数值PDE练习题

数值 PDE

Autumn, 2024

For the system

\[ u_{t}=v_{x}, \]

\[ v_{t}=u_{x}, \]

analyze the truncation error and stability of the scheme

\[ \frac{1}{\tau}\left(u_{j}^{n+1}-\frac{1}{2}(u_{j+1}^{n}+u_{j-1}^{n})\right)=\frac{1}{2h}(v_{j+1}^{n}-v_{j-1}^{n}), \]

\[ \frac{1}{\tau}\left(v_{j}^{n+1}-\frac{1}{2}(v_{j+1}^{n}+v_{j-1}^{n})\right)=\frac{1}{2h}(u_{j+1}^{n}-u_{j-1}^{n}). \]

Autumn, 2024

Write and prove the maximum principle of the centered finite difference scheme for discretizing the equation

\[ u_{xx}+u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0, \quad f<0 \]

under some suitable assumptions.

Spring, 2025

\[ u_{j}^{n+1}=-\frac{1}{2}\nu(1-\nu)u_{j+1}^{n}+(1-\nu^{2})u_{j}^{n}+\frac{1}{2}\nu(1+\nu)u_{j-1}^{n}, \]

where \(\nu=a\tau/h\).

(i) For the Cauchy problem imposed on the real line, show that

\[ ||u^{n+1}||_{2}^{2}=||u^{n}||_{2}^{2}-\frac{1}{2}\nu^{2}(1-\nu^{2})(||\delta_{x}^{+}u^{n}||_{2}^{2}-\langle\delta_{x}^{+}u^{n},\delta_{x}^{-}u^{n}\rangle),\]

where \(||v||_{2}^{2}=\sum_{j}|v_{j}|^{2}\), \(\langle v,w\rangle=\sum_{j}v_{j}w_{j}\), \(\delta_{x}^{-}v_{j}=v_{j}-v_{j-1}\), \(\delta_{x}^{+}v_{j}=v_{j+1}-v_{j}\).

(ii) Suppose \(a>0\) for the problem imposed on \((0,1)\) with homogeneous boundary condition at \(x=0\) (i.e., \(u_{0}^{n}=0\)), give a simple numerical boundary condition for \(x=1\) such that the Lax-Wendroff scheme is stable.

证明

\((i)\)：纯计算。

\((ii)\)：在边界点 \(x_N = 1\) 处，放弃中心差分的 Lax-Wendroff 格式，改用一阶迎风格式：

\[ u_N^{n+1} = u_N^n - \nu (u_N^n - u_{N-1}^n) \]

当 \(a>0\) 时，迎风格式是稳定的（只要 CFL 条件 \(\nu \le 1\) 满足），且只利用了左侧的信息 \(u_N, u_{N-1}\)，符合特征线把信息从内部传向边界的物理事实，不会引入错误的反向波。计算量小，易于实现。

Spring, 2025

For the system

\[ u_{t}=-v_{x}, \]

\[ v_{t}=u_{xx}, \]

analyze the truncation error and stability of the scheme

\[ \frac{u_{j}^{n+1}-u_{j}^{n}}{\tau}=-\frac{1}{2h^{2}}(v_{j+1}^{n}-2v_{j}^{n}+v_{j-1}^{n}+v_{j+1}^{n+1}-2v_{j}^{n+1}+v_{j-1}^{n+1}), \]

\[ \frac{v_{j}^{n+1}-v_{j}^{n}}{\tau}=\frac{1}{2h^{2}}(u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}+u_{j+1}^{n+1}-2u_{j}^{n+1}+u_{j-1}^{n+1}). \]

证明

由 Taylor 展开

\[ \frac{u_{j}^{n+1}-u_{j}^{n}}{\tau}=u_{t}+\frac{\tau}{2}u_{tt}+O(\tau^{2}), \]

\[ \frac{v_{j+1}^{n}-2v_{j}^{n}+v_{j-1}^{n}}{2h^2}=\frac{1}{2}v_{xx}+O(h^2), \]

\[ \frac{v_{j+1}^{n+1}-2v_{j}^{n+1}+v_{j-1}^{n+1}}{2h^2}=\frac{1}{2}v_{xx}+\frac{\tau}{2}v_{xxt}+O(\tau^2+h^2). \]

