 Solution

ID: past-exam-of-the-short-options-paper-shared-by-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2019/s31/3/d/solution

Backward Euler method has update

y^{n + 1} = (I - h M)^{- 1} y^{n}

and amplification eigenvalues

g_{\pm} = (1 - h λ_{\pm})^{- 1}

. Hence

∣ g_{\pm} ∣^{2} = \frac{1}{( 1 + γh ) ^{2} + ( k ^{2} - γ ^{2} ) h ^{2}} = \frac{1}{1 + 2 γh + k ^{2} h ^{2}} \leq 1

for every

h \geq 0

. It is therefore unconditionally stable for this linear system. Even with

γ = 0

, it damps every nonzero numerical mode: stability is obtained at the cost of artificial damping and phase error. Large

h

does not accurately resolve the oscillation period.

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