Section: New Results
Splitting methods with complex times for parabolic equations
Participants : François Castella, Philippe Chartier, Gilles Vilmart.
This is a joint work with S. Descombes, from the University of Nice.
Although the numerical simulation of the heat equation in several space dimension is now well understood, there remain a lot of challenges in the presence of an external source, e.g. for reaction-diffusion problems, or more generally for the complex Ginzburg-Landau equation. From a mathematical point of view, these belong to the class of semi-linear parabolic partial differential equations and can be represented in the general form
When one wishes to approximate the solution of the above parabolic non-linear problem, a method of choice is based on operator-splitting: the idea is to split the abstract evolution equation into two parts which can be solved explicitly or at least approximated efficiently.
For a positive step size h , the most simple numerical integrator is the Lie-Trotter splitting which is an approximation of order 1, while the symmetric version is referred to as the Strang splitting and is an approximation of order 2. For higher orders, one can consider general splitting methods of the form
However, achieving higher order is not as straightforward as it looks. A disappointing result indeed shows that all splitting methods (or composition methods) with real coefficients must have negative coefficients ai and bi in order to achieve order 3 or more. The existence of at least one negative coefficient was shown in  ,  , and the existence of a negative coefficient for both operators was proved in  . An elegant geometric proof can be found in  . As a consequence, such splitting methods cannot be used when one operator, like , is not time-reversible.
In order to circumvent this order-barrier, there are two possibilities. One can use a linear, convex (see  ,  ,  for methods of order 3 and 4) or non-convex (see  ,  where an extrapolation procedure is exploited), combinations of elementary splitting methods like (17 ). Another possibility is to consider splitting methods with complex coefficients ai and bi with positive real parts (see  in celestrial mechanics). In 1962/1963, Rosenbrock  considered complex coefficients in a similar context.
In  , we consider splitting methods, and we derive new high-order methods using composition techniques originally developed for the geometric numerical integration of ordinary differential equations  . The main advantages of this approach are the following:
the splitting method inherits the stability property of exponential operators;
we can replace the costly exponentials of the operators by cheap low order approximations without altering the overall order of accuracy;
using complex coefficients allows to reduce the number of compositions needed to achieve any given order;