因此截断误差为 \(u_t+ \frac{\tau}{2}u_{tt} +v_{xx} + \frac{\tau}{2}v_{xxt} + O(\tau^2+h^2)\)。由条件有 \(u_t+v_{xx}=0\)，\(u_{tt}+v_{xxt}=0\)，因此截断误差为 \(O(\tau^2+h^2)\)。因此该格式对时间和空间都具有二阶精度。第二个格式同理。

对于稳定性分析，设 \(u_j^n = U^n e^{i k j h}\)，\(v_j^n = V^n e^{i k j h}\)，代入差分方程有

\[ \frac{U^{n+1} - U^n}{\tau} = -\frac{1}{2} (-S) (V^n + V^{n+1}) = \frac{S}{2} (V^n + V^{n+1}) \]

\[ \frac{V^{n+1} - V^n}{\tau} = \frac{1}{2} (-S) (U^n + U^{n+1}) = -\frac{S}{2} (U^n + U^{n+1}) \]

其中 \(S=\frac{4}{h^2} \sin^2\left(\frac{k h}{2}\right)\)。设 \(\mu=\frac{\tau S}{2} = \frac{2\tau}{h^2} \sin^2\left(\frac{kh}{2}\right)\)。整理为矩阵形式：

\[ \begin{pmatrix} U^{n+1} \\ V^{n+1} \end{pmatrix} - \begin{pmatrix} U^n \\ V^n \end{pmatrix} = \mu \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \left( \begin{pmatrix} U^{n+1} \\ V^{n+1} \end{pmatrix} + \begin{pmatrix} U^n \\ V^n \end{pmatrix} \right) \]

设 \(W^n = \begin{pmatrix} U^n \\ V^n \end{pmatrix}\)，则上式可写为

\[ W^{n+1} - W^n = \mu J (W^{n+1} + W^n) \]

也就是

\[ W^{n+1} = (I - \mu J)^{-1} (I + \mu J) W^n \]

由于 \(J\) 的特征值为 \(\pm i\)，因此 \((I - \mu J)^{-1} (I + \mu J)\) 的特征值为 \(\frac{1 + i \mu}{1 - i \mu}\) 和 \(\frac{1 - i \mu}{1 + i \mu}\)，它们的模长均为 \(1\)。因此该方法无条件稳定。

Autumn, 2025

Construct the Du Fort-Frankel scheme for the diffusion equation in 2D \(u_t = u_{xx} + u_{yy}\) and discuss its consistency and stability.

证明

\[ \frac{u_{i,j}^{n+1} - u_{i,j}^{n-1}}{2\tau} = \frac{u_{i+1,j}^n - \left(u_{i,j}^{n+1} + u_{i,j}^{n-1}\right) + u_{i-1,j}^n}{h^2} + \frac{u_{i,j+1}^n - \left(u_{i,j}^{n+1} + u_{i,j}^{n-1}\right) + u_{i,j-1}^n}{h^2}\]

设 \(\Delta x = \Delta y = h\)，网格比 \(r = \frac{\tau}{h^2}\)：

\[ u_{i,j}^{n+1} = \frac{1 - 4r}{1 + 4r} u_{i,j}^{n-1} + \frac{2r}{1 + 4r} (u_{i+1,j}^n + u_{i-1,j}^n + u_{i,j+1}^n + u_{i,j-1}^n) \]

相容性：由 Taylor 展开可得

\[ LHS=u_t+\frac{\tau^2}{6}u_{ttt}+O(\tau^4) \]

\[ RHS=u_{xx}+u_{yy}-\frac{\tau^2}{h^2}u_{tt}\cdot 2+O(h^2) \]

故截断误差

\[ LHS-RHS=O(\tau^2+h^2+\frac{\tau^2}{h^2}). \]

若 \(\frac{\tau^2}{h^2}\to 0\)，则该方法相容。否则，该方法不相容。

稳定性：设 \(u_{i,j}^n = \xi^n e^{i(k_x i h + k_y j h)}\)，代入差分方程，得到特征方程

\[ (1 + 4r) \xi^2 - 4r\alpha \xi - (1 - 4r) = 0 \]

