The Lax-Friedrichs scheme for with is u_j^ n+1 = 12 (u_ j+1 ^n + u_ j-1 ^n ) - C 2…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy

Error, Chance and the Machine, Ch Partial differential equations: stability, convergence and solvers

Problem

The Lax-Friedrichs scheme for

u_t + cu_x = 0

with

c > 0

u_j^{n+1} = \tfrac12\bigl(u_{j+1}^n + u_{j-1}^n\bigr) - \tfrac{C}{2}\bigl(u_{j+1}^n - u_{j-1}^n\bigr),\qquad C = \frac{c\,\Delta t}{\Delta x}.

When

\Delta x

and

\Delta t

are refined together at fixed

C

it is consistent, with a residual of order

\Delta x

. ((a)) Substitute

u_j^n = G^ne^{ij\theta}

to find

G(\theta)

and

|G|^2

. Show that

|G| \le 1

for every

\theta

exactly when

C \le 1

, and find the smallest

|G|

C = 0.9

. ((b)) Three runs are refined with

C

held fixed and

\Delta x, \Delta t \to 0

together: forward-time centred-space advection at

C = 0.01

, upwind at

C = 0.8

and Lax-Friedrichs at

C = 0.9

. For each, say whether it converges to the exact solution at a fixed time

T

, and give the reason. ((c)) A colleague runs the centred scheme at

C = 0.01

with

c = 1\,\mathrm{m\,s^{-1}}

on the grid

\Delta x = 0.01\,\mathrm{m}

. The worst mode,

\theta = \pi/2

, grows by only

1.65

over

T = 1\,\mathrm{s}

, and concludes that the scheme converges. Find the growth of that mode over

T = 1\,\mathrm{s}

on the grid

\Delta x = 2\times10^{-4}\,\mathrm{m}

, still at

C = 0.01

, and say what the coarse-grid test could not show.

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