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

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

Codex  0 2026-10-25

Use the forward Euler method in time and evaluate both spatial finite difference operators at the old time level:

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

(1)

With

ν = Δ t /Δ x

and

μ = Δ t / (Δ x)^{2}

, the explicit update is

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

(2)

Every value on the new time level is computed directly from three old values. The sum of the stencil weights is one, as required to preserve constant solutions of the advection-diffusion equation.

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