petsc-3.5.4 2015-05-23
TSSSP
Explicit strong stability preserving ODE solver Most hyperbolic conservation laws have exact solutions that are total variation diminishing (TVD) or total variation
bounded (TVB) although these solutions often contain discontinuities. Spatial discretizations such as Godunov's
scheme and high-resolution finite volume methods (TVD limiters, ENO/WENO) are designed to preserve these properties,
but they are usually formulated using a forward Euler time discretization or by coupling the space and time
discretization as in the classical Lax-Wendroff scheme. When the space and time discretization is coupled, it is very
difficult to produce schemes with high temporal accuracy while preserving TVD properties. An alternative is the
semidiscrete formulation where we choose a spatial discretization that is TVD with forward Euler and then choose a
time discretization that preserves the TVD property. Such integrators are called strong stability preserving (SSP).
Let c_eff be the minimum number of function evaluations required to step as far as one step of forward Euler while
still being SSP. Some theoretical bounds
1. There are no explicit methods with c_eff > 1.
2. There are no explicit methods beyond order 4 (for nonlinear problems) and c_eff > 0.
3. There are no implicit methods with order greater than 1 and c_eff > 2.
This integrator provides Runge-Kutta methods of order 2, 3, and 4 with maximal values of c_eff. More stages allows
for larger values of c_eff which improves efficiency. These implementations are low-memory and only use 2 or 3 work
vectors regardless of the total number of stages, so e.g. 25-stage 3rd order methods may be an excellent choice.
Methods can be chosen with -ts_ssp_type {rks2,rks3,rk104}
rks2: Second order methods with any number s>1 of stages. c_eff = (s-1)/s
rks3: Third order methods with s=n^2 stages, n>1. c_eff = (s-n)/s
rk104: A 10-stage fourth order method. c_eff = 0.6
References
Ketcheson, Highly efficient strong stability preserving Runge-Kutta methods with low-storage implementations, SISC, 2008.
Gottlieb, Ketcheson, and Shu, High order strong stability preserving time discretizations, J Scientific Computing, 2009.
See Also
TSCreate(), TS, TSSetType()
Level:beginner
Location:src/ts/impls/explicit/ssp/ssp.c
Index of all TS routines
Table of Contents for all manual pages
Index of all manual pages
Examples
src/ts/examples/tutorials/ex9.c.html
src/ts/examples/tutorials/ex11.c.html
src/ts/examples/tutorials/ex31.c.html