数值ODE练习题

数值 ODE

2017 Individual

Problem 1. Consider the differential equation

\[ -u'' + \alpha u = f, \quad x \in (0,1), \]

where prime denotes $d/dx$ and $\alpha$ is a constant. We consider a mixed boundary condition

\[ u(0) = 0, \quad u'(1) - b u(0) = 0. \]

This equation is approximated by a standard finite difference method:

\[ \frac{-U_{j-1} + 2U_j - U_{j+1}}{h^2} + \alpha U_j = f_j, \quad j = 1, \ldots, N-1. \]

Here, $N$ is the number of grid points, $h = 1/N$ is the mesh size, $U_j$ is the approximate solution at $x_j := jh$, and $f_j = f(x_j)$. The boundary condition is approximated by

\[ U_0 = 0, \quad \frac{U_N - U_{N-1}}{h} - b U_N = 0. \]

The resulting linear system is $AU = F$ with

\[ \begin{matrix} \beta & -1 & 0 & \cdots \\ -1 & \beta & -1 & \cdots \\ & & \ddots & \\ & & & -1 \\ & & & 0 \end{matrix} \begin{bmatrix} U_1 \\ U_2 \\ \vdots \\ U_{N-1} \\ U_N \end{bmatrix} = \begin{bmatrix} h^2 f_1 \\ h^2 f_2 \\ \vdots \\ h^2 f_{N-1} \\ 0 \end{bmatrix}, \]

where $\beta = 2 + \alpha h^2$.

(a) Write down the complete linear system, including the last row corresponding to the boundary condition at $x=1$.
(b) Analyze the stability and convergence of this finite difference scheme.

2017 Individual

Problem 2. For solving the partial differential equation

\[ u_t + f(u)_x = 0, \quad 0 \leq x \leq 1, \]

where $f'(u) \geq 0$, with periodic boundary condition, we can use the following semi-discrete upwind scheme

\[ \frac{d}{dt} u_j + \frac{f(u_j) - f(u_{j-1})}{\Delta x} = 0, \quad j = 1,2,\ldots,N, \]

with periodic boundary condition $u_0 = u_N$, where $u_j = u_j(t)$ approximates $u(x_j,t)$ at the grid point $x = x_j = j \Delta x$, with $\Delta x = \frac{1}{N}$.

(a) Prove the following $L^2$ stability of the scheme:

\[ \frac{d}{dt} E(t) \leq 0, \]

where $E(t) = \sum_{j=1}^N |u_j|^2 \Delta x$.
(b) Do you believe the above inequality is true for $E(t) = \sum_{j=1}^N |u_j|^{2p} \Delta x$ for arbitrary integer $p \geq 1$? If yes, prove the result. If not, give a counterexample.

2019 Team

Problem 3. (Störmer's Method) Consider the following problems.

(i) Determine the order of Störmer's method,

\[ y_{n+2} - 2y_{n+1} + y_n = h^2 f(t_{n+1}, y_{n+1}), \quad n \geq 0, \]

for solving the second order system of ODE's

\[ y'' = f(t, y), \quad t \geq 0, \]

with the initial conditions $y(0) = y_0$ and $y'(0) = y_0'$.
(ii) Using the second order central differences in space and Störmer's method in time, construct a scheme to solve the wave equation,

\[ u_{tt} = u_{xx}. \]
(iii) Determine the condition for its stability.

证明

$(1)$：设 $y(t)$ 为微分方程的精确解，则由 Taylor 展开可得

\[ y(t_{n+1}) = y(t_n) + h y'(t_n) + \frac{h^2}{2} y''(t_n) + \frac{h^3}{6} y^{(3)}(t_n) + O(h^4), \]

\[ y(t_{n-1}) = y(t_n) - h y'(t_n) + \frac{h^2}{2} y''(t_n) - \frac{h^3}{6} y^{(3)}(t_n) + O(h^4). \]

将上述两式相加，得到

\[ y(t_{n+1}) + y(t_{n-1}) = 2 y(t_n) + h^2 y''(t_n) + O(h^4). \]

