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.