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 cmake -B build -GNinja -DCMAKE_BUILD_TYPE=Release && cmake --build build in Qt/QtWaves. Hopefully everything required is there.

Running

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

Note: If the program fails to start with a segfault, then this could be due to an issue with Qt on Wayland, in which case running with QT_QPA_PLATFORM=xcb ./build/QtWaves should work.

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

The product of FPS, Steps/Frame and Timestep give the ratio of simulation time to real-time, which is shown in parentheses in the Time display. Initial configuration was built for an older laptop with lower specs, so 1:1 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
• Double slit 2 (mixed BCs)
  • Similar to the "Double slit (mixed BCs)" configuration, but with the double slit centred in a 1000x2000 cell rectangle.
  • dx = dy = 0.025
  • wave speed = 1

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.
• Off-centre localised hump 2:
  • A localised hump at a more suitable position for the "Double slit 2 (mixed BCs)" boundary condition.