Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)

Assyr Abdulle, Alexei A. Medovikov. Second Order Chebyshev Methods based on Orthogonal Polynomials. Numerische Mathematik, 90 (1), pp 1-18, 2001. doi: https://dx.doi.org/10.1007/s002110100292

ROCK2: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes optional keyword arguments min_stages, max_stages, and eigen_est. The function eigen_est should be of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

ROCK4(; min_stages = 0, max_stages = 152, eigen_est = nothing)

Assyr Abdulle. Fourth Order Chebyshev Methods With Recurrence Relation. 2002 Society for Industrial and Applied Mathematics Journal on Scientific Computing, 23(6), pp 2041-2054, 2001. doi: https://doi.org/10.1137/S1064827500379549

ROCK4: Stabilized Explicit Method. Fourth order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes optional keyword arguments min_stages, max_stages, and eigen_est. The function eigen_est should be of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

ESERK4(; eigen_est = nothing)

J. Martín-Vaquero, B. Kleefeld. Extrapolated stabilized explicit Runge-Kutta methods, Journal of Computational Physics, 326, pp 141-155, 2016. doi: https://doi.org/10.1016/j.jcp.2016.08.042.

ESERK4: Stabilized Explicit Method. Fourth order extrapolated stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes the keyword argument eigen_est of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

ESERK5(; eigen_est = nothing)

J. Martín-Vaquero, A. Kleefeld. ESERK5: A fifth-order extrapolated stabilized explicit Runge-Kutta method, Journal of Computational and Applied Mathematics, 356, pp 22-36, 2019. doi: https://doi.org/10.1016/j.cam.2019.01.040.

ESERK5: Stabilized Explicit Method. Fifth order extrapolated stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes the keyword argument eigen_est of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

RKC(; eigen_est = nothing)

B. P. Sommeijer, L. F. Shampine, J. G. Verwer. RKC: An Explicit Solver for Parabolic PDEs, Journal of Computational and Applied Mathematics, 88(2), pp 315-326, 1998. doi: https://doi.org/10.1016/S0377-0427(97)00219-7

RKC: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues.

This method takes the keyword argument eigen_est of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.