Problems

1. Explain the difference between hyperbolic, parabolic and elliptic partial differential equations, and give an example of each. What is the important physical distinction between hyperbolic and parabolic equations, on the one hand, and elliptic equations on the other?
2. Describe the von Neumann procedure for analysing the stability of partial differential equations.
3. Describe the physical principle behind the Courant-Friedrichs-Lewy condition as applied to the numerical solution of partial differential equations.
4. Are the following equations hyperbolic, elliptic or parabolic?
   $$\begin{eqnarray*} &\mbox{a)}\qquad& 3{\partial^2 f\over\partial t^2}+ 2{\partia... ...artial t}- {\partial f\over\partial x}\right) = \sin(x + t^3) \end{eqnarray*}$$
   
   
   
   
   

5. The equation
   
   $${\partial f\over\partial t}= \beta {\partial^3 f\over\partial x^3}$$
   
   
   can be represented by the difference equation
   
   $$f_n^{(m+1)} = f_n^{(m)} + \mu\left( f_{n+2}^{(m)} - 3 f_{n+... ...^{(m)}\right) \qquad \mu = {\beta\delta t\over\delta x^3} .$$
   
   
   Derive the truncation error of this difference equation. Write down an alternative difference equation which is 2nd order accurate in both $\delta t$ and $\delta x$.
6. The Dufort-Frankel scheme is a method for the solution of the diffusion equation. Show that the method is unconditionally stable. Discuss the advantages and disadvantages of this method.
7. The diffusion equation in a medium where the diffusion constant $D$ varies in space $\left(D = D(x)\right)$ is
   
   $${\partial f\over\partial t}= {\partial\over\partial x}\left... ...x^2}+ {\partial D\over\partial x}{\partial f\over\partial x}.$$
   
   
   Show that the difference equation
   
   $${f_n^{(m+1)} - f_n^{(m)}\over\delta t} = D_n{f_{n+1}^{(m)... ...) \left(f_{n+1}^{(m)} - f_{n-1}^{(m)}\over 2\delta x\right)$$
   
   
   is not conservative, i.e. $\int f\d x$ is not conserved. Construct an alternative difference scheme which is conservative.
8. Show that the Lax scheme for the solution of the advection equation is equivalent to
   
   $${\partial f\over\partial t}= - u{\partial f\over\partial x}... ...partial^2 f\over\partial x^2} + \mbox{higher order terms} .$$
   
   
   Examine the behaviour of wave-like solutions $f = \exp(\i(k x - \omega t))$ in the Lax scheme and explain the behaviour in terms of diffusion.
9. Describe what is meant by numerical dispersion.
10. Lax-Wendroff method consists of 2 steps, just like Runge-Kutta or Predictor-Corrector. It is given by
    

    $$u_{j+{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{... ...c12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}} = {\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scri... ...scriptscriptstyle{1/2}}}}c{\delta t\over\delta x}\left(u_{j+1}^n - u_j^n\right)$$ (2.49)

    $$u_j^{n+1} = u_j^n - c{\delta t\over\delta x} \left(u_{j+{\mathchoice{{\textst... ...{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}}\right)$$ (2.50)

    
    
    Draw a diagram to illustrate the way this algorithm operates on an $(x,t)$ grid. Show that the algorithm is stable provided the Courant-Friedrichs-Lewy condition is obeyed. Show that the algorithm tends to dampen waves with wavelength $\lambda = 2\delta x$.