Math C42H 3042: Corrections 

Lecture notes, page 13a (between page 13 and 14):
Lax-Friedrichs scheme:
Un+1j = ½(Unj+1 + Unj-1) - ½(k/h) A (Unj+1 - Unj-1)
[First `+' is a `-' in notes, second ½ is missing in notes.]

Lecture notes, page 24:
CFL condition: For the p-wave with speed ap
left end of stencil < characteristic < right end of stencil
i.e. xj + sminh <xj - ap.k < xj + smaxh
implies - sminh > ap.k > - smaxh
implies - smin > ap.k/h > - smax
NOT
smin < ap.k/h < smax

Lecture notes, page 37:
For the scalar Beam-Warming scheme, the dispersion coefficient in the modified equation is mu = h2/6 a (2 - 3 v + v2) = h2/6 a (2 - v)(1 - v) where v = ak/h

Lecture notes, page 59:
For a general convex problem ut +f(u)x = 0, the rarefaction fan solution is

                  { UL      x < f'(UL )t
       u(x,t)  =  { v(x/t)  f'(UL )t < x < f'(UR)t
                  { UR      x > f'(UR)t
       
       
where v(x/t) is a ....

Note that this requires f'(UL) < f'(UR) and applies to both convex and concave f(u). This requires UL < UR for convex f and UL > UR for concave f.

Lecture notes, page 74:
From p.7, the RP solution for constant states uL=Unj-1 and uR=Unj is

       U~n,n+1(x,t) = Unj-1 + sum aprp
                          p: lamn <
                    = Unj - sum aprp
       
       

Exercise sheet 1, question 7:
Un+1j = Unj + kD( Unj+1 - 2 Unj + Unj-1 )/h2
NOT
Un+1j = Unj + kD( Unj + 1 - 2 Unj + Unj-1 )/h2

Sample code: LF_2x2_system.m
Comments at the top should be
% The Lax-Friedrichs scheme is used here is stable for
%               |lambda_p|k/h < 1 , 
% i.e. it handles waves going in either direction
%
% Exercise: Use a symmetrized one-sided scheme instead.
       
NOT
% For the following one-sided scheme, all eigenvalues of A must be 
% the same sign.
%
% Extension: symmetrize this code to handle arbitrary 
% hyperbolic systems
       

Sample code: UP_cauchy.m, LW_periodic.m, LF_2x2_system.m

text(0.05,1.0,['n = 'int2str(n)])
NOT
text(0.05,1.0,['n = 'int2str(k)])

Sample code: LW_periodic.m

k = nu*h/abs(a);
NOT
k = nu*h/a;