因此，有

\[ y(t_{n+1}) - 2 y(t_n) + y(t_{n-1}) = h^2 y''(t_n) + O(h^4). \]

另一方面，由于 $y''(t_n) = f(t_n, y(t_n))$，所以

\[ y(t_{n+1}) - 2 y(t_n) + y(t_{n-1}) = h^2 f(t_n, y(t_n)) + O(h^4). \]

因此，Störmer 方法的局部截断误差为 $O(h^4)$，从而该方法的阶数为 $4 - 2 = 2$。

$(ii)$： $$ \frac{u_j^{n+1} - 2 u_j^n + u_j^{n-1}}{h^2} = \frac{u_{j+1}^n - 2 u_j^n + u_{j-1}^n}{\tau^2}, $$

其中，$u_j^n$ 近似表示 $u(x_j, t_n)$，$x_j = j \Delta x$，$t_n = n \tau$。

$(iii)$：设 $u_j^n = \xi^n e^{i jkh}$，代入差分方程，得到

\[ \frac{\xi^{n+1} - 2 \xi^n + \xi^{n-1}}{h^2} = \frac{e^{i (j+1)kh} - 2 e^{i jkh} + e^{i (j-1)kh}}{\tau^2}. \]

化简后得到特征方程

\[ \xi^2 - 2 \left( 1 - 2 r^2 \sin^2 \left( \frac{kh}{2} \right) \right) \xi + 1 = 0, \]

两个根的乘积

2020 Team

Problem 4. For the initial value problem $y' = f(t, y)$, $y(0) = y_0$ on the interval $[0, T]$, consider the implicit two-step method

\[ y_{n+1} = \frac{4}{3} y_n - \frac{1}{3} y_{n-1} + \frac{2h}{3} f(t_{n+1}, y_{n+1}), \]

with starting value $y_1 = y_0 + h f(t_1, y_0)$, where $h$ is the step size and $t_n = n h$.

(a) What is the order of accuracy of the scheme?
(b) Check the stability of the scheme by analyzing the stability polynomial.
(c) Find the stability region of the scheme.

2020 Team

Problem 5. Suppose the difference scheme $u^{n+1} = B u^n$ is stable, and $C(\Delta t)$ is a bounded family of operators. Show that the scheme

\[ u^{n+1} = (B + \Delta t C(\Delta t)) u^n \]

is stable.

2021 Team

Problem 6. Consider the family of semi-implicit Runge-Kutta methods

\[ \begin{aligned} k_1 &= f(y_n + \beta h k_1), \\ k_2 &= f(y_n + h k_1 + \beta h k_2), \\ y_{n+1} &= y_n + h \left( \left(\frac{1}{2} + \beta\right) k_1 + \left(\frac{1}{2} - \beta\right) k_2 \right). \end{aligned} \]

(a) Determine the order and the principal part of the local truncation error.
(b) Show that if $\beta > \frac{1}{2}$, then the negative real axis $\{z : \text{Re}(z) < 0, \text{Im}(z) = 0\}$ is contained in the region of absolute stability of the method.

2021 Team

Problem 7. (Beam Equation) Consider the Beam equation from mechanics with boundary conditions that model a cantilever beam:

\[ u^{(4)} = f(x), \quad x \in (0,1), \]

with boundary conditions

\[ u(0) = u'(0) = u''(1) = u'''(1) = 0. \]

(a) Recast this equation into a variational problem, stating the trial and test function spaces.
(b) Interpret the variational problem as an energy minimization problem, clearly stating the energy functional. Prove that the variational problem and the energy minimization problems are equivalent.
(c) Develop a CG(3) (cubic continuous Galerkin method) finite element method for this problem.
(d) Prove an a priori error estimate for this method in the energy norm:

\[ \|e\|_E = \left( \int_0^1 (e'')^2 dx \right)^{1/2}, \]

where $e = u(x) - U(x)$, in which $u(x)$ is the exact solution to the variational problem, $U(x)$ is the FEM solution.
(e) Prove an a priori error estimate for this method in the $L^2$ norm:

\[ \|e\|_{L^2} = \left( \int_0^1 e^2 dx \right)^{1/2}. \]