Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S31 / 1

Round-off error arises from representing numbers and carrying out arithmetic at finite precision: storing $1/3$ or subtracting nearly equal numbers can lose information even when the mathematical algorithm is exact. Truncation error arises from replacing a continuous or infinite mathematical operation by a finite approximation, such as discarding higher Taylor series terms in a finite difference. Refining a grid usually reduces truncation error but may increase accumulated round-off error; the two have different origins.

Write $h=\Delta x$, $k=\Delta t$ and ${\delta }_{xx}{u}_i=(u_{i+1}-2u_i+u_{i-1})/h^2$. Insert a smooth exact heat equation solution into the Crank-Nicolson method. Expanding about the midpoint $(x_i,t_n+k/2)$ givesThus the residual, defined as left side minus right side, isSince $u_t=D{u}_{xx}$ implies $u_{ttt}=D{u}_{xxtt}$,This proves second-order numerical consistency in both space and time. About the old time level, uncombined Taylor remainders can include $O(k^3)$; the midpoint expression displays the symmetric leading errors most cleanly. Multiplying by $k$ gives the one-step defect $O(k^3+k{h}^2)$. Stable evolution with suitable initial and boundary data then gives global error $O(k^2+h^2)$; consistency alone does not prove global convergence.

For balanced global error $\epsilon$, a method second order in both variables can take $h\sim {\epsilon }^{1/2}$ and $k\sim {\epsilon }^{1/2}$. With $O(1/h)$ work per time step in one spatial dimension, its cost scales as ${\epsilon }^{-1}$. A method second order in space but first order in time instead needs $k\sim \epsilon$, so its cost is $\boxed{O({\epsilon }^{-3/2})}$ under the same per-step cost assumption. Equivalently at fixed $h$, using $k\sim h^2$ rather than $k\sim h$ needs about $1/h$ more steps. A restrictive stability condition or different linear-solver cost can dominate these accuracy-based estimates.

Use a smooth exact solution such as $u(x,t)=e^{-D{q}^{2}t}\sin (qx)$ on a domain with matching boundary conditions, and compare numerical and exact values at a fixed physical time. To measure spatial order, halve $h$ while making temporal error negligible; the error norm should fall by a factor of four. To measure temporal order, halve $k$ on a sufficiently fine fixed spatial grid and again expect a factor of four. Report observed order $p={\log }_2(E_h/E_{h/2})$ and use several refinements to identify the asymptotic regime.

Check more than one smooth mode, boundary implementation and expected conservation or decay properties. A manufactured solution with an added known source can test nontrivial boundaries and forcing. Repeat in maximum and root-mean-square norms, stop before finite-precision or solver tolerances dominate, and separate stability failures from errors in convergence order. A single small error or a single visually smooth solution does not establish accuracy.