Summary of Time Integrators Available In PETSc#

Table 1 Time integration schemes#

TS Name

Reference

Class

Type

Order

euler

forward Euler

one-step

explicit

\(1\)

ssp

multistage SSP [Ket08]

Runge-Kutta

explicit

\(\le 4\)

rk*

multiscale

Runge-Kutta

explicit

\(\ge 1\)

beuler

backward Euler

one-step

implicit

\(1\)

cn

Crank-Nicolson

one-step

implicit

\(2\)

theta*

theta-method

one-step

implicit

\(\le 2\)

bdf

Backward Differentiation Formulas

one-step

implicit

\(\le 6\)

alpha

alpha-method [JWH00]

one-step

implicit

\(2\)

gl

general linear [BJW07]

multistep-multistage

implicit

\(\le 3\)

eimex

extrapolated IMEX [CS10]

one-step

IMEX

\(\ge 1\), adaptive

dirk

DIRK

diagonally implicit Runge-Kutta

implicit

\(\ge 1\)

arkimex

See IMEX Runge-Kutta schemes

IMEX Runge-Kutta

IMEX

\(1-5\)

rosw

See Rosenbrock W-schemes

Rosenbrock-W

linearly implicit

\(1-4\)

glee

See GL schemes with global error estimation

GL with global error

explicit and implicit

\(1-3\)

mprk

Multirate Partitioned Runge-Kutta

multirate

explicit

\(2-3\)

basicsymplectic

Basic symplectic integrator for separable Hamiltonian

semi-implicit Euler and Velocity Verlet

explicit

\(1-2\)

irk

fully implicit Runge-Kutta

Gauss-Legrendre

implicit

\(2s\)

[BJW07]

J.C. Butcher, Z. Jackiewicz, and W.M. Wright. Error propagation of general linear methods for ordinary differential equations. Journal of Complexity, 23(4-6):560–580, 2007. doi:10.1016/j.jco.2007.01.009.

[CS10]

E.M. Constantinescu and A. Sandu. Extrapolated implicit-explicit time stepping. SIAM Journal on Scientific Computing, 31(6):4452–4477, 2010. doi:10.1137/080732833.

[JWH00]

K.E. Jansen, C.H. Whiting, and G.M. Hulbert. A generalized-alpha method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Computer Methods in Applied Mechanics and Engineering, 190(3):305–319, 2000.

[Ket08]

D.I. Ketcheson. Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM Journal on Scientific Computing, 30(4):2113–2136, 2008. doi:10.1137/07070485X.