其中 \(r=\frac{\tau}{h^2}\)，\(\alpha = \cos(k_x h) + \cos(k_y h)\)。我们需要两根 \(\|\xi_{1,2}\|\leq 1\)。容易验证这是无条件稳定的。

Autumn, 2025

For the wave equation \(u_{tt} = u_{xx}\), analyze the stability of the scheme

\[ \frac{u_j^{n+1} - 2u_j^n + u_j^{n-1}}{\tau^2} = \frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{4h^2} + \frac{u_{j+1}^n - 2u_j^n + u_{j-1}^n}{2h^2} + \frac{u_{j+1}^{n-1} - 2u_j^{n-1} + u_{j-1}^{n-1}}{4h^2}. \]

证明

由 von Neumann 分析，设 \(u_j^n = \xi^n e^{i k j h}\)，代入上式得到特征方程

\[ \xi+\frac{1}{\xi}-2=\frac{a^2}{4}\xi (e^{i k h}+e^{-i k h}-2)+\frac{a^2}{2}(e^{i k h}+e^{-i k h}-2)+\frac{a^2}{4}\frac{1}{\xi}(e^{i k h}+e^{-i k h}-2), \]

即

\[ \frac{(\xi-1)^2}{(\xi+1)^2}=\frac{a^2}{4}(e^{i k h}+e^{-i k h}-2)=-a^2 \sin^2\left(\frac{k h}{2}\right). \]

解得 \(\xi =\frac{1 \pm i I}{1 \mp i I}\)，其中 \(I = a \sin\left(\frac{k h}{2}\right)\)。由于 \(\|\xi\|=1\)，所以该方法无条件稳定。

2019T

Problem 1. 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

$$
y^{\prime\prime}=f(t,y),\quad t\geq 0,
$$

with the initial conditions $y(0)=y_{0}$ and $y^{\prime}(0)=y_{0}^{\prime}$.

(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

2019I

Problem 2. Consider Richardson's difference scheme for the heat equation \(u_t = u_{xx}\):

\[ \frac{1}{2k}(u(x,t+k)-u(x,t-k))=\frac{1}{h^{2}}(u(x-h,t)-2u(x,t)+u(x+h,t)). \]

(i) Show that this scheme has second-order truncation error.
(ii) Use either ODE principles or von Neumann analysis to show that this scheme is unconditionally unstable.
(iii) Demonstrate a minor modification of the left-side of Richardson's scheme that yields a familiar unconditionally stable scheme and prove it.

2018T

Problem 3. For the one-way wave equation

\[ u_{t}+au_{x}=f, \]

consider the multistep scheme given by

\[ \frac{3u_{m}^{n+1}-4u_{m}^{n}+u_{m}^{n-1}}{2k}+a\frac{u_{m+1}^{n+1}-u_{m-1}^{n+1}}{2h}=f_{m}^{n+1}. \]

(i) Show that the scheme is second order accurate.
(ii) Show that the scheme is unconditionally stable.

Hint: (1) apply von Neumann analysis to the scheme with \(f\equiv 0\) and find the characteristic polynomial. (2) show that for all \(k,h\), the characteristic polynomial satisfies the root condition: all roots reside in the unit disk, and all roots on the unit circle are simple. (3) for a root \(r\) of the characteristic polynomial, it would be more convenient to study the form \(\frac{1}{r}=X+iY\) and prove that \(X^{2}+Y^{2}\geq 1\).

2018I

Problem 4. We consider the following convection-diffusion equation

\[ u_{t}+au_{x}=bu_{xx},\quad 0\leq x<1 \]

with an initial condition \(u(x,0)=f(x)\) and periodic boundary condition, where \(a\) and \(b>0\) are constants. The first order IMEX (implicit-explicit) time discretization and second order central spatial discretization are used to give the following scheme:

\[ \frac{u_j^{n+1}-u_j^n}{\Delta t}+a\frac{u_{j+1}^n-u_{j-1}^n}{2\Delta x}=b\frac{u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{\Delta x^2} \]

with a uniform mesh \(x_j=j\Delta x\) with spatial mesh size \(\Delta x\) and time step \(\Delta t\). Here \(u_j^n\) is the numerical solution approximating the exact solution at \(x=x_j\) and \(t=n\Delta t\). Prove that the scheme is \(L^2\) stable under the very mild time step restriction

\[ \Delta t \leq c, \]

with a constant \(c\) which is independent of \(\Delta x\). Can you determine the dependency of \(c\) on the two constants \(a\) and \(b\)?

