Page 164 - Computational Fluid Dynamics for Engineers

5.4 Implicit Methods 151

At n n n At 8E n
1 + -eS-A AQ = AQ ~ (5.4.9)
( 1 + 0 dx ;i+o ( 1 + 0 dx

where / is the identity matrix,

I = (5.4.10)

This three-level Beam and Warming scheme, often called the Z\-form (delta-
n l
form), contains Q ~ , Q n and Q n+1 . For £ = 0 and 9 = 1/2, it reduces to the
one-step trapezoidal (Crank-Nicolson) scheme.
dE n
\ A t d AT AQ n = -At- (5.4.11)
1 + A
To~ x dx
n
With the application of central differences to the spatial derivatives (dA /dx)
n
and (dE /dx) in Eq. (5.4.11), this equation can be written in the tridiagonal
form discussed in subsection 4.4.2 for the unsteady heat conduction equation,
that is,

5 4 12
A A
*£ + fx\ {Ai+lA ^ - ^ ®ti) - ^(S?+i - #-i) ( - - )
-
When Eq. (5.1.4) is linear, that is, A c, then the above tridiagonal system
has the form

EU) (5.4.13)

The application of the one-step trapezoidal scheme to the vector equation
(5.1.4) allows Eq. (5.4.11) to be generalized to the form

dE n
r
AQ -At (5.4.14)
7 + A
2 dx dx
where A is the matrix given by Eq. (5.1.5c). With the application of central
differences to the spatial derivatives, Eq. (5.4.14) can again be written in a
tridiagonal form; the elements a^, bi and Q in the resulting equation, however,
are no longer scalars but blocks, as discussed in Section 4.5. The solution of the
vector equation with the Beam-Warming scheme for two-dimensional flows will
be discussed in Section 12.4.

Example 5.4. Solve the inviscid Burger's equation, Eq. (4.2.8), subject to the initial and
boundary conditions
t = 0, u(x,0)=x 0 < x < l
~ " (E5.4.1)
x = 0, u(0,t) = 0
with the Beam-Warming method (one step trapezoidal scheme). Take Ax = 0.02 and
At = 0.01 and perform the calculations for 200 time steps. Use backward differencing for
the numerical boundary condition at x = 1.