Perimeter Scholars International
Explorations in Gravitational Physics
Numerical Relativity

MW Choptuik, L Lehner & F Pretorius

Project 1: Scalar Wave Evolution on a Schwarzschild Background

• PROJECT HANDOUT [PDF]
• Account confguration
  
• Solving the 1-D wave equation with RNPL: It is recommended that you quickly work through these notes, and use the RNPL source file, w1dcn/w1dcn_rnpl as a template for your implementation.
• Generic Makefile for RNPL applications: Makefile.rnpl
• Source files for w1dcnm, an extension of w1dcn that illustrates the incorporation of external source into RNPL code:
  • w1dcnm_rnpl
  • mass.inc
  • mass_init.inc
  • Distribution: w1dcnm.tar.gz
    
    
• March 16 and March 18 tutorials: Warm-up exercise: Modify w1dcn_rnpl so that it implements a solution of the wave equation on 0 <= x <= 1, but with fixed (Dirichlet) boundary conditions on phi. Note that the basic FD scheme should remain the same:  i.e. continue to work with the wave equation re-written as a system of first-order-in-time PDEs for pp=phi_x and pi=phi_t and figure out how the Dirichlet conditions on phi, namely phi(t,0) = phi(t,1) = 0, can be implemented as boundary conditions for pp(t,0), pp(t,1) and pi(t,0), pi(t,1).  Investigate the convergence of your implementation.
  
  HINT: If you're having difficulty figuring out what the appropriate boundary conditions are for Phi(t,0), Phi(t,1), consider first modifying the RNPL code so that it solves the wave equation in the following first-order-in-time form
  
       phi(t,x)_t = pi
  
       pi(t,x)_t = phi_xx
  
  with the boundary conditions
  
       phi(t,0) = phi(t,1) = 0
  
       pi(t,0) = pi(t,1) = 0
  
  and initial conditions
  
       phi(0,x) = G(x; ...)
  
       pi(0,x) = s dG(x; ...)/dx
   
  where G(x;. ..) is the usual Gaussian profile  (... denotes the real-valued parameters that define the specific gaussian profile, i.e.  overall amplitude, A, etc).
  
  Once you are confident that your RNPL implementation is correct (view convergence testing as mandatory!), examine the behaviour of phi_x(t,0) and phi_x(t,1) using xvs to compute the spatial derivative.  This should give you insight into what boundary condition you should impose on pp_x(t,0) and pp_x(t,1) to solve the exercise.  If it doesn't, take another derivative and consider phi_xx(t,0) and phi_xx(t,1). 
  
  If that still doesn't help, ask Matt or Ben for assistance.
  
  
• March 18 and March 19 tutorials and homework:  Complete Project 1