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3391Central-Difference Richardson Upgrade 1A second-order central-difference estimate of f'(x) is 1.28 at h=0.2 and 1.22 at h=0.1. Using an O(h 2) Richardson extrapolation, what improved estimate do you get?数学简单derivation未尝试面试订阅3392Second-Derivative Richardson Upgrade 2A second-order estimate of f''(x) is 5.8 at h=0.4 and 5.2 at h=0.2. Assuming an O(h 2) truncation error, what Richardson-extrapolated estimate do you get?数学简单derivation未尝试面试订阅3393First-Order Extrapolation From Two Forward MeshesA first-order one-sided derivative estimate is 2.6 at h=0.1 and 2.45 at h=0.05. Assuming an O(h) bias, what extrapolated estimate cancels the leading error?数学简单derivation未尝试面试订阅3394Recover the Leading O(h^2) Error CoefficientSuppose D h = L + c h 2 + O(h 4). If D 0.3 = 4.18 and D 0.15 = 4.045, what is c?数学简单derivation未尝试面试订阅3395Recover the Limit of a First-Order SchemeA first-order approximation obeys D h = L + c h + O(h 2). If D 0.4 = 1.9 and D 0.2 = 1.6, what is the extrapolated estimate of L?数学简单derivation未尝试面试订阅3396Max Mesh Under a Central-Difference Error BudgetA central-difference first-derivative formula has truncation error bounded by M h 2 / 6, and a third-derivative bound gives M=12. How large can h be if you want the truncation error below 0.002?数学中等derivation未尝试面试订阅3397Forward-Difference Mesh Under a Linear Error BoundA forward-difference derivative formula has truncation error bounded by M h / 2 with M=8. How large can h be if the truncation budget is 0.01?数学中等derivation未尝试面试订阅3398Second-Derivative Mesh From a Fourth-Derivative BoundA centered second-derivative formula has error bounded by M h 2 / 12 with M=24. What h keeps the truncation error at or below 0.005?数学中等derivation未尝试面试订阅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