2017T

Problem 5. We have the following partial differential equation

\[ u_{t}=H(u)_{xx},\quad 0\leq x<1 \]

with an initial condition \(u(x,0)=f(x)\) and periodic boundary condition. Here \(0\leq H^{\prime}(u)\leq d\). Consider the following one-step, three-point scheme on a uniform mesh \(x_j=j\Delta x\) with spatial mesh size \(\Delta x\):

\[ u^{n+1}_{j}=u^{n}_{j}+aH(u^{n}_{j-1})+bH(u^{n}_{j})+cH(u^{n}_{j+1}), \]

where \(a,b,c\) are constants which may depend on the mesh ratio \(\mu=\Delta t/\Delta x^{2}\), \(\Delta t\) is the time step, and \(u^{n}_{j}\) approximates the exact solution at \(u(x_j,t^n)\) with \(t^{n}=n\Delta t\).

(i) Find the constants \(a,b,c\) such that the scheme is second order accurate.
(ii) Find the CFL number \(\mu_0\) such that the scheme (with the constants determined by (i)) is stable under the time step restriction \(\mu\leq\mu_0\). Please specify which norm you are using for stability, and prove this stability result.

证明

\((i)\) 由 Taylor 展开，

\[ H(u_{j-1}^n)=H(u_j^n-\Delta x u_x+\frac{\Delta x^2}{2}u_{xx}+O(\Delta x^3))=H(u_j^n)+(-\Delta x u_x+\frac{\Delta x^2}{2}u_{xx})H'(u_j^n)+\frac{(-\Delta x u_x)^2}{2}H''(u_j^n)+O(\Delta x^3), \]

\[ H(u_{j+1}^n)=H(u_j^n+\Delta x u_x+\frac{\Delta x^2}{2}u_{xx}+O(\Delta x^3))=H(u_j^n)+(\Delta x u_x+\frac{\Delta x^2}{2}u_{xx})H'(u_j^n)+\frac{(\Delta x u_x)^2}{2}H''(u_j^n)+O(\Delta x^3). \]

\[ H(u_{j-1}^n)+H(u_{j+1}^n)-2H(u_j^n)=\Delta x^2 u_{xx} H'(u_j^n)+(\Delta x u_x)^2 H''(u_j^n)+O(\Delta x^3). \]

\[ u_j^{n+1}-u_j^n=\Delta t u_t+\frac{\Delta t^2}{2}u_{tt}+O(\Delta t^3)=\mu \Delta x^2 H(u)_{xx}+O(\Delta x^4)=\mu \Delta x^2 (H'(u_j^n) u_{xx}+H''(u_j^n) (u_x)^2)+O(\Delta x^4). \]

比较上面两式的系数，得到

\[ a=c=\mu, \quad b=-2\mu. \]

2017I, 2014I

Problem 6. For solving the following partial differential equation

\[ u_{t}+f(u)_{x}=0,\quad 0\leq x\leq 1 \]

where \(f^{\prime}(u)\geq 0\), with periodic boundary condition, we can use the following semidiscrete 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 \(\Delta x=1/N\).

(i) Prove the following \(L^{2}\) stability of the scheme

\[ \frac{d}{dt}E(t)\leq 0 \quad \text{where} \quad E(t)=\sum_{j=1}^{N}|u_{j}|^{2}\Delta x. \]
(ii) 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.

