petsc-3.3-p7 2013-05-11

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.

Examples

src/ts/examples/tutorials/ex9.c.html

TSSSP

References

See Also

Examples