# Operator Splitting Methods

Operator splitting is a powerful method for numerical investigation of complex models. The basic idea of the operator splitting methods based on splitting of complex problem into a sequence of simpler tasks, called split sub-problems. The sub operators are usually chosen with regard to different physical process. Then instead of the original problem, a sequence of sub-models is solved, which gives rise to a splitting error. In practice, splitting procedures are associated with different numerical methods for solving the sub-problems, which also causes a certain amount of error.

Complex physical processes are frequently modelled by the systems of linear or non-linear partial differential equations. Due to the complexity of these equations, typically there is no numerical method which can provide a numerical solution that is accurate enough while taking reasonable integrational time. In order to simplify the task, operator splitting procedure has been introduced, which is widely used for solving advection-diffusion-reaction problems and Navier-Stokes equation including modelling turbulence and interfaces.

Different algorithms like Lie-Trotter, Strang and Additive splitting were introduced to solve ODEs and PDEs. One of the simplest problems is the initial value Cauchy problem:

Lie-Trotter Splitting

Lie-Trotter splitting is a first order splitting method which solves two sub-problems sequentially on subintervals  where    and   . The different subproblems are connected via the initial conditions as follows:

For , where . The approximated split solution at the point   is defined as .

Algorithm:

• Step 1:

• Step 2:

• Step 3:

• Step 4: if    go to step 1, otherwise stop.

Strang Splitting

One of the most popular and widely used operator splitting method is Strang splitting. By small modification it is possible to make the splitting algorithm second order accurate. The idea is that instead of first solving the first sub-problem for a full-time step, we solve it for a half time step. We then solve the second sub-problem for a full-time step, and then back to the first sub-problem and solve it for a half time step.

where   , and the approximated split solution at the point    is defined as .

Algorithm:

• Step 1:

• Step 2:

• Step 3:

• Step 4:

• Step 5: if   , go to step 1, otherwise stop.