I know that Implicit Midpoint (IM2) is not L-stable either but it is
my understanding that it has a lower truncation error because the
coefficient in front of the error is smaller compared to that of CN.
So I usually prefer IM2 to CN.
I'm curious about the TSGL methods and will try out those integrators
soon. It might be a stretch to do that in the current problem
implementation but I have other coupled problems that could use
adaptive, higher order temporal integration. If there are references
explaining the properties of these methods, their butcher tableau and
such, please do let me know. Also, are these implemented only in
petsc-dev currently since I do not see it in the latest release of
On Tue, Jan 26, 2010 at 1:08 PM, Jed Brown <jed@...> wrote:
> On Tue, 26 Jan 2010 12:42:58 -0600, "Vijay S. Mahadevan" <vijay.m@...> wrote:
>> I'm solving a pure diffusion problem and there is no convection here.
>> But I do understand that time integration makes a big difference and
>> even making delt=1e-10 does not seem to help. The negativity occurs on
>> the first step, the first call to nonlinear residual. When you say
>> trapezoidal rule, are you talking about Implicit midpoint here because
>> CN is based on the trapezoidal rule and is not L-stable (spurious
>> oscillations are not damped).
> Implicit midpoint is also not L-stable, it actually has exactly the same
> stability function as trapezoid:
> (1 + z/2) / (1 - z/2)
> Implicit Euler, BDF2, and various implicit Runge-Kutta schemes are
> L-stable. Also, if you're feeling adventurous, I'd love to hear how
> your system works with TSGL (in PETSc-dev). These are A- and L-stable
> methods of order and stage order 1 through 5, with adaptive controllers
> (though the adaptive controllers may not robust, they haven't had much
> For oscillations, as in hyperbolic systems and I think not the issue
> here, you may need a strong stability preserving integrator. There do
> not exist SSP integrators without a CFL constraint and order greater
> than 1 (implicit Euler is the only SSP method without a CFL constraint).