1. Hyperbolic Equations
Pure IVP:
Lax-Wendroff method...

2. Preliminaries:
Input:
Integer p, lower spatial index at tjmax
Integer q, upper spatial index at tjmax
Integer jmax, number of time steps
real k, time step
real h, space step
real a, coefficient
function f(x), initial condition
function ss(x,t), source term
functions (sx(x,t) and st(x,t), respectively)

3. Define Grid:
s = k/h
check stability
Initialize (outputting initial condition)
t = 0
nmin = p – jmax
nmax = q + jmax
for n = nmin, nmin+1, …, nmax
xn = n*h
V(n) = f(xn)
endloop
output t
for n = p, p+1, …, q
output V(n)

4. Time Loop:
for j = 1, 2, …, jmax
nmin = nmin+1
nmax = nmax-1
for n = nmin, nmin+1, …, nmax
U(n) = V(n)-0.5*s*a*(V(n+1)-V(n-1))
+ s**2*a**2/2*(V(n-1)-2*V(n)+V(n+1)
+ k*ss(n*h,t)+k**2/2*(st(n*h,t)-a*sx(n*h,t))
endloop
t = t + k
V(n) = U(n)
output t
for n = p, p+1, …, q
output V(n)

5. Wendroff implicit method for the IBVP
Preliminaries:
Input:
Integer nmax, maximum x index
Integer jmax, number of time steps
real k, time step
real h, space step
real a, coefficient
function f(x), initial condition
function g(t), boundary condition
function ss(x,t), source term

6. Define Grid:
s = k/h
Initialize (outputting initial condition)
t = 0
xmax = nmax*h
for n = 0, 1, …, nmax
xn = n*h
V(n) = f(xn)
endloop
output t
for n = p, p+1, …, q
output V(n)

7. Time Loop:
for j = 1, 2, …, jmax
t = t + k
U(0) = g(t)
q = (1-s*a)/(1+s*a)
for n = 0, 1, …, nmax-1
U(n+1) = V(n)+q*V(n+1)-q*U(n)
+2*k*ss(xmax+h/2, t-k/2)/(1+s*a)
endloop
for n = 0, 1, …, nmax
V(n) = U(n)
output t
output U(n)

8. The Wendroff method (among others) does not require specification of a downstream boundary condition, but others, like the Lax-Wendroff do. This can be done either numerically or with some physical reasoning.
Numerical:
characteristics (skip)
finite difference approximation
Issues: stability and accuracy

9. Conservation Law Form
recall...

10. Conservation Law Form
Conservation law form if FD equation can be cast as:
where Q is a numerical approximation of the flux at “cell” boundaries:

11. Conservation Law Form
write eqn. as:
Lax-Friedrichs method
where