7.2 Heat Problems 407
the replacement equations are found to be (forn=4)
u 1 (m+ 1 )=ru 0 (m)+( 1 − 2 r)u 1 (m)+ru 2 (m),
u 2 (m+ 1 )=ru 1 (m)+( 1 − 2 r)u 2 (m)+ru 3 (m),
u 3 (m+ 1 )=ru 2 (m)+( 1 − 2 r)u 3 (m)+ru 4 (m),
u 4 (m+ 1 )= 2 ru 3 (m)+(
1 − 2 r−^12 rγ)
u 4 (m).(13)
(Remember thatu( 1 ,t), corresponding tou 4 , is an unknown. The boundary
condition has been incorporated into the equation foru 4 (m+ 1 ).) Again, the
second stability requirement is satisfied automatically; but the first rule re-
quires that
1 − 2 r−^1
2rγ≥0orr≤^1
2 +^12 γ. (14)
EXERCISES
Solve Eqs. (4)–(6) numerically withf(x)=x,asinthetext( x= 1 /4,
r= 1 /2), but takeu 4 ( 0 )=0. Compare your results with Table 4.
Solve Eqs. (4)–(6) numerically withf(x)=x, x= 1 /4,u 4 ( 0 )=1, as in
the text, but user= 1 /4. Compare your results with Table 4. Be sure to
compare results at corresponding times.
For the problem in Eqs. (10)–(12), find the longest stable time step when
γ=1, and compute the numerical solution with the corresponding value
ofr.
Solve the problem in Eqs. (10)–(12) with x= 1 /4,r= 1 /2andγ=0, for
mup to 5.
For each problem in the following exercises, set up the replacement equations
forn=4, compute the longest stable time step, and calculate the numerical
solution for a few values ofm.
∂^2 u
∂x^2 =∂u
∂t, u(^0 ,t)=u(^1 ,t)=t, u(x,^0 )=0.∂^2 u
∂x^2 −u=∂u
∂t, u(^0 ,t)=u(^1 ,t)=1, u(x,^0 )=0.- ∂
(^2) u
∂x^2 =
∂u
∂t−1, u(^0 ,t)=u(^1 ,t)=0, u(x,^0 )=0.