 Solution

ID: past-exam-of-the-numerical-solution-of-differential-equations-course-of-the-university-of-oxford/2010/a/3/b/ii/solution

The convex-combination formula proved in part (i) gives

0 \leq U_{j}^{m + 1} \leq sup_{ℓ} U_{ℓ}^{m}

for every

j

. Taking the supremum and using nonnegativity gives

∥ U^{m + 1} ∥_{ℓ^{\infty}} \leq ∥ U^{m} ∥_{ℓ^{\infty}} \leq ∥ U^{0} ∥_{ℓ^{\infty}} .

This is the discrete maximum principle for the nonlinear upwind step and proves the requested conditional

ℓ^{\infty}

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