Past exam of the numerical solution of differential equations course of the University of Oxford / 2010 / A / 3 / b

Write

M_{0} = sup_{x} u_{0} (x) < \infty

and assume

λ (1 + p M_{0}^{p - 1}) \leq 1

. Suprema are understood where the printed maxima on the infinite domain are not attained. We prove the invariant interval

0 \leq U_{j}^{m} \leq M_{0}

simultaneously with the stated speed bound.

Suppose the interval property holds at time

m

a_{j}^{m} = \frac{f ( U _{j}^{m} ) - f ( U _{j - 1}^{m} )}{U _{j}^{m} - U _{j - 1}^{m}}

is between

1

and

1 + p M_{0}^{p - 1}

; when the two states coincide define it as

f^{'} (U_{j}^{m})

. The update is the convex combination

U_{j}^{m + 1} = (1 - λ a_{j}^{m}) U_{j}^{m} + λ a_{j}^{m} U_{j - 1}^{m} .

Both coefficients are nonnegative and sum to one, so the same interval property holds at time

m + 1

. It is true initially and hence for all times by induction. Therefore

λ j sup ∣1 + p (U_{j}^{m})^{p - 1} ∣ \leq λ (1 + p M_{0}^{p - 1}) \leq 1 for every m .

The propagated nonnegativity is essential; it keeps the flux increasing on the entire range of the numerical solution.

ii

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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