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= 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.

Lie-Trotter splitting
The simplest splitting is the Lie-Trotter splitting


 * $$\mathbf{y}=e^{L_1 (\mathbf{y})x} e^{L_2 (\mathbf{y})x} \quad \text{or} \quad \mathbf{y}=e^{L_2 (\mathbf{y})x} e^{L_1 (\mathbf{y})x}. $$

The splitting error is of first order in $$\Delta x$$.

Strang Splitting
The Strang splitting, also called symmetric Lie-Trotter splitting, is named after Gilbert Strang and is a second order splitting.


 * $$\mathbf{y}=e^{L_1 (\mathbf{y})x/2} e^{L_2 (\mathbf{y})x}e^{L_1 (\mathbf{y})x/2} \quad \text{or} \quad \mathbf{y}=e^{L_2 (\mathbf{y})x/2} e^{L_1 (\mathbf{y})x}e^{L_2 (\mathbf{y})x/2}. $$

Further comments
There are also splitting methods of higher order and different number of compositions.

Splitting methods can also be applied to nonlinear differential equations.