Waves

Wave simulator for the nonlinear wave equation, u_tt=c^2*(u_xx+u_yy)-V'(u), using Lobatto IIIA-IIIB discretisation in space (with 2, 3, or 4 stages for now) to get explicit ODEs in time (see my PhD thesis or Section 3 of https://doi.org/10.1137/140958050 for details), which are then integrated using leapfrog to get a high-order fully explicit multisymplectic integrator. The observed global numerical order in space for an r-stage discretisation is 2 for r=2 or r+1 for r>2, while the order in time is 2 for leapfrog (other time-stepping methods may give higher order in time). Additionally, the explicit ODEs are local (only depending on neighbouring cells), which allows various boundary conditions to be handled simply.

A Qt frontend to the integrator can be found in the Qt folder. As of version 3.0, the integrator has been implemented in OpenCL using boost.compute. This allows for a significant performance gain sufficient to run a 1 million element simulation using 3 stages in x and y.

Build Instructions

Run make in Qt/QtWaves. Hopefully everything required is there.

Running

Run ./QtWaves in the Qt/QtWaves folder and choose the desired configuration in the GUI.

Some configurations have a much higher number of cells than others, see the configurations below.

The SPF number in the Time box indicates the number of steps being taken per frame, so adjust the Timestep to get something approaching real-time. Initial configuration was built for an older laptop with lower specs, so real-time simulations will generally require much lower time-steps.

Configurations:

Boundary Conditions

• Periodic square:
  • 200x200 cells square domain, centered at the origin with periodic boundary conditions.
  • dx = dy = 0.1
  • wave speed = 1
• Dirichlet, square:
  • 200x200 cells square domain with Dirichlet boundary conditions.
  • dx = dy = 0.1
  • wave speed = 1
• Dirichlet, circle:
  • Circle of radius 100 cells, centered at the origin with Dirichlet boundary conditions.
  • dx = dy = 0.1
  • wave speed = 1
• Dirichlet, circle with a cusp:
  • Circle of radius 1303 cells (totalling ~1 million cells) with Dirichlet boundar conditions.
  • dx = dy = 0.1
  • wave speed = 1
• Dirichlet, intersecting circles:
  • Two slightly intersecting circles within a 200x200 cell square with Dirichlet boundary conditions.
  • dx = dy = 0.1
  • wave speed = 10
• Double slit (mixed BCs)
  • A double slit in a 1000x1000 cell square with Dirichlet boundary conditions at the slit and two edges, periodic boundary conditions on the side edges.
  • dx = dy = 0.025
  • wave speed = 0.3

Initial Conditions

• Single frequency:
  • A traveling cosine wave.
• Continuous spectrum:
  • A wide Gaussian hump with a continuous spectrum of frequencies centred at the origin.
• Localised hump:
  • A narrow Gaussian hump limited to an area around the origin.
  • u=exp(f(x,y))-1, where f(x,y)=a*(1-x^2/bx^2-y^2/by^2)
• Two localised humps:
  • As above, but two of them at the centre of two intersecting circles boundary condition.
• Off-centre localised hump:
  • Just one of the two localised humps.
• A localised wave:
  • A Gaussian hump in one dimension only with the other dimension being constant to make a wave.
  • Good starting condition for the double slit boundary condition.