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# 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.