Math C42H 3042: Assignment 1 

Date due: Friday, 29th August

Instructions:

  1. (6 marks) The linearized one-dimensional Euler equations can be reduced to \[ %\frac{\partial\ }{\partial t} \left( \begin{array}{c} \rho_t \\ u_t \\ p_t \end{array} \right) + \left( \begin{array}{ccc} u_0 & \rho_0 & 0 \\ 0 & u_0 & \frac{1}{\rho_0} \\ 0 & \rho_0 c^2_0 & u_0 \\ \end{array} \right) % \frac{\partial\ }{\partial x} \left( \begin{array}{c} \rho_x \\ u_x \\ p_x \end{array} \right) = 0 \] where \vspace{-2mm} \[ \begin{array}{lc} {\rm density} = \rho_0 + \epsilon \rho(x,t), & \ \ \rho_0, u_0, p_0 {\rm \ constant}, (\rho_0>0, p_0>0) \\ {\rm velocity} = u_0 + \epsilon u(x,t), & \epsilon \ll 1\ , \ \ c_0 = \sqrt{\frac{\gamma p_0}{\rho_0}} \ {\rm and} \\ {\rm pressure} = p_0 + \epsilon p(x,t), & % {\rm and\ assume} \ u_0 \ge 0 \ {\rm and}\ u_00 \end{array} \right. \] \vspace{-5mm} at some time $T>0$. Find and sketch the solution to the ``one-dimensional burst balloon'' problem \vspace{-1mm} \[ p^0(x) = \left\{ \begin{array}{ll} 1 & {\rm for}\ |x| < X \\ 0 & {\rm for}\ |x| > X \end{array} \right. \] \vspace{-5mm} at a time, $T$, long after the balloon ``pops'' ($c_0 T \gg X$). \item (6 marks) Fromm's scheme can be derived as the average of the Lax-Wendroff and Beam-Warming schemes. Using this fact (or otherwise), find the order of the scheme. Comment on the smoothness of initial data necessary for numerical solution to converge at this rate. Use von Neumann stability analysis to find the stability restriction for Fromm's scheme for the scalar advection equation. [{\footnotesize Assuming you find the correct symbol and $\nu=ak/h$ is the Courant number, the modulus of the symbol is given by \\ % \vspace{-1mm} \rule{10mm}{0mm} $ |\lambda(\beta)|^2 = 1 + \frac{1}{2}\nu(\cos^2(\beta) - 1)^2(\nu-1)(\nu^2\cos(\beta)+\nu^2 -\nu\cos(\beta)-\nu + 2). ] $ %where $\nu=ak/h$ is the Courant number.] % \vspace{-5mm} } How does this compare with the CFL condition for Fromm's scheme? % 2 2 2 % 1/2 v (cos(a) - 1) (v - 1) (v cos(a) + v - v cos(a) - v + 2) Interpret Fromm's scheme as an advected interploation scheme - what is the interpolation profile and what is the resulting flux function? \item (6 marks) Make a numerical error comparison of Fromm's scheme with the Lax-Wendroff and Beam-Warming schemes for the scalar advection equation for discontinuous (``top-hat'') initial data. By varying the resolution, (varying $k$ and $h$ appropriately) for the range of Courant numbers $0.5<\nu<0.8$, conculde that Fromm's scheme is worth a factor of two in Use linearity to find an appropriate dispersive modified equation for Formm's scheme. Sketch graphs of dispersion coefficient versus Courant number for the modified equations for the three schemes. Comment on what you expect to happen when $\nu=0.5$. % {\footnotesize Fromm's scheme flux function? % Fromm's scheme on the slope-limiter diagram?} Identify Fromm's scheme on the slope-limiter diagram. Describe a slope-limiter which gives a Total Variation Diminishing scheme and and uses Fromm's scheme \[ \tilde{U}^n(x) = U^n_j + \frac{(x-x_j)}{h} \Delta_j U^n \ {\rm for}\ x_{j-\frac{1}{2}}0 \\ 0 & {\rm if}\ \Delta^-_j U^n.\Delta^+_j U^n \le 0 \end{array} \right. \] \item (12 marks) Numerically solve the linearized ``one-dimensional burst balloon'' problem described Q.1 with a second order scheme. Use $(\rho_0,u_0,p_0) = (1,0,0.5,1.0)$, $\gamma=1.44$ and $X=0.05$. 50\% for a symmetric scheme, 75\% for a flux-slpit scheme and 100\% for a slope-limited, flux-split scheme. \end{enumerate} \end{document}