A stiff linear system has A = pmatrix -804 & 796 796 & -804 pmatrix

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

A stiff linear system

\dot{\mathbf y}=\mathsf A\mathbf y

has

\mathsf A = \begin{pmatrix}-804 & 796\\ 796 & -804\end{pmatrix}\,\text{s}^{-1},

whose eigenvalues are

-8\,\text{s}^{-1}

and

-1600\,\text{s}^{-1}

(check this from the trace and determinant). It is integrated by backward Euler with fixed step

h=0.05\,\text{s}

for

N=1000

steps, so the same matrix

\mathsf M=\mathsf I-h\mathsf A

appears in

\mathsf M\mathbf y_{n+1}=\mathbf y_n

at every step. ((a)) Using the backward-Euler stability function

R(z)=1/(1-z)

, evaluate the multiplier for the fast eigenvalue and comment on it in the light of L-stability. ((b)) More generally, for a stiff system of size

n=50

with a fixed Jacobian integrated by backward Euler for

N=2000

steps, recall from Chapter 2 that an LU factorisation of an

n\times n

matrix costs about

\tfrac23n^3

flops and each forward/back solve costs about

2n^2

flops. Give the ratio of the flop count for re-factorising

\mathsf M

afresh at every step to the flop count for factorising it once and reusing that factorisation for every solve. ((c)) A colloidal bead in the same laboratory follows the overdamped Langevin equation

\mathrm dx = -\dfrac{x}{\tau}\,\mathrm dt + \sqrt{2D}\,\mathrm dW

, with

\tau = 8\ \mathrm{ms}

and

D = 0.6\ \mu\mathrm{m^2\,s^{-1}}

, so that the true stationary variance is

D\tau

(Chapter 9). Using the Euler-Maruyama stationary variance

\sigma^2/(1 - h/2\tau)

of this chapter, find the largest step

h

for which the simulated variance exceeds the true one by at most 2 per cent, and the number of steps needed to simulate

1\ \mathrm{s}

at that step.

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