2016T

Problem 7. For solving the following partial differential equation

\[ u_{t}+u_{x}=0,\quad -\infty\leq x\leq\infty \]

with compactly supported initial condition, we consider the following one-step, three-point scheme on a uniform mesh \(x_{j}=j\Delta x\) with spatial mesh size \(\Delta x\):

\[ u^{n+1}_{j}=au^{n}_{j}+bu^{n}_{j-1}+cu^{n}_{j-2},\quad j=\ldots,-1,0,1,\ldots \]

where \(a,b,c\) are constants which may depend on the mesh ratio \(\lambda=\Delta t/\Delta x\). Here \(\Delta t\) is the time step, and \(u^{n}_{j}\) approximates the exact solution at \(u(x_{j},t^{n})\) with \(t^{n}=n\Delta t\).

(i) Find the constants \(a,b,c\) such that the scheme is second order accurate.
(ii) Find the CFL number \(\lambda_0\) such that the scheme (with the constants determined by (i)) is stable in \(L^{2}\) under the time step restriction \(\lambda\leq\lambda_0\).
(iii) If the PDE is defined on \((0,\infty)\) with an initial condition compactly supported in \((0,\infty)\) and a boundary condition \(u(0,t)=g(t)\), how would you modify the scheme so that it can be applied? Can you prove the stability and accuracy of your modified scheme?

2016I

Problem 8. Consider the implicit leapfrog scheme

\[ \frac{u_{m}^{n+1}-u_{m}^{n-1}}{2k}+a\left(1+\frac{h^{2}}{6}\delta^{2}\right)^{-1}\delta_{0}u_{m}^{n}=f_{m}^{n} \]

for the one-way wave equation \(u_{t}+au_{x}=f\). Here \(\delta^{2}\) is the central second difference operator, and \(\delta_{0}\) is the central first difference operator.

(1) Show that the scheme is of order \((2,4)\) (second order in time, fourth order in space).
(2) Show that the scheme is stable if and only if \(|\frac{ak}{h}|<\frac{1}{\sqrt{3}}\).

2015I

Problem 9. Solve the following linear hyperbolic partial differential equation

\[ u_{t}+au_{x}=0,\quad t\geq 0, \]

where \(a\) is a constant. Using the finite difference approximation, we can obtain the forward-time central-space scheme as follows,

\[ \frac{u_{m}^{n+1}-u_{m}^{n}}{k}+a\frac{u_{m+1}^{n}-u_{m-1}^{n}}{2h}=0, \]

where \(k\) and \(h\) are temporal and spatial mesh sizes.

(i) Show that when we fix \(\lambda=k/h\) as a positive constant, the forward-time central-space scheme is consistent with equation \(u_t + a u_x = 0\).
(ii) Analyze the stability of this method. Is the method stable with \(\lambda=k/h\) being fixed as a constant?
(iii) How would the answer change if you are allowed to make \(\lambda=k/h\) small?
(iv) Would this be a good scheme to use even if you can make it stable by making \(\lambda\) small? If not, please provide a simple modification to make this scheme stable by keeping \(\lambda\) fixed.

2014I

Problem 10. For solving the following heat equation on interval

\[ u_{t}=u_{xx},\quad 0\leq x\leq 1 \]

with boundary condition \(u(0)=u_{0},\ u(1)=u_{1}\), we first discretize the interval \([0,1]\) into \(N\) subintervals uniformly, that is, the mesh size \(h=1/N\). We choose a temporal step size \(k\) and approximate the solution \(u(jh,nk)\) by \(U_{j}^{n}\), \(j=1,\ldots,N-1,n=0,1,2,\ldots\). Using the backward Euler method in time and central finite difference in space, the discrete function \(U_{j}^{n}\) satisfies:

\[ U_{j}^{n+1}-U_{j}^{n}=\lambda(U_{j-1}^{n+1}-2U_{j}^{n+1}+U_{j+1}^{n+1}),\quad j =1,\ldots,N-1, \]

where \(\lambda=k/h^{2}\), and \(U_0^{n+1}=u_0\), \(U_N^{n+1}=u_1\).

