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ID: past-exam-of-the-numerical-solutions-of-partial-differential-equations-course-of-the-university-of-oxford/2023/3/a/solution

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Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 3 / a / Solution by

Codex  0 2026-10-25

Use the Backward Euler method in time and the prescribed central finite differences in space. With

j \in Z

and

m \geq 0

, the implicit finite difference method for the advection-diffusion equation is

\frac{U _{j}^{m + 1} - U _{j}^{m}}{Δ t} + \frac{U _{j + 1}^{m + 1} - U _{j - 1}^{m + 1}}{2Δ x} = a \frac{U _{j + 1}^{m + 1} - 2 U _{j}^{m + 1} + U _{j - 1}^{m + 1}}{( Δ x ) ^{2}}, U_{j}^{0} = u_{0} (j Δ x) .

Equivalently, for

ν = Δ t /Δ x

and

μ = Δ t / (Δ x)^{2}

, the new time level solves

(ν /2 - a μ) U_{j + 1}^{m + 1} + (1 + 2 a μ) U_{j}^{m + 1} + (- ν /2 - a μ) U_{j - 1}^{m + 1} = U_{j}^{m} .

The spatial terms are evaluated at the new time level, making this an implicit scheme. On the whole line there are no finite-endpoint boundary conditions; in the next part its solution is chosen in the l2 sequence space.

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