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3399Balancing Truncation and Rounding ErrorSuppose a numerical derivative has total error model E(h)=3h 2 + 10 -4 /h. What h minimizes this approximation?数学中等derivation未尝试面试订阅3400Minimum Error at the Balanced MeshUsing E(h)=3h 2 + 10 -4 /h and the optimal h from first-order balancing, what is the minimum modeled error?数学中等derivation未尝试面试订阅3401Explicit Heat Stability Cap 1For an explicit heat-equation update, diffusion coefficient kappa = 0.5, and grid spacing Delta x = 0.1, what is the largest stable time step under the standard Delta t <= Delta x 2 / (2 kappa) rule?数学中等derivation未尝试面试订阅3402Explicit Heat Stability Cap 2For an explicit heat update with kappa = 1.2 and Delta x = 0.05, what is the largest stable Delta t under the same rule?数学中等derivation未尝试面试订阅3403Upwind Advection CFL LimitAn explicit upwind advection scheme has speed u = 2 and grid spacing Delta x = 0.03. What is the largest stable Delta t under the CFL rule Delta t <= Delta x / |u|?数学中等derivation未尝试面试订阅3404Advection-Diffusion Explicit Time-Step Rule 1Use the simple sufficient rule Delta t <= 1 / (|u|/Delta x + 2 kappa / Delta x 2) for an explicit advection-diffusion update. If u = 1, kappa = 0.2, and Delta x = 0.1, what is the resulting Delta t cap?数学中等derivation未尝试面试订阅3405Advection-Diffusion Explicit Time-Step Rule 2Using the same sufficient explicit rule Delta t <= 1 / (|u|/Delta x + 2 kappa / Delta x 2), what is the Delta t cap when u = 0.5, kappa = 0.1, and Delta x = 0.05?数学中等derivation未尝试面试订阅3406Zero-Flux Ghost PointAt a left boundary, a zero-flux Neumann condition is approximated by (u 1 - u -1 )/(2h)=0. What relation does this impose on the ghost point u -1 ?数学中等derivation未尝试面试订阅3407Ghost Point Under a Positive Boundary SlopeAt x=0, approximate u x(0)=3 using (u 1-u -1 )/(2h) with h=0.1 and u 1=1.4. What is the ghost point value u -1 ?数学中等derivation未尝试面试订阅3408Boundary Curvature From a Nonzero Slope ConditionSuppose u(0)=4, u 1=5, h=0.25, and the left-boundary slope satisfies u x(0)=-2 approximated by (u 1-u -1 )/(2h). What second derivative estimate do you get from (u -1 - 2u 0 + u 1)/h 2?数学中等derivation未尝试面试订阅3409Boundary Curvature Under Zero FluxSuppose u(0)=7, u 1=6, h=0.5, and the left boundary has zero flux. What second derivative estimate follows from the ghost-point stencil (u -1 -2u 0+u 1)/h 2?数学中等derivation未尝试面试订阅3410Second-Order Forward Stencil at a BoundaryUse the second-order forward stencil (-3f(0)+4f(h)-f(2h))/(2h) with h=0.1, f(0)=1, f(0.1)=1.24, and f(0.2)=1.52. What derivative estimate do you get at 0?数学中等derivation未尝试面试订阅3411Why Richardson Extrapolation HelpsWhy does Richardson extrapolation often improve an estimate even when you never see the exact derivative value?数学中等essay未尝试面试订阅3412Why Smaller h Is Not Always BetterWhy can making h smaller eventually worsen a finite-difference estimate?数学中等essay未尝试面试订阅3413Why Diffusion Stability Scales With Delta x SquaredWhy does the explicit heat-equation time-step restriction shrink like Delta x squared rather than just like Delta x?数学中等essay未尝试面试订阅3414Why Boundary Conditions Change the StencilWhy can the same derivative operator require a different finite-difference stencil near a boundary than in the interior?数学中等essay未尝试面试订阅3415Why Central Differences Are Usually More Accurate Than One-Sided OnesWhy do central differences often achieve a higher order of accuracy than one-sided differences using similar nearby points?数学中等essay未尝试面试订阅3416Entropy of a One-Heavy Rest-Uniform Source 1A discrete source has one special symbol of probability 0.5 and 4 other symbols sharing the remaining probability uniformly. What is the entropy in bits?数学简单derivation未尝试面试订阅3421Entropy Gain From Splitting a Quarter-Mass SymbolA source contains a symbol of probability 1/4. If you split that symbol into four equally likely sublabels while leaving everything else unchanged, by how many bits does the entropy increase?数学简单derivation未尝试面试订阅3422Entropy Gain From Splitting a 30% State Into ThreeA source has a single state of probability 0.3. If that state is refined into three equally likely sublabels, how many bits of entropy are added?数学简单derivation未尝试面试订阅