Show that

\[ \frac{1}{2}\sum_{j=1}^{N-1}((U^{n+1}_{j})^{2}-(U^{n}_{j})^{2}) \leq-\lambda\sum_{j=1}^{N-2}(U^{n+1}_{j+1}-U^{n+1}_{j})^{2} -\frac{\lambda}{2}((U^{n+1}_{1})^{2}+(U^{n+1}_{N-1})^{2})+\frac{\lambda}{2}(u_{0}^{2}+u_{1}^{2}). \]

2013T

Problem 11. The wave guide problem is defined as

\[ u_{t}+u_{x}=0,\quad v_{t}-v_{x}=0 \]

with the boundary condition

\[ u(-1,t)=v(-1,t),\quad v(1,t)=u(1,t) \]

and the initial condition

\[ u(x,0)=f(x),\quad v(x,0)=g(x). \]

The upwind scheme for the guide problem is defined as

\[ \begin{aligned} \frac{u^{n+1}_{j}-u^{n}_{j}}{\Delta t}+\frac{u^{n}_{j}-u^{n}_{j-1}}{\Delta x}&=0,\quad j=-N+1,\ldots,N;\\[5pt] \frac{v^{n+1}_{j}-v^{n}_{j}}{\Delta t}-\frac{v^{n}_{j+1}-v^{n}_{j}}{\Delta x}&=0,\quad j=-N,\ldots,N-1; \end{aligned} \]

with the boundary condition

\[ u^{n+1}_{-N}=v^{n+1}_{-N},\quad v^{n+1}_{N}=u^{n+1}_{N} \]

where \(u^{n}_{j}\) and \(v^{n}_{j}\) approximate \(u(x_{j},t^{n})\) and \(v(x_{j},t_{n})\) respectively at the grid point \((x_{j},t_{n})\), with \(x_{j}=j\Delta x\), \(t^{n}=n\Delta t\), \(\Delta x=\frac{1}{N}\).

(i) For the solution to the wave guide problem with the above boundary condition, prove the energy conservation

\[ \frac{d}{dt}\int_{-1}^{1}(u^{2}+v^{2})dx=0. \]
(ii) For the numerical solution of the the upwind scheme, if we define the discrete energy as

\[ E^{n}=\sum_{j=-N+1}^{N}(u^{n}_{j})^{2}+\sum_{j=-N}^{N-1}(v^{n}_{j})^{2}, \]

prove the discrete energy stability \(E^{n+1}\leq E^{n}\) under a suitable time step restriction \(\frac{\Delta t}{\Delta x}\leq\lambda_{0}\). You should first find \(\lambda_{0}\).
(iii) Under the same time step restriction, is the numerical solution stable in the maximum norm? That is, can you prove

\[ \max_{-N\leq j\leq N}\max(|u^{n+1}_{j}|,|v^{n+1}_{j}|)\leq\max_{-N\leq j\leq N}\max(|u^{n}_{j}|,|v^{n}_{j}|)? \]

2012I

Problem 12. Describe the forward-in-time and center-in-space finite difference scheme for the one-way wave equation:

\[ u_{t}+u_{x}=0. \]

(i) Conduct the von Neumann stability analysis and comment on their stability property.
(ii) Under what condition on \(\Delta t\) and \(\Delta x\) would this scheme be stable and convergent?
(iii) How many ways you can modify this scheme to make it stable when the CFL condition is satisfied?

2011T

Problem 13. We use the following scheme to solve the PDE \(u_{t}+u_{x}=0\):

\[ u^{n+1}_{j}=au^{n}_{j-2}+bu^{n}_{j-1}+cu^{n}_{j} \]

where \(a,b,c\) are constants which may depend on the CFL number \(\lambda=\frac{\Delta t}{\Delta x}\). Here \(x_{j}=j\Delta x\), \(t^{n}=n\Delta t\) and \(u^{n}_{j}\) is the numerical approximation to the exact solution \(u(x_{j},t^{n})\) with periodic boundary conditions.

(i) Find \(a,b,c\) so that the scheme is second order accurate.
(ii) Verify that the scheme you derived in Part (i) is exact (i.e. \(u^{n}_{j}=u(x_{j},t^{n})\)) if \(\lambda=1\) or \(\lambda=2\). Does this imply that the scheme is stable for \(\lambda\leq 2\)? If not, find \(\lambda_{0}\) such that the scheme is stable for \(\lambda\leq\lambda_{0}\).

