Two chemical species interconvert rapidly while slowly leaking away

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

Two chemical species interconvert rapidly while slowly leaking away, according to c˙1=−1001c1+1000c2,c˙2=1000c1−1001c2.\dot c_1 = -1001c_1 + 1000c_2, \qquad \dot c_2 = 1000c_1 - 1001c_2. The Jacobian of this linear system has eigenvalues λ1=−1 s−1\lambda_1 = -1\,\mathrm{s^{-1}} and λ2=−2001 s−1\lambda_2 = -2001\,\mathrm{s^{-1}}. ((a)) State the stiffness ratio of this system. ((b)) Using the explicit stability limit h<2/∣λfast∣h < 2/|\lambda_{\rm fast}|, estimate the number of forward Euler steps needed to integrate to t=5 st = 5\,\mathrm{s} (five slow timescales) at the largest stable step.

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