• Project 1: The 1D Advection Equation

      We discuss the leap-frog differencing scheme and related issues of convergence in the simple (but instructive) example of the 1D advection equation ut=ux. The domain is taken to be periodic in x (topology of a circle).

      The numerical solution is computed using C, FORTRAN and RNPL and compared to the exact solution. We check for convergence and calculate the truncation error. A plot displaying the topology is also included.

      • Document PDF
      • Surface plots showing the solution and error
      • Plots of tests
      • Mpegs
        • Small Animated Solution of 1-d Advection Equation on a Periodic Grid (34K)
        • C and RNPL Solution for Level=6, NC=3, Lambda=0.8 (531K)
        • C and RNPL Solution for Level=9, NC=3, Lambda=0.8 (448K)
        • Convergence factor for the handcoded solution (164K)
        • Convergence Test: Level=6 minus Level=7 (543K)
        • Convergence Test: Level=8 minus Level=9 (491K)

  • Project 2: Massless Scalar Field on a Black Hole Background

      We consider a massless scalar field propagating on a Schwarzschild background in ingoing Eddington-Finkelstein (IEF) coordinates. The IEF system has the time coordinate chosen so that the ingoing tangent vector is null. In addition, the metric is nondegenerate even at the horizon of the black hole. Thus, the coordinate system is particularly convenient for numerical work, since the horizon is null and no special boundary condition is needed at r=2M.

      The numerical solution is created using RNPL and a parameter survey in pulse width is conducted. For most pulses some mass falls into the black hole, and by integrating the ``conserved'' mass function over the entire region we can calculate how much mass due to the scalar field is actually there and thus find the absorption coefficient as a function of pulse width.

      • Document PDF
      • Convergence factor and mass absorption
      • Mpegs
        • Gradient of scalar field for width 2M (224K)
        • Mass for width 2M (228K)
        • Gradient of scalar field for width 10M (324K)
        • Mass for width 10M (386K)