MATHMM04A-001 -EXAM-14

.4 //可以尝试提问，但只有在最佳的四个解决方案上获得的成绩才会

1.（a）求解二阶差分方程，
2an + 2 + 3an + i + 2on = 2

n —► oo？
（b）对于微分方程，

（c）初值问题，

（i）通过考虑y（tn）或其他的泰勒展开，表明
Runge-Kutta方法，

yn + \ = yn + hf（yn，tn）+〜\ f（yn + ^ hf {yn，tn），tn + -hj-f {yn，tn）\。

0 {h3）。
（ii）在没有证据的情况下减记任何明确的，数值的稳定性条件

（iii）根据以下公式推导Runge-Kutta方法的稳定性条件

2.（a）在双曲方程中，
du du Tjr + a = 0，在斧头处

（i）写下等式的合适的一阶有限差分近似
xn = nAx，tm = mAt，使用正向时间导数。这里

Courant表示数值算法稳定性的条件

（ii）需要对…进行有限差分近似的变化

（b）扩散方程，
du d2u

（i）验证Crank-Nicolson近似值，

MATHMM04A-001 -EXAM-14

.4// questions may be attempted but only marks obtained on the best four solutions will
count.
The use of an electronic calculator is not permitted in this examination.
1. (a) Solve the second-order difference equation,
2an+2 + 3an+i + 2on = 2,
with the initial conditions a0 = 0, a,\ = 1. Does the solution have a limit as
n —► oo?
(b) For the differential equation,
write down, separately, a first-order forward and a first-order backward Euler
approximation. Which of the two approximations leads to an implicit numerical
algorithm?
(c) The initial-value problem,
is solved on a uniform grid tn = nh with the step size h and the values of y(tn)
approximated as yn.
(i) By considering Taylor expansions of y(tn) or otherwise, show that the
Runge-Kutta method,

yn+\ =yn + hf(yn,tn) + ~\f(yn + ^hf{yn,tn),tn + -hj – f{yn,tn) \ .
gives the solution of the differential equation with a local truncation error of
0{h3).
(ii) Writedown, without proof, the stability condition for any explicit, numerical
method of the form, yn+l = F{yn,tn).
(iii) Derive the stability condition for the Runge-Kutta method in terms of the
step size h and the magnitude of \df/dy\.

2. (a) In the hyperbolic equation,
du du Tjr + a= 0, at ax
the constant o is positive.
(i) Write down a suitable first-order finite-difference approximation of the equa
tion on the grid xn = nAx, tm = mAt, using a forward time derivative. Here
the step sizes, Ax and At, are constant. Derive the Courant-Friedrichs-Lewy
condition for the stability of the numerical algorithm in terms of the Courant
number, C = aAt/Ax.
(ii) What changes need to be made in the finite-difference approximation to
ensure stability in the case a < 0?
(b) The diffusion equation,
du d2u
is solved in the region t ^ 0, 0 < x ^ L. The function, u(x,t). is specified on
the boundaries as follows,
(i) Verify that the Crank-Nicolson approximation,

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