 满足拉普拉斯算子 f 等于 f 的二阶导数加上二乘以 f 的一阶导数除以 r，即二除以 r 立方减去二除以 r 立方，结果在原点以外为零。
Cartesian forms apply only to Cartesian dependences.
笛卡尔形式仅适用于笛卡尔依赖关系。
Separation of variables turns a boundary-value problem into ordinary modes.
分离变量法将边值问题转化为普通模态。
For a string fixed at both ends, write u equals X of x times T of t.
对于两端固定的弦，设 u 等于 X(x) 乘以 T(t)。
The wave equation gives X double prime over X equals one over c squared times T double prime over T equals negative k squared.
波动方程给出 X 二阶导数除以 X 等于 1/c² 乘以 T 二阶导数除以 T 等于负 k 平方。
The fixed ends force X equals sine of n pi x over L.
端点固定条件迫使 X 等于 sin(nπx/L)。
And the allowed frequencies are omega sub n equals n pi c over L, and f sub n equals n c over 2 L.
允许的频率为 ω_n 等于 nπc/L，f_n 等于 nc/(2L)。
Each mode must satisfy the boundary conditions before any sum over modes.
每个模态在求和之前必须满足边界条件。
An explicit Euler update for y prime equals negative lambda y is y sub n plus 1 equals open parenthesis one minus lambda h close parenthesis times y sub n.
y′ = −λy 的显式欧拉更新为 y_{n+1} = (1 − λh)y_n。
This is bounded when the absolute value of one minus lambda h is less than or equal to one, that is, zero is less than h which is less than two divided by lambda, for lambda greater than zero.
当 |1 − λh| ≤ 1 时有界，即 0 < h < 2/λ（其中 λ > 0）；渐近衰减需满足 0 < h < 2/λ。
Asymptotic decay requires zero is less than h which is less than two divided by lambda. With lambda equal to ten per second, h must not exceed point two seconds. If h equals point two five seconds, it multiplies by negative one point five each step and diverges with alternating sign.
取 λ = 10 /s 时，步长 h 不得超过 0.2 s；若 h = 0.25 s，则每步乘以 −1.5，导致符号交替发散。
Stability is a property of the recurrence, not of the true solution.
稳定性是递推关系的性质，而非真实解的性质。
Check a numerical answer three ways before trusting it.
在信任数值答案之前，请通过三种方式进行验证。
First, check the units of every term.
首先，检查每一项的单位。
Lambda h must be dimensionless.
lambda h 必须是无量纲的。
Second, check convergence under refinement.
其次，检查加密后的收敛性。
Halve h and compare.
将 h 减半并进行比较。
Explicit Euler error is first order.
显式欧拉法的误差是一阶的。
Third, check a limiting case with a known exact answer.
第三，检查具有已知精确解的极限情况。
For y prime equals negative 4 y, with y sub 0 equals 8 and h equals 0.2, five steps give 8 times 0.2 to the fifth power, which is 0.00256, versus the exact value of 8 e to the minus 4, which is approximately 0.1465.
对于 y 撇等于负 4y，其中 y 下标 0 等于 8，h 等于 0.2，五步计算结果为 8 乘以 0.2 的五次方，即 0.00256，而精确值为 8e 的负 4 次方，约等于 0.1465。
This is stable but inaccurate, so refine.
该结果稳定但不准确，因此需要加密。
A spherical radial field has divergence one over r squared, times the derivative with respect to r of r squared times A sub r.
球对称径向场的散度等于 r 平方分之一，乘以 r 平方与 A_r 之积对 r 的导数。
For A sub r equals C r, it gives 3 C, not C.
对于 A_r = Cr，结果为 3C，而非 C。
Fixed string ends require X of 0 equals X of L equals 0, hence X sub n equals sine of n pi x over L with positive integer n.
固定弦端要求 X(0) = X(L) = 0，因此 X_n = sin(nπx/L)，其中 n 为正整数。
Euler for y dot equals negative lambda y gives y sub n plus 1 equals 1 minus lambda h times y sub n.
对 ẏ = −λy 使用欧拉法得 y_{n+1} = (1 − λh)y_n。
For positive h, bounded absolute stability permits h less than or equal to 2 over lambda; decay to zero requires the strict inequality 0 less than h less than 2 over lambda.
当 h > 0 时，有界绝对稳定性允许 h ≤ 2/λ；而衰减至零需严格不等式 0 < h < 2/λ。
At the upper endpoint the numerical solution alternates without decay.
在上端点处，数值解振荡但不衰减。
Stability does not establish accuracy.
稳定性不能保证精度。
Relate the original teaching diagram to the governing method.
将原始教学图表与主导方法联系起来。
Read its labels and conditions before comparing quantities.
在比较数量之前，先阅读其标签和条件。
Apply explicit and implicit Euler to y dot equals negative lambda y, where lambda is positive.
对 y 点等于负 lambda y（lambda 为正数）分别应用显式欧拉法和隐式欧拉法。
Derive both amplification factors.
推导两者的放大因子。
Find the step range for nonnegative monotone explicit decay, and decide whether implicit Euler is exact for arbitrarily large steps.
找出显式衰减保持非负且单调的步长范围，并判断隐式欧拉法是否对任意大的步长都精确。
Explicit factor is one minus lambda h.
显式因子为 1 减 λh。
Nonnegative monotone decay needs zero less than or equal to h, which is less than or equal to one over lambda.
非负单调衰减需满足 0 ≤ h ≤ 1/λ；在等号处，一步即达到零，这与精确指数函数不同。
At equality, one step reaches zero, unlike the exact exponential. Implicit Euler solves y sub n plus 1 equals y sub n minus lambda h y sub n plus 1, so its factor is one over one plus lambda h, between zero and one for every positive h.
隐式欧拉法求解 y_{n+1} = y_n - λh y_{n+1}，因此其因子为 1/(1 + λh)，对于任意正数 h 均介于零与一之间。
It is stable but only first-order accurate. Refine h at fixed final time and compare with e to the minus lambda t.
该方法稳定但仅具一阶精度；请在固定最终时间下细化 h，并与 e^{-λt} 进行比较。
Award factors, monotonicity range, endpoint, implicit stability and accuracy refinement distinction.
评分点包括：因子、单调性范围、端点、隐式稳定性以及精度与细化的区别。
Applying the Cartesian Laplacian to a radial function, summing modes that violate the boundary conditions, or calling a bounded-looking run converged without refining the step.
将笛卡尔拉普拉斯算子应用于径向函数，叠加违反边界条件的模式，或在未细化步长的情况下将看似有界的运行判定为收敛。
Match operator, boundaries and stability condition to the actual problem.
请将算子、边界和稳定性条件与实际问题相匹配。
separation of variables: Solving a partial differential equation by writing the unknown as a product of single-variable factors.
分离变量法：通过将未知量写为单变量因子的乘积来求解偏微分方程。
stability condition: The step-size restriction that keeps a numerical recurrence from amplifying errors.
稳定性条件：防止数值递推放大误差的步长限制。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆含义，然后更改原始传递中的一个参数并测试极限情况。