Numerics

Numerical Analysis toolkit centred around PDEs, for demonstration and understanding purposes not production

Use procedure:

• Initialise a new instance of the PDE class, all calculations and variables will be managed by this object

  Inputs:

•      dt - timestep
  

•      tlim - time to solve up to
  

•      dx - x spacing, by default a uniform grid is used
  

•      xlims - np.array([lower, upper]) bounds of x domain
  

• instance.setSystem: Set the governing equation of the PDE in the form dU/dt = lambda U,x,t: some function

• instance.setInitConds: Set the initial conditions of U in the form lambda x: some function

• instance.solve: Integrate the system to time tlim using the input integrator (ie RK4, an explicit 4th order Runge Kutta method)

Note on Chebyshev pseudo-spectral differentiator:

• To use this differentiator, the local x coords must be set to the Gauss-Lobatto-Chebyshev collocation points using instance.chebyx(N) where N is the number of points

Known issues:

• 6th order error finite difference matrices can behave badly near domain edges
• fft derivative can cause steady state errors as it implicitly assumes periodic behaviour outside of domain
• Chebyshev differentiator is unstable in current iteration, particularly for higher orders and at the domain boundary, most likely due to Gibb's phenomenon
• More of a use detail, but fft and Chebyshev differentiators rely on compact support. If in doubt, use the finite differencing differentiator

Future implementations:

• Support for boundary conditions
• More integration schemes
• Potentially support for multiple spatial dimensions