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

Related tags

Data AnalysisNumerics
Overview

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
Owner
George Whittle
George Whittle
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