SSPRK33

class pathsim.solvers.ssprk33.SSPRK33(*solver_args, **solver_kwargs)

Bases: ExplicitRungeKutta

Three-stage, 3rd order optimal SSP Runge-Kutta method.

The unique optimal three-stage, 3rd order SSP scheme. Commonly paired with WENO and ENO spatial discretisations for hyperbolic conservation laws.

Characteristics

• Order: 3

• Stages: 3

• Explicit, fixed timestep

• SSP coefficient \(\mathcal{C} = 1\)

Note

The standard SSP time integrator for method-of-lines PDE discretisations inside ODE blocks. If the spatial operator is TVD under forward Euler, this method preserves that property at the same timestep restriction. When stability is borderline, SSPRK34 allows roughly twice the timestep at the cost of one extra stage.

References

[1]

Shu, C.-W., & Osher, S. (1988). “Efficient implementation of essentially non-oscillatory shock-capturing schemes”. Journal of Computational Physics, 77(2), 439-471. :doi:`10.1016/0021-9991(88)90177-5`

[2]

Gottlieb, S., Shu, C.-W., & Tadmor, E. (2001). “Strong stability-preserving high-order time discretization methods”. SIAM Review, 43(1), 89-112. :doi:`10.1137/S003614450036757X`

[3]

Gottlieb, S., Ketcheson, D. I., & Shu, C.-W. (2011). “Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations”. World Scientific. :doi:`10.1142/7498`

interpolate(r, dt)

Interpolate solution after successful timestep as a ratio in the interval [t, t+dt].

This is especially relevant for Runge-Kutta solvers that have a higher order interpolant. Otherwise this is just linear interpolation using the buffered state.

Parameters:

• r (float) – ration for interpolation within timestep

• dt (float) – integration timestep

Returns:

x – interpolated state

Return type:

numeric, array[numeric]