Recall that a scheme is stable if there exist constants \(M\) and \(C\), which are independent of the mesh sizes \(\Delta x\) and \(\Delta t\), such that \(\|u^{n}\|\leq Me^{CT}\|u^{0}\|\) for all \(\Delta x\), \(\Delta t\) and \(n\) such that \(t^{n}\leq T\). You can use either the \(L^{\infty}\) norm or the \(L^{2}\) norm to prove stability.

2010T

Problem 14. When considering finite difference schemes approximating partial differential equations (PDEs), for example, the scheme

\[ u^{n+1}_{j}=u^{n}_{j}-\lambda(u^{n}_{j}-u^{n}_{j-1}) \]

where \(\lambda=\frac{\Delta t}{\Delta x}\), approximating the PDE \(u_{t}+u_{x}=0\), we are often interested in stability, namely

\[ \|u^{n}\|\leq C\|u^{0}\|,\quad n\Delta t\leq T \]

for a constant \(C=C(T)\) independent of the time step \(\Delta t\) and the spatial mesh size \(\Delta x\). Here \(\|\cdot\|\) is a given norm, for example the \(L^{2}\) norm or the \(L^{\infty}\) norm, of the numerical solution vector \(u^{n}=(u^{n}_{1},u^{n}_{2},\ldots,u^{n}_{N})\). The mesh points are \(x_{j}=j\Delta x\), \(t^{n}=n\Delta t\), and the numerical solution \(u^{n}_{j}\) approximates the exact solution \(u(x_{j},t^{n})\) of the PDE with a periodic boundary condition.

(i) Prove that the scheme \(u^{n+1}_{j}=u^{n}_{j}-\lambda(u^{n}_{j}-u^{n}_{j-1})\) is stable in the sense of (14) for both the \(L^{2}\) norm and the \(L^{\infty}\) norm under the time step restriction \(\lambda\leq 1\).
(ii) Since the numerical solution \(u^{n}\) is in a finite dimensional space, Student A argues that the stability (14), once proved for a specific norm \(\|\cdot\|_{a}\), would also automatically hold for any other norm \(\|\cdot\|_{b}\). His argument is based on the equivalency of all norms in a finite dimensional space, namely for any two norms \(\|\cdot\|_{a}\) and \(\|\cdot\|_{b}\) on a finite dimensional space \(W\), there exists a constant \(\delta>0\) such that

\[ \delta\|u\|_{b}\leq\|u\|_{a}\leq\frac{1}{\delta}\|u\|_{b}. \]

Do you agree with his argument? If yes, please give a detailed proof of the following theorem: If a scheme is stable, namely (14) holds for one particular norm (e.g. the \(L^{2}\) norm), then it is also stable for any other norm. If not, please explain the mistake made by Student A.

2010T

Problem 15. We have the following 3 PDEs

\[ \begin{aligned} u_{t}+Au_{x} &=0, (15)\\ u_{t}+Bu_{x} &=0, (16)\\ u_{t}+Cu_{x} &=0,\quad C=A+B. (17) \end{aligned} \]

Here \(u\) is a vector of size \(m\) and \(A\) and \(B\) are \(m\times m\) real matrices. We assume \(m\geq 2\) and both \(A\) and \(B\) are diagonalizable with only real eigenvalues. We also assume periodic initial condition for these PDEs.

(i) Prove that (15) and (16) are both well-posed in the \(L^{2}\)-norm. Recall that a PDE is well-posed if its solution satisfies

\[ \|u(\cdot,t)\|\leq C(T)\|u(\cdot,0)\|,\quad 0\leq t\leq T \]

for a constant \(C(T)\) which depends only on \(T\).
(ii) Is (17) guaranteed to be well-posed as well? If yes, give a proof; if not, give a counter example.
(iii) Suppose we have a finite difference scheme \(u^{n+1}=A_{h}u^{n}\) for approximating (15) and another scheme \(u^{n+1}=B_{h}u^{n}\) for approximating (16). Suppose both schemes are stable in the \(L^{2}\)-norm. If we now form the splitting scheme \(u^{n+1}=B_{h}A_{h}u^{n}\) which is a consistent scheme for solving (17), is this scheme guaranteed to be \(L^{2}\) stable as well? If yes, give a proof; if not, give a counter example.