Problems

1. Write down definitions of the terms order of accuracy, truncation error, conditional stability as applied to the numerical solution of ordinary differential equations.
2. Write down 1st order accurate finite difference approximations for
   
   $$\mbox{a) }{\d f\over\d x}\qquad \mbox{b) }{\d^2 f\over\d x... ...mbox{d) }{\partial\over\partial t}{\partial f\over\partial x}$$
   
   
   Hint: the result has to be something like $a y_{n+1} + b y_n + c y_{n-1}$. Expand the $y$s around $y_n$ and choose the coefficients to eliminate contributions from unwanted terms in the expansion. N.B. This question refers to the accuracy of the approximation for the derivatives given, not to the accuracy of $f$.
3. Derive expressions for the truncation error of the following difference approximations.
   
   $$\begin{array}{l} \begin{array}{ll} \mbox{a) }\displaystyle{... ...ac{1}{12} f_{n-2} \over\delta x}} \end{array} \end{array}$$
   
   

4. The torsion of a bar is described by the differential equation
   
   $${\d^4\theta\over\d x^4} + \tau\theta = 0.$$
   
   
   Show how to re-express this as a system of first order differential equations.
5. Write down an expression for solving the differential equation
   
   $${\d y\over\d t} + {\gamma\over y} = 0$$
   
   
   by Euler's method and show under what conditions the method is stable. Write and test a short program (it should only require a few lines) to test the method. Vary $\delta t$, $\gamma$ and $y(t=0)$ to check the validity of the stability condition you have derived.
6. Using equation 1.2.5 show that the Euler method is stable for a vector equation provided all the eigenvalues of $\bss{G}$ have modulus less than or equal to unity.
7. Show that equation 1.37 gives the correct stability condition for both the Runge-Kutta and Predictor-Corrector methods. Why do you think this is a good method for damped oscillatory equations? (The last part doesn't have to be mathematically rigorous).