Operator Splitting Method with Applications
In the modelling of complex time-depending physical phenomena the
simultaneous effect of several different sub-processes has to be
described. The operators describing the sub-processes are as a
rule simpler than the whole spatial differential operator.
Operator splitting is a widely used procedure in numerical
solution of such problems. The point in operator splitting is the
replacement of the original model with one in which appropriately
chosen groups of the sub-processes, described by the model, take
place successively in time. This de-coupling procedure allows us
to solve a few simpler problems instead of the whole one.
In the talk several splitting methods will be constructed
(sequential splitting, Strang splitting, weighted splitting,
additiv splitting, iterated splitting). We discuss the accuracy
(local splitting error) of the methods. We also examine the effect
of the choice of the numerical method chosen to the numerical
solution of the sub-problems in the splitting procedure.
We list the main benefits and drawbacks of this approach.
Dr. Robert Horvath
University of West-Hungary
Institute of Mathematics and Statistics
Sopron, Hungary
Application of the operator splitting technique
in the staggered grid finite difference solution
of the Maxwell equations
In the last two decades, a lot of efforts was invested in order to
construct unconditionally stable numerical schemes to the numerical
solution of the Maxwell equations. The main goal was the improvement
of the classical Yee-method, which was published in 1966. The Yee-method
is second order accurate both in space and time. However, in some real
life problems like computer chip design or modelling the effects of
microwaves on a human brain, its strict stability condition does not
make possible the fast computation of the numerical results.
It has been discovered that the newly constructed schemes together
with the Yee-scheme all fit into the framework of a general technique:
the so-called operator splitting. The look at the methods through the
glasses of operator splitting reveals the fundamental properties of
the schemes and shows the way of construction of further efficient
methods.
In this presentation, the operator splitting technique is applied to
the investigation of the staggered grid finite difference numerical
solutions of the lossy and not source free Maxwell equations. This
broader viewing angle helps to understand the schemes more deeply and
to construct new efficient schemes.