MM.1 · Coordinate operators, separated modes and stable updates

Writing the Laplacian of a radial function as f″ drops a term; the operator belongs to the coordinates, not the symbol.

Prerequisites: Algebra, differentiation and vector components.

• Use the divergence 散度 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

Choose coordinates by symmetry before writing operators. For a radial function in spherical coordinates, ∇²f=f″+(2/r)f′, and the divergence of a radial field A=A_r r̂ is (1/r²)d(r²A_r)/dr. Check: A=k r̂/r² has divergence (1/r²)dk/dr=0 away from the origin, and f=1/r satisfies ∇²f=f″+2f′/r=2/r³−2/r³=0 away from the origin. Cartesian forms apply only to Cartesian dependences.

Separation of variables 分离变量法 turns a boundary-value problem into ordinary modes. For a string fixed at both ends, write u=X(x)T(t); the wave equation gives X″/X=(1/c²)T″/T=−k², the fixed ends force X=sin(nπx/L), and the allowed frequencies are ω_n=nπc/L, f_n=nc/(2L). Each mode must satisfy the boundary conditions before any sum over modes.

An explicit Euler update for y′=−λy is y_{n+1}=(1−λh)y_n, bounded when |1−λh|≤1, i.e. 0<h≤2/λ for λ>0; asymptotic decay needs 0<h<2/λ. With λ=10 /s, h must not exceed 0.2 s; h=0.25 s multiplies by −1.5 each step and diverges with alternating sign. Stability is a property of the recurrence, not of the true solution.

Check a numerical answer three ways before trusting it: units of every term (λh must be dimensionless), convergence under refinement (halve h and compare; explicit Euler error is first order), and a limiting case with a known exact answer. For y′=−4y with y0=8 and h=0.2, five steps give 8×0.2⁵=0.00256 versus exact 8e^(−4)≈0.1465: stable but inaccurate, so refine.

A spherical radial field has divergence

For $A_r=Cr$, it gives 3C, not C. Fixed string ends require $X(0)=X(L)=0$, hence $X_n=\sin(n\pi x/L)$ with positive integer n. Euler for $\dot y=-\lambda y$ gives $y_{n+1}=(1-\lambda h)y_n$. For positive h, bounded absolute stability permits $h\leq2/\lambda$; decay to zero requires the strict inequality $0. At the upper endpoint the numerical solution alternates without decay. Stability does not establish accuracy.

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. Match operator, boundaries and stability condition 稳定性条件 to the actual problem.

separation of variables: Solving a partial differential equation by writing the unknown as a product of single-variable factors.

stability condition: The step-size restriction that keeps a numerical recurrence from amplifying errors.