Math C42H 3042: Corrections

Lecture notes, page 13a (between page 13 and 14):

Lax-Friedrichs scheme:
Uⁿ⁺¹_j = ½(Uⁿ_j+1 + Uⁿ_j-1) - ½(k/h) A (Uⁿ_j+1 - Uⁿ_j-1)
[First `+' is a `-' in notes, second ½ is missing in notes.]

Lecture notes, page 24:

CFL condition: For the p-wave with speed a_p
left end of stencil < characteristic < right end of stencil
i.e. x_j + s_minh <x_j - a_p.k < x_j + s_maxh
implies - s_minh > a_p.k > - s_maxh
implies - s_min > a_p.k/h > - s_max
NOT
s_min < a_p.k/h < s_max

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 u_t +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'(U_L) < f'(U_R) and applies to both convex and concave f(u). This requires U_L < U_R for convex f and U_L > U_R for concave f.

Lecture notes, page 74:

From p.7, the RP solution for constant states u_L=Uⁿ_j-1 and u_R=Uⁿ_j is

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

Exercise sheet 1, question 7:

Uⁿ⁺¹_j = Uⁿ_j + kD( Uⁿ_j+1 - 2 Uⁿ_j + Uⁿ_j-1 )/h²
NOT
Uⁿ⁺¹_j = Uⁿ_j + kD( Uⁿ_j + 1 - 2 Uⁿ_j + Uⁿ_j-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;