Splitting methods

Splitting methods are numerical methods for solving ordinary differential equations (ODEs) or e.g. the time dependence of a parabolic or hyperbolic partial differential equation (PDE). The essential idea is to split the differential operator into a sum of operators and integrate them separately. The individual operators are sometimes much easier to integrate.

As an example, let us consider the differential equation

{\frac {d\mathbf {y} }{dx}}=L_{1}(\mathbf {y} )+L_{2}(\mathbf {y} )

where $L_{1}$ , $L_{2}$ are differential operators. The formal solution of the ODE is given by

\mathbf {y} =e^{\int L_{1}(\mathbf {y} )+L_{2}(\mathbf {y} )dx}

or if the operators do not depend on $x$ by

\mathbf {y} =e^{(L_{1}(\mathbf {y} )+L_{2}(\mathbf {y} ))x}.

The splitting methods now approximate the solution by a product of terms $e^{L_{1}(\mathbf {y} )x}$ , $e^{L_{2}(\mathbf {y} )x}$ which is not the same as the correct solution, if the operators do not commute (cf. Baker-Campbell-Hausdorff formula). The error introduced in this step is called the splitting error.